Copyright by Seong-Won Na 2006 - The University of Texas at Austin

Copyright by Seong-Won Na 2006

The Dissertation Committee for Seong-Won Na certifies that this is the approved version of the following dissertation:

On a Class of Two-Dimensional Inverse Problems: Wavefield-Based Shape Detection and Localization and Material Profile Reconstruction

Committee:

Loukas F. Kallivokas, Supervisor Omar Ghattas Spyros A. Kinnas Kenneth H. Stokoe II John L. Tassoulas Carlos Torres-Verdin Dan L. Wheat


by Seong-Won Na, B.S.; M.S.

DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY

THE UNIVERSITY OF TEXAS AT AUSTIN May 2006

Dedicated to my family.

Acknowledgments

I would like to express my sincere gratitude and appreciation to my advisor Prof. Loukas F. Kallivokas, for his valuable guidance and support throughout my Ph.D. studies. It has been a great pleasure to work with him, and this dissertation would not have been possible without his inspiration and encouragement. I would also like to thank Professors Omar Ghattas, Spyros A. Kinnas, Kenneth H. Stokoe II, John L. Tassoulas, Carlos Torres-Verdin and Dan L. Wheat for serving on my dissertation committee and for their helpful suggestions. Finally, I would like to express my special gratitude to my family. I thank my wife, Dohee, for her love and patience. Special thanks are extended to my sisters, Yoo-Eun and Yoo-Shin, and their families, for their constant encouragement. Most importantly, I thank my parents for their endless care and support, and would like to dedicate this dissertation to them.

Seong-Won Na The University of Texas at Austin January 2006

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Publication No. Seong-Won Na, Ph.D. The University of Texas at Austin, 2006 Supervisor: Loukas F. Kallivokas

In this dissertation we discuss the numerical treatment of two classical inverse problems: firstly, we are interested in the shape detection and localization problem that arises when it is desirable to identify the location and shape of an unknown object embedded in a host medium using response measurements at remote stations. Secondly, we are concerned with the reconstruction of a medium’s material profile given, again, scant response data. For both problems we use acoustic (or equivalent) waves, to illuminate the interrogated object/medium; however, the mathematical/numerical treatment presented herein extends directly to other wave types. There is a wide, and ever widening, spectrum of possible applications that stand to benefit: of particular interest here are geotechnical applications that arise during site characterization efforts.

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To tackle both inverse problems we adopt the systematic framework of governing-equation-constrained optimization. Accordingly, misfit functionals are augmented with appropriate regularization terms, and with the weak imposition of the equations describing the physics of the wave interrogation. The governing equations may be either of the partial-differential or integral kind, subject only to user preference or problem bias. The framework is flexible enough to accommodate various misfit norms and regularization terms. We seek solutions that minimize the augmented functional by requiring that the first-order optimality conditions vanish at the optimum, thereby giving rise to Karush-Kuhn-Tucker-type systems. We then solve the associated state, adjoint, and control problems with a reduced-space approach. To alleviate the theoretical and numerical difficulties inherent to all inverse problems that are present here as well, we seek to narrow the solution feasibility space by adopting special schemes. In the shape detection and localization problem we adopt amplitude-based misfit functionals, and a frequencyand directionality-continuation scheme, somewhat akin to multigrid methods, that, thus far, have lend robustness to the inversion process. The mathematical details are based on integral equations, where, in addition, the control problem is cast in the elegant framework of total or material derivatives that allow computational speed-up when compared to finite-difference-based gradient schemes. Similarly, in the material profile reconstruction problem we adopt a time-dependent regularization scheme that exhibits superior performance to classical Tikhonov-type regularizations and is shown to be capable

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of recovering both sharp and smooth material distributions, while being relatively insensitive to the choice of initial guesses and regularization factors. These schemes constitute particular contributions of this work. We describe the mathematical framework and report numerical results. Specifically, with respect to the shape detection and localization problem we report on the two-dimensional case of sound-hard objects embedded in fullspace; with respect to the material profile reconstruction problem, we report results on the one-dimensional case of horizontally-layered systems, and on the two-dimensional case of finite or infinite-extent domains. We discuss the algorithmic performance in the presence of both noise-free and noisy data and provide recommendations for possible extensions of this work.

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Table of Contents

Acknowledgments

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Abstract

vi

List of Tables

xii

List of Figures

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Chapter 1. Introduction 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . 1.2 Problem Definitions . . . . . . . . . . . . . . . 1.2.1 Shape detection and localization (SDL) 1.2.2 Material profile reconstruction (MPR) . 1.3 Review of Related Research . . . . . . . . . . 1.4 Dissertation Outline . . . . . . . . . . . . . . .

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Chapter 2. Mathematical Modeling Framework 2.1 The Forward Problem . . . . . . . . . . . . . . . 2.2 The Inverse Problem . . . . . . . . . . . . . . . 2.3 Regularization Schemes . . . . . . . . . . . . . . 2.3.1 TSVD regularization . . . . . . . . . . . . 2.3.2 Tikhonov regularization . . . . . . . . . . 2.4 The First-Order Optimality Conditions . . . . . 2.5 Solution Approach . . . . . . . . . . . . . . . . . 2.5.1 Reduced-space method . . . . . . . . . . 2.5.2 Gradient-based schemes . . . . . . . . . . 2.5.2.1 Steepest descent . . . . . . . . . 2.5.2.2 Conjugate-gradient . . . . . . . . 2.5.2.3 Inexact line search . . . . . . . . ix

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Chapter 3. Shape Detection and Localization (SDL) 3.1 The Forward Problem . . . . . . . . . . . . . . . . . . 3.1.1 Integral equation-based solution . . . . . . . . 3.2 The Inverse Problem . . . . . . . . . . . . . . . . . . 3.3 Augmented Functional . . . . . . . . . . . . . . . . . 3.4 Boundary Shape Evolution - Total Differentiation . . 3.5 The First-Order Optimality Conditions . . . . . . . . 3.5.1 First KKT condition . . . . . . . . . . . . . . . 3.5.2 Second KKT condition . . . . . . . . . . . . . 3.5.3 Third KKT condition . . . . . . . . . . . . . . 3.6 Inversion Process . . . . . . . . . . . . . . . . . . . . 3.6.1 Frequency-continuation scheme . . . . . . . . . Chapter 4. Numerical Experiments - SDL 4.1 Example I - Circular Scatterer . . . . . . . 4.2 Example II - Penny-Shaped Scatterer . . . 4.3 Example III - Potato-Shaped Scatterer . . 4.4 Example IV - Kite-Shaped Scatterer . . . . 4.5 Example V - Arbitrarily-Shaped Scatterer . 4.6 Summary of SDL Numerical Experiments .

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Chapter 5. Material Profile Reconstruction (MPR) 5.1 The Forward Problem . . . . . . . . . . . . . . . . . 5.2 The Inverse Problem . . . . . . . . . . . . . . . . . 5.3 Regularization Schemes . . . . . . . . . . . . . . . . 5.3.1 Tikhonov regularization . . . . . . . . . . . . 5.3.2 Time-dependent regularization . . . . . . . . 5.4 Augmented Functional . . . . . . . . . . . . . . . . 5.5 The First-Order Optimality Conditions . . . . . . . 5.5.1 First KKT condition . . . . . . . . . . . . . . 5.5.2 Second KKT condition . . . . . . . . . . . . 5.5.3 Third KKT condition . . . . . . . . . . . . . 5.5.4 Fourth KKT condition . . . . . . . . . . . . . x

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5.6 Inversion Process . . . . . . . . . . . . . . . . 5.6.1 State and adjoint semi-discrete forms . . 5.6.2 Temporal discretization . . . . . . . . . 5.6.3 Model parameter updates . . . . . . . . 5.6.3.1 Time-dependent regularization . 5.6.3.2 Tikhonov regularization . . . . 5.7 Inversion in 1D Truncated Domains . . . . . . 5.8 Inversion in 2D Truncated Domains . . . . . .

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Chapter 6. Numerical Experiments - MPR 6.1 On Smooth Profiles . . . . . . . . . . . . . . . . . . 6.1.1 Regularization factor effects . . . . . . . . . . 6.1.2 Initial estimates effects . . . . . . . . . . . . 6.1.3 Noise effects . . . . . . . . . . . . . . . . . . 6.2 On Sharp Profiles . . . . . . . . . . . . . . . . . . . 6.3 Simultaneous Inversion of Modulus and Damping . 6.4 Inversion in 1D Truncated Domains . . . . . . . . . 6.5 Inversion in 2D Truncated Domains . . . . . . . . . 6.5.1 Embedded inclusion in homogeneous medium 6.5.2 Embedded inclusion in layered medium . . . 6.6 Summary of MPR Numerical Experiments . . . . .

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Chapter 7. Conclusions 167 7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 7.2 Pointers to Future Research . . . . . . . . . . . . . . . . . . . 169 Appendix

171

Bibliography

175

Vita

185

xi

List of Tables

4.1 4.2 4.3 4.4

Estimated Estimated Estimated Estimated

model model model model

parameters parameters parameters parameters

of of of of

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Example Example Example Example

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List of Figures 1.1 1.2 1.3

Forward and inverse problems . . . . . . . . . . . . . . . . . . Shape detection and localization problem (SDL) . . . . . . . . Material profile reconstruction problem (MPR) . . . . . . . .

3.1

Scattering from a sound-hard object embedded in full-space and sampling stations . . . . . . . . . . . . . . . . . . . . . . . . . Scattered pressure amplitude distribution around a kite-shaped rigid scatterer; insonification angle α = −45o ; multiple frequencies; a = kite height; kite parametrization: (x(θ), y(θ)) = (cos(θ) + 0.65(cos 2θ − 1), −10 + 1.5 sin θ), θ = 0 . . . 2π . . . . Model problem with a circular scatterer . . . . . . . . . . . . . Distribution of misfit functionals J1 and J for the problem shown in Fig. 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary shape evolution under a velocity transformation field

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3.3 3.4 3.5 4.1 4.2

Example I configuration; detection of a circular scatterer . . . Convergence patterns of the model parameters of a circular scatterer using a single-frequency and a frequency-continuation scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Convergence path of a circular scatterer using the frequencycontinuation scheme . . . . . . . . . . . . . . . . . . . . . . . 4.4 Example II configuration; detection of a penny-shaped scatterer 4.5 Convergence patterns of the model parameters of a penny-shaped scatterer using the frequency-continuation scheme . . . . . 4.6 Convergence path of a penny-shaped scatterer using the frequency-continuation scheme . . . . . . . . . . . . . . . . . . . 4.7 Example III configuration; detection of a potato-shaped scatterer 4.8 Convergence path of a potato-shaped scatterer using the frequency-continuation scheme . . . . . . . . . . . . . . . . . . . 4.9 Example IV configuration; detection of a kite-shaped scatterer 4.10 Convergence path of a kite-shaped scatterer using the frequency-continuation scheme . . . . . . . . . . . . . . . . . . . . . xiii

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42 43 44 47 64 66 67 68 70 71 73 74 75 77

4.11 Example V configuration; detection of arbitrarily-shaped scatterer 79 4.12 Convergence path of arbitrarily-shaped scatterer . . . . . . . . 81 5.1 5.2

MPR problem configuration: soil deposits on bedrock . . . . . (a)Semi-infinite domain; and (b)truncated finite domain using absorbing boundary . . . . . . . . . . . . . . . . . . . . . . . .

6.1 6.2 6.3 6.4

Smooth target profile of α . . . . . . . . . . . . . . . . . . . . Source signal and its frequency content . . . . . . . . . . . . . Measured data, um (0, t) . . . . . . . . . . . . . . . . . . . . . Target, initial, and estimated profile of α(x) using the timedependent regularization scheme . . . . . . . . . . . . . . . . . Misfit error using the time-dependent regularization scheme . Convergence patterns of α(x) using the time-dependent regularization scheme . . . . . . . . . . . . . . . . . . . . . . . . . Target, initial and estimated profile of α(x) using the Tikhonov regularization scheme . . . . . . . . . . . . . . . . . . . . . . . Convergence pattern of estimated profile of α(x) using the Tikhonov regularization scheme . . . . . . . . . . . . . . . . . . Misfit error using the Tikhonov regularization scheme . . . . . Target, initial and estimated profile of α(x) using the timedependent regularization scheme . . . . . . . . . . . . . . . . . Misfit error using the time-dependent regularization scheme . Target, initial and estimated profile of α(x) using the Tikhonov regularization scheme . . . . . . . . . . . . . . . . . . . . . . . Measured data contaminated by noise . . . . . . . . . . . . . . Target, initial and estimated profile of α(x) in case of timedependent regularization scheme . . . . . . . . . . . . . . . . . Misfit error using the time-dependent regularization scheme . Target, initial and estimated profile of α(x) using the Tikhonov regularization scheme . . . . . . . . . . . . . . . . . . . . . . . Measured data contaminated by different levels of noise . . . . Target, initial and estimated profile of α(x) using the timedependent regularization scheme . . . . . . . . . . . . . . . . . Target, initial and estimated profile of α(x) using the Tikhonov regularization scheme . . . . . . . . . . . . . . . . . . . . . . .

6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19

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86 116 122 123 123 125 126 127 128 129 130 131 132 133 135 136 137 138 140 141 142

6.20 Target, initial and estimated profile of α(x) using the timedependent regularization scheme . . . . . . . . . . . . . . . . . 144 6.21 Target, initial and estimated profile of α(x) using the Tikhonov regularization scheme . . . . . . . . . . . . . . . . . . . . . . . 145 6.22 Measured data . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.23 Target, initial and estimated profiles of α(x) and β(x) using the time-dependent regularization scheme . . . . . . . . . . . . . . 147 6.24 Problem configuration: layered medium with truncation boundary149 6.25 Target and initial profile of α(x) . . . . . . . . . . . . . . . . . 149 6.26 Measured data . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.27 Target, initial and estimated profile of α(x) using the timedependent regularization scheme . . . . . . . . . . . . . . . . . 150 6.28 Target profile: an inclusion embedded in homogeneous medium 152 6.29 Configurations of sources and receivers: the locations of sources and receivers are marked with H and , respectively . . . . . . 153 6.30 Finite element discretizations . . . . . . . . . . . . . . . . . . 154 6.31 Source signals and their frequency components . . . . . . . . . 155 6.32 Results using 119 sources and receivers . . . . . . . . . . . . . 157 6.33 Results using different configurations of sources and receivers . 158 6.34 Results using sources with different frequency content . . . . . 159 6.35 Results using different mesh densities . . . . . . . . . . . . . . 160 6.36 Target profile: an inclusion embedded in a layered medium . . 161 6.37 Sources and their frequency components . . . . . . . . . . . . 162 6.38 Configurations of sources and receivers: the locations of sources and receivers are marked with H and , respectively . . . . . . 163 6.39 Results using sources containing different frequency contents . 164

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Chapter 1 Introduction

Inverse problems are of considerable practical interest in various areas of science and engineering due, in part, to the ever broadening spectrum of important applications that range from medical, to geophysical, to target identification, to structural health monitoring investigations. Among engineering applications, inverse problems arise most commonly in the non-destructive identification or assessment of engineered systems.

1.1

Preliminaries

Typically, the primary goal in an inverse problem is the recovery of a few unknown model parameters based on the measured response to a, mostly, known excitation. This is in juxtaposition to the typical modeling of physical or engineered systems – the so-called forward problem – where the goal is to obtain the response of the system based on knowledge of the input to the system (typically the applied loads), the system properties (typically material behavior), and the boundary conditions. In an inverse problem setting some of these usually known parameters may be unknown; however, what is typically known is partial information on the response of the system. Schematically, Fig. 1.1

1

encapsulates the path to modeling the response of a physical or engineered system in both its forward and inverse character. Forward Problem Input

Structure/physical domain (idealized via a few model parameters) (geometry, material properties, BCs, etc)

Response

Inverse Problem

Figure 1.1: Forward and inverse problems From a mathematical standpoint, well-posed forward problems enjoy unique solutions that often depend continuously on the model parameters. By contrast, in most inverse problems, a complete description of the response is not available, and the model parameters are only partially known. In addition, in practice, the measured data are often contaminated by noise. As a result, a solution to the inverse problem, that is, a complete description of the unknown model parameters may not exist. Even if a solution were to exist, it may not be unique: multiple model configurations may present themselves as equally best fits to the available data and the physics of the problem. From a numerical perspective, this inherent ill-posedness of inverse problems may manifest in the ill-conditioning of matrices associated with the underlying numerical method, thus further exacerbating the search for a solution. The common source of the inherent difficulties of inverse problems can be always traced to incomplete information. Thus, enriching the available information, by, for example, augmenting the response data set, is usually a 2

sensible approach to narrowing the solution feasibility space, and thus potentially allowing for solution existence and uniqueness. However, there are only a few application domains where such a strategy may be applicable. In medical imaging, for example, where typically density information is sought through scanning (e.g. using CT-scanners), the density of the response measurements could be driven to extremes through multiple source-receiver pairs. In addition the response data can be collected at great density effectively resulting in nearly “continuous” information; the initial guess is typically quite accurate, by exploiting prior anatomical information; the experimental environments can be relatively well-controlled; and in many cases, simple relations between the model parameters and measured data are applicable (e.g. Beer’s law). However, in most other practical applications, such an advantageous setting is unusual (e.g. in geotechnical site characterization). In most cases, the number and location of the source and measurement points are limited, and the measured data are contaminated by noise; information along the bounding surfaces of interrogated objects is typically only partially known. In addition, the initial estimates might be far from the true ones. In such cases, which represent the majority of inverse problems encountered in practice, various regularization schemes have been devised and used in order to tackle the illposedness. Roughly, by regularization schemes we collectively refer to the incorporation of a-priori information in the mathematical model. Such apriori information tends to narrow the solution feasibility space. For example, if one searches for objects buried in the soil (e.g. unexploded ordnances) –a

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classical inverse scattering problem– where the objects’ shape conforms to of few shapes known a-priori, it is possible to direct the search to this limited set of shapes by requiring that the volume of objects resulting as solutions to the problem be bound to the volume of the family of the a-priori known shapes. Of interest in this work is a particular class of inverse problems, typically referred to as inverse scattering problems. The specific focus is on two closely-related problems: a shape detection and localization problem (SDL), and a material profile reconstruction problem (MPR). In the first problem, we assume that an object is fully embedded, without loss of generality, in a full-space occupied by a homogeneous, so-called, acoustic fluid. The object is assumed to be sound-hard and its location and shape are unknown. The system of the acoustic fluid with the embedded object is insonified and the response is collected at a few sparsely spaced stations arranged in the presumed backscattered region of the interrogated object. The goal is to locate the object and describe its shape, based on the measurements. In the second problem, an inhomogeneous medium whose material composition is unknown is again insonified, aiming at the reconstruction of its profile in terms of its spatially varying wave velocity and, potentially, any material attenuation parameters. In both cases waves are used as the probing agents. In such problems, the typical modeling starting point is the formation of a misfit functional between the measured response at a few locations and a computed response corresponding to an initial estimate of the model parameters, whether the latter pertain to shape parameters describing the sought 4

object of the first problem, or the material properties of the second problem. In this context, the aforementioned regularization scheme that aims at the narrowing of the solution feasibility space, appears as a penalty term imposed via a regularization factor in an augmented misfit functional. Even though a regularization scheme may “stabilize” the problem, it does not guarantee solution exactness, since the solution depends on the regularization factor which controls the amount of the penalty. To stabilize the problem without sacrificing the quality of the solution, it is necessary that optimal values of the regularization factor be sought; available procedures include the works in [14, 37, 41, 48, 73]. However, the computational cost associated with these schemes, due largely to their iterative character, can be quite high, which further exacerbates the cost associated with the typical highly iterative solution approaches to nonlinear inverse problems. Once the misfit functional is formed (with or without the regularization terms), the goal is to minimize it, thus seeking a best fit to the available data, subject to the underlying physics of the problem. As is often the case, a central difficulty revolves around the presence of multiple local minima – a direct consequence of the incomplete information or the inherent ill-posedness. One possible strategy is to use a global minimum search scheme. However, such problems typically suffer from the “curse of dimensionality” [7], whereby the number of misfit evaluations needed to estimate the model parameters grows exponentially with the number of parameters needed to be estimated. By contrast, local minimum search schemes require fewer solutions of the for-

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ward problem, but work well only when the initial estimates are within the neighborhood (or attraction basin) of the, yet unknown, global minimum. It is thus desirable to attempt to formulate the minimization problem in a manner that may a-priori lead to the enlargement of the attraction basin. Moreover, in large-scale nonlinear inverse problems, there is considerable computational cost associated with the large number of required forward problem solutions, typically at the heart of the inversion process, especially when finite difference schemes are employed for computing search directions (per iteration updates to the model parameters). It is thus also desirable to improve on the schemes needed for evaluating the search directions. In this work and for both the shape detection and localization problem, and the material profile reconstruction problem, we use a governing-equationconstrained optimization approach as a starting point for formulating the inverse problem. We use both partial-differential-equation- (PDE)- and integralequation (IE)-constrained approaches to illustrate the considerable flexibility of the modeling framework. For example, within this framework, we are able to directly and elegantly compute search directions without resorting to finite difference schemes. In an attempt to enlarge the attraction basin and improve on the regularization schemes and thereby on the chances of finding the global minimum we adopt problem-specific strategies: (i) for the shape detection and localization problem, we use an amplitude-based misfit functional within the context of frequency- and/or directionality-continuation schemes; (ii) for the material profile reconstruction problem, we focus on the use of

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a time-dependent regularization scheme, which imposes no bias on solution smoothness and does not exhibit significant sensitivity with respect to the choice of the regularization factor. The details of the proposed schemes are described in the following chapters.

1.2

Problem Definitions

We consider two different problems, a shape detection and localization problem, and a material profile reconstruction problem; detailed descriptions of both problems follow. 1.2.1

Shape detection and localization (SDL)

We are interested in recovering the shape and location of an insonified scatterer based on scant measurements of its response when excited by impinging acoustic plane waves. We treat, without loss of generality, the case of a sound-hard scatterer (rigid object, tantamount to a cavity) embedded in a two-dimensional domain (full-plane case) (Fig. 1.2). However, the approach we discuss herein can be applied to more complex (and more realistic) problems with only minor modifications to account for, for example, the threedimensional case, half-plane or half-space case, elastodynamic waves, and/or any combination of the above. Given the many practical applications, the problem has received considerable attention in the literature; among the many reviews on the topic we mention the works in [21, 56, 62]. One may roughly classify the approaches 7

Receivers xm, usm u

s

u

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inc

n

Acoustic Fluid

Scatterer S

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Figure 1.2: Shape detection and localization problem (SDL) that have been followed into methods that rely on optimization-based schemes (e.g. [5, 9, 31, 34, 35, 40, 51–53, 55, 58, 76]), and methods that do not explicitly seek to minimize a misfit functional (e.g. [22, 23, 26, 64, 65]). The advantage of the latter category methods is that the shape reconstruction can be carried out without necessarily relying on a-priori information, whereas, when optimization methods are used, the solution feasibility space may be considerably narrowed due to a-priori knowledge –almost a necessity for robust solution schemes. As mentioned, in this work we favor optimization methods for the generality they offer, and explore continuation algorithms that have, thus far, provided robust results. We remark that a considerable body of work exists where solutions are sought based on complete or almost-complete information: in the context of the shape detection and localization problem of interest herein, complete information refers to, for example, scattered pres-

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sure data that circumscribe the scatterer, whether in the near- or the far-field. Even if such measurements were collected on a relatively coarse discretization of a boundary surface surrounding the scatterer, they can still be viewed as complete information. However, in many engineering applications, obtaining complete information is practically impossible. In this work, we focus exclusively on incomplete information with a bias towards applications, where the information is typically collected on only a small part in the neighborhood of the scatterer. Closely related to the approach presented in this dissertation is the work of Bonnet and Guzina (e.g. [9, 31, 34, 35]), where the authors treated inverse obstacle problems within an acoustic or elastodynamic host medium using scant measurements. Their approach is based on integral equations and the elegant framework of total or material derivatives, similar to what we follow herein. However, in their work, convergence to the global optimum highly depends on the quality of the initial guess, given the highly oscillatory nature of the misfit functional. Herein, to overcome, we have adopted specialized schemes (e.g. continuation schemes and amplitude-based misfits), thereby improving upon their work. In addition, here the problem, as will be shown, is mathematically cast in the familiar Karush-Kuhn-Tucker (KKT) framework, that adds to the framework’s generality.

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1.2.2

Material profile reconstruction (MPR)

As a second problem we consider the reconstruction of the material profile of a soil deposit based on surficial measurements collected as the response of the soil to dynamic loads imparted on its surface (Fig. 1.3). In essence, we seek to reconstruct the distribution of the wave velocity (shear wave velocity in the case of SH-waves), assuming constant, or known, mass density (both parameters are not simultaneously recoverable for homogeneous problems). In addition, we are equally interested in recovering the distribution of attenuation parameters, subject to an appropriate description of the soil’s damping properties. In this work, we consider both one- and two-dimensional cases. The

Source N Receivers G

W : linear elastic isotropic heterogeneous soil Ga Figure 1.3: Material profile reconstruction problem (MPR) same problem, albeit at considerably different length scales, is also of primary importance to seismic hazard mitigation efforts, to soil-structure-interaction problems, and to geophysics applications (e.g. discovery of hydrocarbon deposits). An often-used technique for the geotechnical site investigations is 10

based on the SASW method (spectral-analysis-of-surface-waves) [67], which in turn relies on the analysis of surface Rayleigh-type waves for determining the dispersion curve that leads to the shear wave velocity profile. The method however is limited in several ways, not the least of which is the difficulty with the reconstruction of the dissipative soil characteristics. In this work, we discuss a PDE-constrained optimization approach for recovering the material profile of the soil deposit, including the spatial distribution of elastic modulus as well as of attenuation parameters. Again, we start with a classical misfit functional to describe the difference between computed and measured response, and augment it via the weak imposition of the governing PDEs, similar to the approach followed by Ghattas and his collaborators ([1–4]). Of significance here, as in most inverse problems, is the regularization scheme added to the augmented functionals. We discuss classical approaches such as a Tikhonov scheme, but focus on time-domain formulations and favor a coupled tempo-spatial formulation that treats the model parameters as time-dependent to improve the algorithmic robustness and to eliminate the dependence of the solution on the choice of the regularization factor; this constitutes a departure from the works in [1–4].

1.3

Review of Related Research

In this work, we consider inverting for the shape and/or material profile based on the complete wavefield. These, so-called wave inversion problems, present added difficulties that go beyond the inherent ill-posedness associated with 11

inverse problems. The first one is the high computational cost due to the nonlinearity of the problem. Because of the system’s nonlinear dependence on the model parameters (which might be the parameters defining the shape and the location of the scatterer, or the profile of the material properties in our case), an iterative scheme is required to invert for the model parameters. The second one is the multiple local minima of the misfit functional. Due to the narrow attraction basin coming from the highly oscillatory characteristic of the misfit functional (typical for these problems), special schemes to widen the attraction basin need be adopted. The simplest way to deal with a nonlinear inverse problem is to linearize the relationship between the data and model parameters based on appropriate assumptions. However, due to the assumptions imposed, such approximations introduce limitations on their applicability and usefulness. The geometrical optics idea, rooted on ray tracing, is possibly the simplest approach to detect an obstacle embedded within a homogeneous background host. Accordingly, a smooth background velocity distribution is assumed and ray theory is adopted (the classical example is radar-based detection). The approach is also used in medical ultrasonic devices to delineate tissue boundaries by calculating the distance between the source and tissue boundaries based on the speed of sound in the tissue and the reflection travel time [15, 30, 44, 45, 74]. The linear approximation of the relationship between the response and the system’s model parameters is also used in the material property recon12

struction problem. For example, in computed tomography (CT), tissue densities are recovered based on measurements of ray attenuation when the rays travel through the tissue. By assuming straight ray-paths between sources and receivers, the relationship between the density and the attenuation can be linearized and the density can be resolved via the solution of linear equations. Unlike the methods above in which straight ray-paths are assumed, diffraction tomography takes into account the diffraction effects for single weak scattering through the Born [10] or Rytov approximations [10, 43]. The use of the Born approximation (in a homogeneous background medium) leads to relations between the spatial Fourier transform of the scattered field and the spatial Fourier transform of the object [28, 29, 75]. However, due to the assumption of a single weak scattering, both Born and Rytov approximations are not applicable in the case of multiple scattering where there is strong wave velocity contrast [19, 66]. In addition, this method requires the dense measurements to allow the spatial Fourier transform to be computed accurately. Direct iterative methods work directly with the coupled nonlinear equations of the inverse scattering problem, usually iterating back and forth between the data and domain equations. Two common iterative schemes are the Born iterative method (BIM), and the distorted-Born iterative method (DBIM). To illustrate, consider the Helmholtz equation describing scattered or total pressure in an acoustic fluid: ∆u(x) + k 2 (x)u(x) = −S(x),

13

(1.1)

in which S(x) is a radiating source, and u(x) denotes the total field. Then, the total field can be expressed via an integral equation as: Z inc

u(x) = u (x) + ω

2

G0 (x, x0 )δm(x0 )u(x0 )dx0 ,

(1.2)

Ω

in which uinc (x) denotes the incident field, δm(x) = [c−2 (x) − c−2 0 (x)] denotes the velocity function to be reconstructed, Ω denotes the domain of a scatterer, and G0 is the background’s Green’s function which is the solution to: ∆G0 (x, x0 ) + k02 (x)G0 (x, x0 ) = −δ(x − x0 ).

(1.3)

Here, k0 denotes an estimate for the background’s wavenumber. It should be noted that G0 (x, x0 ) is known in closed form [59] for a homogeneous medium with k0 . Thus, in (1.2) there are two unknowns, the total field u(x) and the velocity (slowness) function δm(x). Hence, (1.2) is a linear equation for the total field u(x), while it is a nonlinear equation for δm(x). The basic idea of the BIM is to alternatingly fix one of them, and solve for the other. To begin, the unknown velocity function δm(x) is set to zero. Hence, the total field becomes equal to the incident field uinc (x). Then, δm(x) is computed numerically as a linear inverse problem. Next, δm(x) can be used to provide a new estimate for the total field u(x0 ) using the integral of (1.2), and the process is repeated iteratively until convergence is achieved [36]. By contrast to the BIM in which the background mean velocity is fixed and only the total field is updated based on the previous estimate of the background velocity, in the DBIM, the mean velocity is continuously updated.

14

Consequently, both the Green’s function and the total field also need to be updated. DBIM converges faster than the BIM but is more sensitive to noise than the BIM. Even though both BIM and DBIM appear attractive, they still have limitations when there is a strong contrast between the scatterer and the background host. In this case, a linear approximation is not applicable: it is necessary to adopt a nonlinear iterative optimization scheme using complete waveforms. Full waveform inversion uses misfit functionals combined with the governing equations [1, 12]. Since this method makes no a-priori assumptions on the scattering field, it can account for both sharp contrasts and heterogeneous hosts. However, the method is computationally expensive due to the need for solving the system equations in every iteration, and due to the need to tackle the multiple local minima difficulty. As mentioned, one approach to find the global minimum among multiple local minima is to adopt a global optimization technique; however, such methods are generally not effective for inverting for more than a few parameters. A better approach is to reformulate the leastsquares problem to enlarge the basin of attraction of the global minimum. A local optimization technique can be used to minimize the reformulated objective function, provided that the initial guess is inside the attraction basin of the global minimum. A successful reformulation method is the migration-based travel-time (MBTT) inversion method [16–18, 20]. Instead of minimizing the least-squares 15

data misfit with respect to a slowness model, one can minimize with respect to the propagator (smooth component of slowness) and time reflectivity (rough component of slowness). It has been shown that the propagator and the reflector have different effects on the objective function. The purpose of the MBTT is to decouple these components and solve for them separately. It has been shown that in some cases MBTT performs better than the classical least-squares (CLS) method [18]. However, when the initial estimate of the material properties is inside the attraction basin, CLS outperforms MBTT. This contrast motivates mixed methods, which use a combination of these two methods. The achievement of MBTT is to enlarge the attraction basin, but without significant improvement in CLS. Another drawback is that the MBTT performance depends on the quality of the initial guess; when the initial guess is not inside the attraction basin, the method fails as any local optimization method. Another approach to address the multiple local minima is the differential semblance method [33, 57, 68–70]. This method relaxes the requirement that the model fit the data precisely by adding the differential semblance term to the data misfit functional; the differential semblance term results in a smooth misfit functional. Another approach to treat the multiple local minima is the multiscale continuation scheme [1–4, 12]. In this scheme, the approximations to the objective function are minimized over a sequence of increasingly finer discretization of state and parameter spaces, which keeps the sequence of minimizers 16

within the basin of attraction of the global minimum. In [1–4], total variation regularization is also employed combined with a multiscale continuation scheme. Total variation regularization is used to address ill-posedness and preserve sharp material interfaces. In addition, to overcome the difficulties of large, dense, expensive to construct, indefinite Hessians, the authors combine Gauss-Newton linearization with matrix-free inner conjugate gradient (Krylov) iterations. As briefly discussed previously, the work herein follows, to an extent, some of the recent developments pertaining to both the shape detection and material reconstruction problems. There are several points of departure: both problems, as will be discussed, are cast within the same governing-equationconstrained optimization framework; amplitude-based misfits and continuation schemes are adopted to overcome solution multiplicity for the SDL problem, whereas a time-dependent regularization scheme is adopted in the MPR problem that is shown to be superior to classical Tikhonov while preserving sharp interfaces. The details are presented in the following chapters.

1.4

Dissertation Outline

This dissertation is organized as follows: in Chapter 2 we discuss the mathematical modeling framework in a problem-independent manner, that is, without particularizing to either of the two problems considered herein, while allowing flexibility with respect to the numerical method (PDEs or IEs, and finite elements or boundary elements). In Chapter 3 we discuss the mathematical 17

specifics of the shape detection and localization problem, and in Chapter 4 we report on the algorithmic performance and the numerical experiments for the same problem. In Chapter 5 we formulate the material reconstruction problem; we report the associated numerical experiments in Chapter 6. Finally, in Chapter 7 we summarize the main contributions and suggest extensions of this work that could be the subject of future research.

18

Chapter 2 Mathematical Modeling Framework

In this chapter, we describe the systematic framework of governing-equationconstrained optimization that we adopted for tackling inverse problems. We discuss the modeling steps in the abstract, that is, without particularizing the formulation to either of the two inverse problems of interest herein. The problem-specific details pertaining to the shape detection and localization, and material profile reconstruction problems are discussed in later chapters. It is noteworthy however that, upon discretization, both problems require the recovery of a finite set of parameters: in the shape detection/localization problem, the inversion (or model, or design) variables represent the parameterization of the unknown scatterer’s bounding surface, whereas in the profile reconstruction problem the inversion variables are material parameters consistent with the spatial discretization of the interrogated medium. Yet, despite the distinct differences, the generality of the framework allows for the like treatment of both problems. In fact, as will be shown, partially illustrative of the generality of the framework is the accommodation of approaches that are based on either partial differential or integral equations, that, in turn, give rise to finite-element- or boundary-element-based discretization schemes,

19

respectively.

2.1

The Forward Problem

Before focusing on the inverse problem, we define first the forward problem. The forward problem describes the relationship between the input source for a given system and the corresponding response of the system and, in general, it is given by a governing partial differential equation (PDE) or, equivalently and where appropriate, by an integral equation (IE). The solution of the inverse problem greatly depends on how well the forward problem is defined; throughout this work we assume that the forward problem is well-posed. Thus, without loss of generality: Forward problem For a given set of model parameters p, find u such that: L(u; p) = 0, in Ω, B(u; p) = 0, on ∂Ω,

(2.1)

where L denotes the forward differential operator (inclusive of any forcing terms), and B denotes the boundary operator (for time-dependent problems initial conditions need also be included); Ω denotes the physical domain of interest, bound by ∂Ω (we assume, without loss of generality, simply-connected domains Ω, bound by smooth surfaces ∂Ω). p denotes the set of model parameters (e.g. boundary nodal coordinates, or material properties), and u denotes the state variable. For example, in the shape detection problem, u may denote 20

scattered or total pressure wavefield. Clearly, in the forward problem p is assumed known; we remark that Ω and ∂Ω, in the context of the shape detection problem, should be viewed as Ωp and ∂Ωp , that is, the model parameters p are coefficients of the domain Ω and its boundary parameterization.

2.2

The Inverse Problem

Contrary to the forward problem, in the inverse problem we seek to determine the unknown model parameters p based on response data um 1 measured at Ns distinct locations within Ω or on ∂Ω; um denotes the vector of size Ns of the measured values of u. We start by defining a misfit functional, that is, we express the difference, using a norm of choice, between um and u, where the latter denotes the vector of computed responses at the same locations u is measured, using estimates for the set of model parameters p. Specifically: Inverse problem Minimize J: J(u, p) = E(u − um ),

(2.2)

subject to the state equations (2.1): L(u; p) = 0, in Ω, B(u; p) = 0, on ∂Ω,

(2.3)

where J denotes the misfit functional, and E(·) the misfit norm operator. A 1

The subscript m stands for “measured”.

21

classical choice for E is the least-squares norm: N

s 1X E(u − um ) = (u(xi ) − um (xi ))2 . 2 i=1

(2.4)

Though throughout this work we used least-squares or least-squares-like norms, candidate norms are not confined to least-squares; indeed, square-rootlike norms, or others, are possible. Next, to resolve the constrained minimization problem (2.2)-(2.3), we introduce an augmented functional that includes the original misfit, but adds the state equations as side conditions via Lagrange multipliers. Accordingly, let: A(u, λ, p) = J(u, p) + λΩ L(u; p) + λ∂Ω B(u; p).

(2.5)

Here A is the augmented functional, λΩ denotes the Lagrange multipliers related to the domain operator L, with λ∂Ω denoting the domain multipliers’ restriction on ∂Ω. Then the constrained minimization problem (2.2)-(2.3) reduces to the unconstrained (2.5), where we now seek to minimize A. Clearly, the convexity of A in (2.5) is not guaranteed, as is typically the case with inverse problems. In addition, there is no a-priori guarantee that there is a solution, or that the solution is unique. As mentioned in Chapter 1, to alleviate the ill-posedness we resort to regularization schemes in an attempt to narrow the solution feasibility space and tame, to the extent possible, ill-conditioning of the discrete systems arising from the approximation/discretization resulting from (2.5). The inclusion of a regularization term results in a modified functional; consequently, (2.5) becomes: A(u, λ, p) = J(u, p) + λΩ L(u; p) + λ∂Ω B(u; p) + Rp R(p), 22

(2.6)

where R(·) denotes the regularization term, and Rp is the regularization factor (a positive scalar). Specific forms the regularization term may take are discussed in the next section.

2.3

Regularization Schemes

There are at least two different strategies associated with regularization terms, both aiming at the taming of the ill-posedness, the solution multiplicity, and the ill-conditioning. We illustrate the two categories by example: In the context of the shape detection and localization problem, the regularization term may provide bounds and limitations on the form of the sought shapes, by, for example, restricting the volume (or area) of the sought obstacles. It may also allow for the rejection of non-physical shapes: for example, it may guard against self-intersecting objects which may otherwise arise as viable solutions of the inverse scattering problem. In a fully-parameterized boundary, where the nodal boundary coordinates appear as the inversion parameters, the regularization term may allow for the rejection of jagged-edge boundaries by imposing penalties on the curvature of the boundary. On the other hand, in the context of the material profile reconstruction problem, a regularization term may allow for the rejection of highly oscillatory components of the material properties, which invariably are non-physical, by imposing a smooth variation in space. At the same time, a regularization term, while rejecting rapid variations, it may, and should, also allow for the recapturing of sharp material interfaces, wherever they may exist. 23

Whereas the treatment of the first category of regularization terms is relatively straightforward, the second category merits explanation. The classical and most-often tried regularization approach, in the context of material reconstruction, is the Tikhonov form [72]. In this work, we have also used it, primarily for purposes of comparison against a time-dependent regularization scheme that, based on the numerical experiments, appears to exhibit robust performance. To illustrate the Tikhonov regularization, we discuss it in the context of discrete systems; to this end, we first describe the truncated singular value decomposition (TSVD) method for treating ill-conditioned systems, and then build on it to explain the Tikhonov regularization. 2.3.1

TSVD regularization

To explain the TSVD, first, let us define the unconstrained least-squares problem given by the discrete form: min kGm − dk, G ∈ Rm×n , m ≥ n,

(2.7)

in which matrix G is the, in general, m × n ill-conditioned system matrix, d denotes the force (or data) vector, and m the (solution) model vector to be recovered. Here, the system matrix G can be decomposed using singular value decomposition (SVD) as in: G = U ΣV T ,

(2.8)

where the left and right singular matrices U = [u1 , u2 , · · · , un ] ∈ Rm×n and V = [v 1 , v 2 , · · · , v n ] ∈ Rn×n have orthogonal columns (U T U = V T V = I), 24

and the matrix Σ has diagonal form, i.e., Σ = diag(σ1 , σ2 · · · , σn ) ∈ Rn×n . The diagonal elements σi ’s are the singular values of G, and they are ordered such that: σ1 ≥ σ2 ≥ · · · σn ≥ 0.

(2.9)

We remark that one or more small values of σi implies that G is nearly rank deficient (near singular), and the vectors v i associated with the small σi are (numerical) null vectors of G. Thus, a matrix in a discrete ill-posed problem is always highly ill-conditioned, and its numerical null space is spanned by vectors with many sign changes. From the least-squares solution to (2.7), mest is given by: £ ¤−1 mest = GT G GT d.

(2.10)

mest = (V Σ−1 U T )d.

(2.11)

There results:

If there are k non-zero singular values, the estimated model vector mest can be obtained as:

 m

est

  =V  

1 T u d σ1 1 1 T u d σ2 2

.. . 1 T u d σk k

   . 

(2.12)

Thus, from (2.12) it can be clearly seen that small singular values have a large effect on the least-squares solution and, as a result, a small amount of noise in the data d, or any round-off error, can induce large perturbation in the solution.

25

We say that the discrete Picard condition[38] is satisfied when the values of uTi d decay to zero faster than the singular values of σi . This is an indication that the pseudo-inverse solution will not be highly sensitive to noise. If the Picard condition is not satisfied, then the solution is likely to be extremely sensitive to the noise in the data. In the TSVD method [39], small singular values are ignored and, as a result, the solutions are stabilized. However, TSVD requires the SVD of the system matrix G, which is computationally expensive. 2.3.2

Tikhonov regularization

Another way to stabilize the ill-conditioned system is to impose a side-constraint; Tikhonov regularization [72] is based on this penalty term. Accordingly, the discrete problem described in (2.7) is replaced by: © ª min kGm − dk + RkL(m − m0 )k ,

(2.13)

where R is the regularization factor that controls the amount of weight associated with the imposition of the side constraint. m0 is the expected model vector and L is a differential (derivative) operator. For the zeroth-order Tikhonov regularization, also referred to as the standard form, L is replaced by the identity matrix I. For the special case of L = I and m0 = 0, the GT G in the least-squares solution given by (2.10) is replaced by GT G + RI: its eigenvalues are now in the range of [R, R + kGk2 ]. Hence the corresponding condition number is less than (R + kGk2 )/R and, it becomes smaller as the regularization factor R increases. This results in the estimates given below (also referred 26

to as “damped least-squares solution”). £ ¤−1 mest = GT G + RI GT d.

(2.14)

Clearly, the Tikhonov scheme favors (or imposes) small perturbations of the model parameters, since the penalty term becomes smaller for the small perturbations of model parameters, whereas it increases with higher perturbations. In addition the regularization factor R controls the weight of the side constraint (also referred to as the “model norm”) relative to the misfit norm. Therefore a large R favors a small model (solution) semi-norm at the cost of large data norm, while a small R (i.e. small amount of regularization) has the opposite effect. The value of R should, therefore, be chosen with care. By contrast to the TSVD method which requires the SVD of the system matrix making it prohibitively expensive for large systems, Tikhonov regularization is applicable to the large scale problems as well as to iterative nonlinear inverse problems. However, since this scheme favors smooth solutions, it works well only for a smooth distribution of the target profile; furthermore, the solution is very sensitive to the choice of the regularization factor. There are various studies on the choice of the regularization factor [14, 37, 41, 48, 73], however, most of them are related to linear problems; there is not much work on the selection of the regularization factor in the context of nonlinear inverse problems. In this work, we use, as will be shown, a time-dependent regularization scheme which uses temporal derivatives of the model parameters. The method 27

has two main advantages over the Tikhonov regularization. The first one is that it is applicable to sharply varying target profiles since it does not use the information on the spatial derivative as the Tikhonov scheme does. In addition, by virtue of not using spatial derivatives, it works better for inexact initial estimates than the Tikhonov-type regularization. In addition, in practice, it appears relatively insensitive to the choice of the regularization factor. The details are described in Chapter 5.

2.4

The First-Order Optimality Conditions

To resolve the inverse problem, we now return to the augmented functional A defined in (2.5) or (2.6). To attempt to find a minimum, we seek first to satisfy the first-order optimality conditions. That is, we require that the first-order variations of A with respect to the primary parameters, the state variables u, the Lagrange multipliers λ, and the model parameters p vanish at the optimum (this, of course, need not necessarily represent a minimum, since convexity of A is not guaranteed.). Accordingly:    δλ A  δu A = 0.   δp A at optimum

(2.15)

The above conditions are also known as the Karush-Kuhn-Tucker (KKT) conditions [47, 54]. It can then be shown, that δλ A = G(u, p) = 0,

28

(2.16)

where G(u, p) in (2.16) refers to the discrete version of (2.3)2 . In other words, the first variation of the augmented functional A with respect to the Lagrange multipliers recovers the original problem, which, henceforth, we refer to as the state problem. Similarly, the first variation of the augmented functional with respect to the state variables u results in: δu A = G∗ (u, λ, p) = 0.

(2.17)

We, henceforth, refer to (2.17) as the adjoint problem, and to the Lagrange multipliers as the adjoint variables, respectively. We remark that the system matrices G∗ (·, ·, ·) and G(·, ·) of the adjoint and state problems, respectively, are identical to each other owing to the self-adjoint nature of the original operator. Thus, from a numerical perspective, tackling the solution of (2.16) and (2.17) requires a single inversion of the system matrix. Next, from the variation of the augmented functional with respect to the model parameters p, there results: δp A = C(u, λ, p) = 0.

(2.18)

We, henceforth, refer to (2.18) as the control problem. Simultaneous solution of (2.16)-(2.18) will yield the model parameters p that minimize the misfit norm of the measurements, while satisfying the physics of the problem, owing to the PDE side constraints. 2

The specifics of the numerical scheme that results in the system matrix G are addressed on a problem-specific basis in later chapters.

29

2.5

Solution Approach

In principle, the state, adjoint and control problems can be solved as a coupled problem (full-space method). However, in this case, the computational cost and required storage space per iteration are considerable due to the large system matrix size. By contrast to the full-space method, here we opt for a reduced-space method that maps the optimization problem to the space of the model parameters, thereby eliminating the state and adjoint variables. We remark that for the solution of the state and the adjoint problem (either as a coupled system or individually) any numerical scheme may be used (finite differences, finite elements, etc.). The details follow. 2.5.1

Reduced-space method

We start by solving the state problem (2.16) to obtain the state variables u, for the initially assumed model parameters p0 , thereby satisfying the first KKT condition δλ A = 0. Then, we solve the adjoint problem (2.17), using the state variables u computed in the first step, to obtain the adjoint variables λ that satisfy the second KKT condition δu A = 0. Then, there remains to seek to update the model parameters, so that the last KKT condition – the control problem – δp A = 0 be satisfied. Having computed the state and adjoint variables from the previous steps, δp A can be obtained straightforwardly from (2.18). Notice that δp A is equivalent to the gradient of the misfit functional ∇p J3 since the constraint terms of the augmented functional vanish due to the 3

Augmented with the gradient of the regularization term, if such a term is present.

30

state problem. Therefore, to update the model parameters p it is sufficient to use any any gradient-based scheme with δp A or ∇p J as the gradients. The entire reduced-space procedure can be summarized as follows: • Step 1: Set the initial guess for the model parameters to pest . • Step 2: Using pest , solve the state problem (2.16) for the state variable u, thereby satisfying the first KKT condition δλ A = 0. • Step 3: Using pest and the computed u, solve the adjoint problem (2.17) for λ, thereby satisfying the second KKT condition δu A = 0. • Step 4: Using pest and the computed u and λ, compute δp A using (2.18), or equivalently ∇p (J + Rp R). • Step 5: Update pest . • Step 6: Iterate from step 2 to step 5 until convergence, i.e., until δp A = 0. 2.5.2

Gradient-based schemes

To satisfy the KKT conditions, we employ a gradient-based local minimum search technique within a reduced-space method. As described, by solving the state and adjoint problems separately, the state variables u and the adjoint variables λ satisfying the two KKT conditions δλ A = 0 and δu A = 0 are obtained. Then, using the computed state and adjoint variables, the variation of the augmented functional with respect to the model parameters δp A can be

31

computed; it can be subsequently used as the search direction in a gradientbased minimum search scheme. Here, δp A is equivalent to the gradient of the misfit functional with respect to the model parameters ∇p J, since the constraint terms in the augmented functional given in (2.5) vanish by virtue of the state problem. Using the gradient of the misfit functional, we could search iteratively for the minimum in the space of the model parameters. The iteration scheme is given by: pk+1 = pk + θk dk ,

(2.19)

in which pk is the model parameter estimated at the k-th iteration; dk denotes the search direction at k-th iteration, and θk the step length. Among various candidate schemes for choosing the search direction, the steepest descent method and the conjugated-gradient method are discussed in the following. In addition, we also discuss the inexact line search scheme for choosing the step length. 2.5.2.1

Steepest descent

The steepest descent method, also referred to as Cauchy’s method, is based on a linear model of the objective function, and, as a result, the negative gradient is chosen as the search direction as in: d = −∇L(p).

(2.20)

Here d is the search direction at given estimates p, and L(·) denotes the cost function which, in our case, is the misfit functional. The detailed algorithm 32

using this method is summarized in the following Algorithm 1.

Algorithm 1 Steepest Descent Method 1: 2: 3: 4: 5: 6: 7: 8: 9: 10:

Set tolerance TOL Set k = 0 and ∇L = I Choose an initial guess p0 while (|∇L(pk )|∞ ≥ TOL) do Compute ∇L(pk ) Set dk = −∇L(pk ) Choose search length θk using Algorithm 3 (inexact line search) Update new estimates by pk+1 = pk + θk dk k=k+1 end while

A disadvantage of the method is its inherent sluggishness near the minimum due to the corrections vanishing as ∇L goes to zero. 2.5.2.2

Conjugate-gradient

The conjugate-gradient method uses conjugate directions instead of the local gradient for going downhill. The steepest descent method tends to be effective far from the minimum but becomes less effective as the estimates approach the minimum, whereas Newton’s method can be unreliable far from the minimum but is very efficient as the estimates approach the minimum. The conjugate-gradient method is reliable far from the minimum, and will accelerate as the sequence of iterations approaches the minimum. In 1952, Hestenes and Stiefel [42] published an efficient iterative technique for solving system of linear equations, which is essentially the method of conjugate gradients. They viewed the set of linear equations as elements of the gradient vector of a

33

quadratic to which they sought the minimum. Later, Fletcher and Reeves [32] proved the quadratic convergence and extended the method to non-quadratic functions. A modified version of Fletcher-Revees’s work is given by Polak and Ribiere [63] for more general objective functions. In Algorithm 2, the detailed procedure of the conjugate-gradient method is presented; variants of the conjugate-gradient method are summarized next. Algorithm 2 Conjugate-Gradient Method 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11:

Set tolerance TOL Set k = 0, ∇L = I and p−1 = 0 Choose an initial guess p0 while (|∇L(pk )|∞ ≥ TOL) do Compute ∇L(pk ) Compute βk (see (2.21)-(2.23)) Set dk = −∇L(pk ) + βk pk−1 Choose search length θk using Algorithm 3 (inexact line search) Update new estimates by pk+1 = pk + θk dk k=k+1 end while

where possible choices for βk include: • Fletcher-Reeves: k∇L(pk )k2 . k∇L(pk−1 )k2

(2.21)

(∇L(pk ) − ∇L(pk−1 )) · ∇L(pk ) . (∇L(pk ) − ∇L(pk−1 )) · ∇L(pk−1 )

(2.22)

(∇L(pk ) − ∇L(pk−1 )) · ∇L(pk ) . (∇L(pk ) − ∇L(pk−1 )) · pk−1

(2.23)

βk = • Polak-Ribiére: βk = • Hestenes-Stiefel: βk =

34

2.5.2.3

Inexact line search

To apply the gradient-based minimum search scheme based on (2.19), we need to decide the step length as well as the search direction. For a given search direction dk the ideal choice would be the global minimizer of the univariate function given by: min = L(pk + θdk ). θ>0

(2.24)

This is a one-dimensional minimization, called exact line search. However, usually it is too expensive, and hence a simpler method for computing θk is required. In Algorithm 3, an inexact line search algorithm is described. In line 2 of Algorithm 3, it is confirmed whether the Armijo condition (also referred to as the sufficient decrease condition) is satisfied. Algorithm 3 Inexact Line Search 1: Choose θ0 , ρ and µ 2: while (L(pk + θdk ) < L(pk ) + µθpTk ∇L(pk )) do 3: θ ← ρθ 4: end while We used the reduced-space method, with an implementation of the Polak-Ribiére conjugate-gradient flavor, and an inexact line search for updating the model parameters. The specifics for both the SDL and MPR problems follow.

35

Chapter 3 Shape Detection and Localization (SDL)

In Chapter 2 we reviewed a general framework for resolving an inverse problem based on a governing-equation-constrained optimization approach. In this chapter we discuss the specific forms the framework takes for detecting the shape and location of a scatterer embedded in an acoustic host medium. Both the formulation and implementational details are presented in this chapter, whereas results of the numerical experiments are reported in Chapter 4.

3.1

The Forward Problem

We are concerned with the classical time-harmonic boundary-value problem that is governed by the Helmholtz equation. Let Γ be a (smooth) closed surface with exterior Ω ⊂ R2 as shown in Fig. 3.1. The exterior domain Ω is occupied by a linear, inviscid, and compressible (acoustic) fluid, characterized by wave velocity c. Γ is the bounding surface of an immovable rigid (soundhard) obstacle S. When S is insonified by an incident plane wave field uinc , the scattered field can be recovered as the solution to the following classical

36

Receivers xm, usm u

s

u

W

inc

n

Acoustic Fluid

Scatterer S

G

Figure 3.1: Scattering from a sound-hard object embedded in full-space and sampling stations problem (in two dimensions): ∆us (x) + k 2 us (x) = 0, x ∈ Ω, ∂uinc (x) ∂us (x) =− , x ∈ Γ, ∂n µ ∂n ¶ √ ∂us s lim r − iku = 0. r→∞ ∂r

(3.1) (3.2) (3.3)

In these equations us denotes scattered pressure; x is the position vector; n is the outward unit normal on Γ (pointing to the interior of S); ∆ is the Laplace operator; k is the wavenumber (k =

ω , c

with ω denoting circular frequency).

Condition (3.3), in which r is radial distance, is the Sommerfeld radiation condition. The incident field uinc describes incoming plane waves, i.e.: uinc = e−ik(x cos α+y sin α) ,

37

(3.4)

in which α is the angle formed between the normal to the traveling wave front and the global x-coordinate axis. 3.1.1

Integral equation-based solution

With the definitions of the forward problem given by (3.1)-(3.3), and for a smooth boundary Γ, the solutions to the forward problem can be obtained by the following standard integral representations: ·

¸ ∂us − D[us ], in Ω, u =S ∂n s

(3.5)

where S and D are the single- and double-layers defined for any smooth function q, as1 : Z S[q](x) = q(y)G(x, y) dΓ(y), x ∈ Ω, y ∈ Γ, Z Γ ∂G(x, y) D[q](x) = q(y) dΓ(y), x ∈ Ω, y ∈ Γ, ∂ny Γ

(3.6) (3.7)

with G(z) denoting the fundamental solution, or Green’s function, i.e., i (2) G(z) = H0 (kz), 4

(3.8)

where z = |x−y| denotes the distance between a point x within Ω and a point (2)

y on Γ; H0 denotes the zeroth order Hankel function of the second kind, and √ i = −1 denotes the imaginary unit. Equation (3.5) provides the scattered 1

We use Euler script letters (e.g. S) for domain representations of the layers, and roman letters (e.g. S) for their boundary counterparts.

38

field in Ω; by taking into account the following jump relations, lim

Ω3x→x∈Γ

lim

S[q](x) = S[q](x), or S[q] = S[q],

Ω3x→x∈Γ

1 D[q](x) = − q(x) + D[q](x), or 2 1 D[q] = − q + D[q], 2

(3.9)

(3.10)

in which Z S[q](x) = q(y)G(x, y) dΓ(y), x, y ∈ Γ, Z Γ ∂G(x, y) D[q](x) = q(y) dΓ(y), x, y ∈ Γ, ∂νy Γ

(3.11) (3.12)

it follows from (3.7) that: · s¸ ∂u 1 s u −S + D[us ] = 0, on Γ. 2 ∂n

(3.13)

Using the sound-hard Neumann condition (3.2), the above boundary integral representation can be recast as: · inc ¸ ∂u 1 s u +S + D[us ] = 0, on Γ. 2 ∂n

(3.14)

The solution of the inverse problem, as it will be shown, entails a number of solutions of the forward problem, each corresponding to a shape perturbation of the boundary Γ; let us denote with Γξ each such boundary instantiation, implying a dependence on a, yet to be defined, scalar parameter ξ. Then, equation (3.14) provides the basis for the numerical solution of the forward problem, for any boundary instantiation Γξ . We remark that (3.14) corresponds to the exterior acoustics problem, for which, it is well known (e.g. [13]) 39

that there exists a set of distinct frequencies, corresponding to eigenfrequencies of the interior problem (non-physical), for which (3.14) becomes singular. A number of schemes to alleviate the difficulty have been reported [8, 11, 13]; though mindful of the difficulty, here we have not implemented any special scheme to address it. In practice, we avoid interrogating frequencies that coincide with the fictitious singular ones.

3.2

The Inverse Problem

We are concerned with establishing the location of the scatterer S, as well as with describing its boundary Γ. Our problem is driven by measurements (Fig. 3.1). In such cases, classical lines of investigation suggest the construction of a misfit functional between the measured and computed fields. For example, one candidate choice is: J1 (Γξ ; ξ) =

Ns 1X |us (xm , ξ) − usm (xm )|2 , 2 m=1 |usm (xm )|2

(3.15)

where Ns is the number of measurement stations, with xm denoting the location of the stations. usm (x) is the measured scattered field at xm , and us (xm , ξ) denotes the forward solution computed for a boundary perturbation Γξ , also at the same locations xm . J1 defines the misfit in the least-squares sense of the complex-valued scattered field normalized with respect to the measured field; it is a reasonable starting point. However, in a recent article [46], we presented arguments in favor of an amplitude-based misfit functional, defined

40

as:

Ns 1X ||us (xm , ξ)| − |usm (xm )||2 J(Γ ; ξ) = . 2 m=1 |usm (xm )|2 ξ

(3.16)

In [46], we argued that J1 becomes highly oscillatory even for moderate frequencies, and presents the optimizer with multiple minima whose basin of attraction continues to narrow as the frequency increases. By contrast, the amplitude-based J, even though it is missing the enforcement of equality in the phase-angles between the measured and computed fields, is considerably less oscillatory, thereby lending hope that local optimization methods may arrive at the global optimum. In physical terms, J exploits the fact that around obstacles embedded in a homogeneous full-space the scattered amplitude distribution is a rather smooth-varying field, as, for example, can be seen in Fig. 3.2 for the case of a kite-shaped obstacle insonified by plane waves (α = −45o ) at four different frequencies (this assertion will not be true for inhomogeneous hosts). The following simple example provides further numerical evidence in support of our favoring J over J1 . We consider the simple case of a circular scatterer with unit radius (a = 1) insonified by a plane wave forming a (−45◦ ) angle with the x-axis as shown in Fig. 3.3, where the measurements are taken at three stations also shown in the figure (schematic is in scale). We consider the coordinates of the center as the unknown parameters. For the true values (0, 0), we solve the forward problem given by (3.14) to arrive at the distribution of the misfit functionals for variations of the center location of the assumed scatterer anywhere in the dotted square region and for a fixed radius (Fig. 3.3). We then 41

0

0.2

0

−2

0.18

−2

−4

0.16

−4

−6

0.14

−6

1.4

−8

0.12

−8

−10

0.1

−10

−12

0.08

−12

−14

0.06

−14

−16

0.04

−16

−18

0.02

−18

0

−20 −10

y

y

1.2

1

0.8

0.6

0.4

0.2

−20 −10

−8

−6

−4

−2

0 x

2

4

6

8

10

−8

−6

−4

(a) ka = 0.3 2.5

2

4

6

8

0

10

0

−2

2.5

−2

−4

2

−4

−6

2

−6

−8

1.5

−10

−8 y

y

0 x

(b) ka = 3.0

0

−12

1

1.5

−10 −12

−14

1

−14

−16

0.5

−16

−18 −20 −10

−2

0.5

−18 −8

−6

−4

−2

0 x

2

4

6

8

10

0

−20 −10

(c) ka = 15.0

−8

−6

−4

−2

0 x

2

4

6

8

10

0

(d) ka = 30.0

Figure 3.2: Scattered pressure amplitude distribution around a kite-shaped rigid scatterer; insonification angle α = −45o ; multiple frequencies; a = kite height; kite parametrization: (x(θ), y(θ)) = (cos(θ) + 0.65(cos 2θ − 1), −10 + 1.5 sin θ), θ = 0 . . . 2π

42

plot the functionals J1 and J as the scatterer’s center moves within the dotted square (Fig. 3.4). In other words, the plots represent cross-sections of the misfit functional for fixed radius (equal to the true one)2 . Figure 3.4 depicts the

(-5,20) (0,20) (5,20)

Stations

45

o

u

inc

Acoustic Fluid W

(6,6) y G a=1

Circular Scatterer S x

(-6,-6)

Figure 3.3: Model problem with a circular scatterer values of the two misfit functionals for three different interrogating frequencies (ka = 0.1, 1.0, 5.0) over the space of feasible values for the scatterer’s center’s coordinates between −6 and 6 (both x and y). As can be seen in Figs. 3.4(a),(c) and (e), J1 exhibits oscillatory nature for higher frequencies; this effectively results in multiple minima which are difficult to distinguish from the global minimum. In fact, as the frequency increases, we observe that the basin of attraction of the global minimum narrows considerably when compared with the lower frequencies. As a result, with local optimization schemes, it would 2

The scalar misfit functional depends on three parameters: its representation occupies four dimensions. Thus, the three-dimensional plot resulting from fixing one of the parameters (the radius) represents a cross-section in four-dimensional space.

43

0.22 2.5

0.2 0.18 2

0.16

0.25

10

0.14

1.5 5

Misfit Error

Misfit Error

0.2 0.15

0.12

0.1 0.1 0.05

1

0 −6 −4 2 0

6 −4

4 −2

0.08

0 −6

6

−2

0.5

2 0

0 2 4

Y

−6

−4 6

X

(a) ka = 0.1, J1

0.02

−2 4

−4 6

0.04

0 2

−2

X

0.06

4

Y

−6

(b) ka = 0.1, J

7 0.06

6 0.05

5 10

0.25 0.04

3 0 −6

6 −4

Misfit Error

Misfit Error

0.2

4 5

0.15 0.1

0 −6

2

4 −2

0.03

0.05

6 −4

2 0

1

0 2

2 0

−2 4

−2 4

−4

0

Y

−6

6

X

(c) ka = 1.0, J1

0.01

0 2

−4 6

X

0.02

4 −2

0

Y

−6

(d) ka = 1.0, J

0.1

0.09

6

0.08

5 10

0.07

0.25

5 3 0 −6

6 −4

Misfit Error

Misfit Error

0.2

4

2

4 −2 0 2

0.05

0.04

0 −6

6

1

2 0

−6

X

0.01

−2 4

Y

(e) ka = 5.0, J1

0.02

0 2

−4 6

0.03

4 −2

−2 4

X

0.05

0.1

−4

2 0

0.06

0.15

−4 6

−6

Y

0

(f) ka = 5.0, J

Figure 3.4: Distribution of misfit functionals J1 and J for the problem shown in Fig. 3.3

44

be difficult for an optimizer to escape the neighborhood of a local minimum in order to converge to the global one. By contrast, it is observed that J presents fewer local minima (Figs. 3.4(b),(d) and (f)). Mindful of these observations, we favor the use of J even though it is based solely on amplitude information. With the choice of (3.16), we seek next the minimization of J subject to the strong form (3.1)-(3.3) written for the domain and boundary perturbations Ωξ and Γξ , respectively. Accordingly, as per the discussion in Chapter 2, we define an augmented functional A, and seek next to satisfy the first-order optimality conditions.

3.3

Augmented Functional

The weak imposition of the strong form (3.1)-(3.3) via Lagrange multipliers λ(xξ , ξ), per the discussion in Chapter 2, allows casting the constrained optimization problem as an unconstrained problem and yields the following augmented functional, for which we next seek a minimum: Ns ||us (xm , ξ)| − |usm (xm )||2 1X 2 m=1 |usm (xm )|2 ½Z h i ξ s ξ 2 s ξ +Re λ(x , ξ) ∆u (x , ξ) + k u (x , ξ) dΩξ Ωξ Z h ∂us (xξ , ξ) ∂uinc (xξ , ξ) i − + dΓξ λ(xξ , ξ) ∂n ∂n ξ ¾ ZΓ h ∂us (xξ , ξ) i ξ s ξ ∞ − λ(x , ξ) − iku (x , ξ) dΓ . ∂n Γ∞

A(us , λ, ξ) =

(3.17)

In (3.17), only the real part of the weak imposition of (3.1)-(3.3) appears, since this is sufficient for ensuring that the strong form is satisfied, while con45

veniently allowing for a real-valued functional, which greatly facilitates the computational process. Seeking the minimization of A in (3.17) is tantamount to ensuring, simultaneously, the matching of the measured to the computed response, and the satisfaction of the governing equations for the true shape of the interrogated scatterer. To this end, we seek to satisfy the KKT first-order optimality conditions, by requiring that the variations of A with respect to λ, us , and ξ vanish. However, since here the model parameters amount to parameterizations of boundary shapes Γξ that change during the search iterations, we discuss first the necessary relations for addressing the mathematical details of the subsequent development.

3.4

Boundary Shape Evolution - Total Differentiation

We adopt the concept of a moving boundary to describe the boundary shape evolution between successive updates of the boundary parameterization. In other words, we assume that between shape updates the boundary evolves or moves according to a transformation velocity v (Fig. 3.5). Of course, this velocity is fictitious, but provides the proper context for computing derivatives and integrals over a domain (Ω) and boundary (Γ) that keep changing as the estimates for the location and shape of the scatterer get updated. In general, the transformation velocity has two components (for a planar curve), one tangential, and one normal to the boundary. It is the imposition of this velocity field on the boundary that forces the boundary shape to evolve. Here, we assume that the boundary evolution is due only to the normal velocity compo-

46

nent vn . We remark that this choice is not restrictive, under the assumption of relatively small boundary perturbations (see [61]). As shown in Fig. 3.5, let Γ be the boundary of a domain Ω, that evolve to Γξ and Ωξ , respectively, under the velocity field vn . Let x be a point on the

G

x

x

x

n v x W

Transformation velocity

x

G W

Figure 3.5: Boundary shape evolution under a velocity transformation field boundary Γ, as shown in Fig. 3.5. Then, under the action of the velocity field, Γ evolves to Γξ , and x becomes such that: Γ 3 x → x + ξ vn (x) n(x) ≡ xξ ∈ Γξ ,

(3.18)

where, for a given normal velocity field, the scalar parameter ξ is all that is needed to characterize the evolving shape (clearly, from (3.18), ξ is such that ξ ≡ 0 on Γ). Next, we are concerned with the derivatives (sometimes termed total or material or Eulerian or shape) of a scalar field and the derivatives of line and domain integrals defined over Γξ and Ωξ , respectively. Let f (xξ , ξ)

47

denote a scalar field defined over Ωξ . Then: · ¸ ∗ Df (xξ , ξ) = f (x, 0) Dξ ξ=0 · ¸ ∂f (xξ , ξ) ξ = + ∇f (x , ξ) · n(x) ∂ξ · ¸ ∂f ˙ = f + vn , ∂n where f˙ =

∂f . ∂ξ

(3.19)

The total derivatives of integrals over Γξ and Ωξ are similarly

defined as [61]: ¸ ¸ · Z Z · ∗ D ξ ξ f (x , ξ) dΓ = f (x, 0) + f (x, 0)divs v dΓ Dξ Γξ Γ ξ=0 ¸ Z · ∂f = f˙ + vn − 2κf vn dΓ, (3.20) ∂n Γ · ¸ ¸ Z Z · ∗ D ξ ξ = f (x , ξ) dΩ f +f divv dΩ Dξ Ωξ Ω ξ=0 ¸ Z · ˙ = f + ∇f · v + f divv dΩ Ω ¸ Z · ¡ ¢ ˙ = f + div f v dΩ Ω Z Z ˙ = f dΩ + f vn dΓ. (3.21) Ω

Γ

In (3.20), use was made of the relation divs n = −2κ, where κ denotes the curvature of the boundary Γ, div denotes the divergence operator, and divs denotes the boundary or surface divergence operator. In the last step of (3.21) use was also made of the divergence theorem. Lastly, it can also be shown (see Appendix) that: ·

D ∂f Dξ ∂n

¸ ξ=0

µ ∗ ¶ ∗ ∗ ∂f ∂f ∂vn . = =(∇f ) ·n = ∇ f ·n − ∂n ∂n ∂n 48

(3.22)

3.5

The First-Order Optimality Conditions

We turn next to the computation of the first-order optimality conditions. To this end we return to (2.15), and recast it per the specifics of the SDL problem; there results:

   δλ A  δus A = 0.   δξ A

(3.23)

Notice above that the variation with respect to ξ is equivalent to the variation with respect to the shape perturbation parameters. We derive next the KKT conditions. 3.5.1

First KKT condition

By taking the variation of A, as defined in (3.17), with respect to the Lagrange multiplier, there results: ½Z

³ ´ δλ ∆us + k 2 us dΩξ

δλ A = Re Ωξ

Z

∂uinc ´ ξ dΓ ∂n ∂n Γξ ¶ ¾ µ s Z ∂u s ∞ − − iku dΓ . δλ ∂n Γ∞ −

δλ

³ ∂us

+

(3.24)

Setting δλ A = 0 recovers the state problem ∀ ξ: State problem: ∆us (x) + k 2 us (x) = 0, s

x ∈ Ωξ ,

(3.25)

inc

∂u ∂u (x) = − (x), x ∈ Γξ , ∂n µ∂n ¶ √ ∂us s lim r − iku = 0. r→∞ ∂r 49

(3.26) (3.27)

3.5.2

Second KKT condition

Similarly, by taking the variation of A with respect to the state variable, there results: ¯ ¯ Ns ¯ s X |u | − |usm |¯ s δ|u | δus A = |usm |2 m=1 ½Z ¡ ¢ +Re λ ∆δus + k 2 δus dΩξ ξ µ ¶ ¾ ZΩ Z ∂δus ∂δus ξ s ∞ λ dΓ − λ − ikδu dΓ − . ∂n ∂n Γ∞ Γξ

(3.28)

Using integration by parts for the first integral in (3.28) yields: ¯ ¯ Ns ¯ s X |u | − |usm |¯ s δus A = δ|u | s |2 |u m m=1 ½Z ³ ´ ¡ ¢ +Re ∇ · λ∇δus − ∇λ · ∇δus + λk 2 δus dΩξ ξ ¶ ¾ µ ZΩ Z ∂δus ∂δus ξ s ∞ − dΓ − − ikδu dΓ . λ λ ∂n ∂n Γξ Γ∞

(3.29)

Using the divergence theorem for the first term of the first integration, and applying the integration by parts to the second term of the first integration yields: ¯ ¯ Ns ¯ s X |u | − |usm |¯ s δus A = δ|u | s |2 |u m m=1 ½Z Z ∂δus ∞ ∂δus ξ dΓ + λ dΓ +Re λ ∂n ∂n ξ Γ∞ Z ZΓ ³ ¡ s ¢´ ξ s λk 2 δus dΩξ + δu ∆λ − ∇ · δu ∇λ dΩ + ξ Ωξ µ ¶ ¾ ZΩ Z ∂δus ξ ∂δus s ∞ − λ dΓ − λ − ikδu dΓ . ∂n ∂n Γξ Γ∞ 50

(3.30)

Finally, using integration by parts on the second term of the third integration of (3.30), there results the variation of the augmented functional with respect to the state variable us : ½X Ns

u¯s ³ |usm | ´ 1 − |usm |2 |us | Zm=1 ³ ´ δus ∆λ + k 2 λ dΩξ + ξ ¶ ¾ µ ZΩ Z ∂λ s ∂λ ξ ∞ s − δu dΓ − δu − ikλ dΓ , (3.31) ∂n ∂n Γξ Γ∞

δus L = Re

δus

where an overbar (¯ us ) denotes complex conjugate. Next, setting δus A = 0 recovers the adjoint problem: Adjoint problem: Ns X u¯s ³ |usm | ´ ∆λ + k λ = − , 1 − s |2 s| |u |u m m=1 2

x ∈ Ωξ ,

∂λ = 0, x ∈ Γξ , ∂n √ lim r(λr − ikλ) = 0.

r→∞

(3.32) (3.33) (3.34)

Notice that the adjoint problem is nearly identical to the state problem: the governing operator is the same, however the forcing term in the adjoint problem, provided by the right-hand-side of (3.32), depends on the state variable us at the measurement stations. The boundary condition on the surface of the scatterer is also affected as per (3.33).

51

3.5.3

Third KKT condition

Finally, the variation of A with respect to ξ results in: " N µ ¶# s s s X u ¯ |u | δξ A = Re u˙ s s 2 1 − ms |um | |u | m=1 "Z ¡ ¢ D λ ∆us + k 2 us dΩξ Re + Dξ Ωξ µ s ¶ µ s ¶ ¸ Z Z ∂u ∂uinc ∂u ξ s ∞ − λ λ + dΓ − − iku dΓ ∂n ∂n ∂n Γξ Γ∞ " N ¶# µ s X ¯s |usm | s u = Re u˙ s 2 1 − s |um | |u | m=1 "Z µ # ¶ Z D ∂uinc ξ s 2 s ξ + Re dΓ , (3.35) − ∇λ · ∇u + k λu dΩ − λ Dξ ∂n Ωξ Γξ where we used: Z

Z

∂us ξ dΓ + ∇ · λ∇u dΩ = λ ∂n Ωξ Γξ s

Z

ξ

λ Γ∞

∂us ∞ dΓ . ∂n

(3.36)

Next, using the total derivatives derived in (3.20), (3.36) becomes: " N ¶# µ s s s X u¯ |u | δξ A = Re u˙ s s 2 1 − ms |um | |u | m=1 "Z ¢ ¡ ˙ s − k 2 λu˙ s dΩ −Re ∇λ˙ · ∇us + ∇λ · ∇u˙ s − k 2 λu Ω

Z

¡ ¢ D + ∇λ · ∇us − k 2 λus vn dΓ + Dξ Γ

52

Z µ Γξ

∂uinc λ ∂n

¶

# dΓξ .

(3.37)

In (3.37), the last term, using (3.21), can be expanded as follows: ¶ ¶ µ ¶ Z µ Z ( µ D ∂uinc ∂ ∂uinc ∂ ∂uinc ξ λ dΓ = λ + vn λ Dξ Γξ ∂n ∂ξ ∂n ∂n ∂n Γ ) ¶ µ ∂uinc − 2κ λ vn dΓ ∂n µ ¶ Z ( ∂λ ∂uinc ∂uinc ∂ ∂uinc ˙ + vn = λ +λ ∂n ∂ξ ∂n ∂n ∂n Γ ) µ ¶ ∂ ∂uinc ∂uinc − 2κλ + vn λ vn dΓ. (3.38) ∂n ∂n ∂n From (3.19), which provides the total derivative for a scalar field, the following relationship is valid: µ ¶ µ ¶ µ ¶ D ∂uinc ∂ ∂uinc ∂ ∂uinc = + vn . Dξ ∂n ∂ξ ∂n ∂n ∂n

(3.39)

Substituting (3.39) into (3.38), yields: D Dξ

Z µ Γξ

∂uinc λ ∂n

Z (

¶ ξ

dΓ

µ ¶ uinc D ∂uinc ˙ = λ +λ ∂n Dξ ∂n Γ ) ∂λ ∂uinc ∂uinc + vn − 2κλ vn dΓ. ∂n ∂n ∂n

(3.40)

We focus next on the second term of the right-hand side of (3.40); using (3.22) there results:

µ ¶ ∗ inc D ∂uinc ∂u ∂vn ∂uinc = − . Dξ ∂n ∂n ∂n ∂n

(3.41)

Using (3.19) for uinc , the total derivative of the incident pressure can be written as: ∗ inc

u = u˙ inc + vn 53

∂uinc . ∂n

(3.42)

However, u˙ inc = 0 since the incident wave pressure is independent of the variation of the boundary Γ; thus, taking the normal derivative of (3.42) yields: ∗ inc

∂u ∂vn ∂uinc ∂ 2 uinc = + vn . ∂n ∂n ∂n ∂n2

(3.43)

In addition, from the fact that uinc satisfies ∆uinc + k 2 uinc = 0, the following relationship can be obtained: ¡ ¢ ∂ 2 uinc ∂uinc − 2κ + ∇s · ∇s uinc + k 2 uinc = 0, 2 ∂n ∂n

(3.44)

where the Helmholtz operator was written for a tangential-normal coordinate system, and ∇s denotes the surface gradient. Substitution of (3.44) into (3.43) yields: ¸ · ∗ inc ¡ inc ¢ ∂u ∂vn ∂uinc ∂uinc 2 inc = + vn 2κ − ∇s · ∇s u −k u . ∂n ∂n ∂n ∂n Next, inserting (3.45) in (3.41) results in the following relation: µ ¶ µ ¶ ¡ ¢ D ∂uinc ∂uinc 2 inc = 2κ −k u vn − vn ∇s · ∇s uinc . Dξ ∂n ∂n

(3.45)

(3.46)

Finally, substituting (3.46) in (3.40), and then the latter in (3.37) yields: " N ¶# µ s s s X u ¯ |u | δξ A = Re u˙ s s 2 1 − ms |um | |u | (m=1 ¶ ¶ Z µ Z µ s 2 s s 2˙ s ˙ − Re ∇λ · ∇u˙ − k λu˙ dΩ + ∇λ · ∇u − k λu dΩ Z µ

Ω

¶

Z

Ω

∂uinc dΓ ∇λ · ∇us − λk 2 us vn dΓ + λ˙ ∂n Γ Γ ¶ Z Z µ ¢ ¡ ∂uinc 2 inc −k u vn dΓ − λvn ∇s · ∇s uinc dΓ + λ 2κ ∂n Γ Γ ¶ ) Z µ inc inc ∂λ ∂u ∂u + vn − 2κλvn dΓ . ∂n ∂n ∂n Γ +

54

(3.47)

The following relations can be used to simplify (3.47): ∇λ · ∇us = ∇s λ · ∇s us on Γ, Z Z ¡ ¢ ∂uinc s 2˙ s ˙ ˙ ∇λ · ∇u − k λu dΩ + λ dΓ = 0. ∂n Γ Γ

(3.48) (3.49)

Equation (3.48) holds since the normal derivative of the adjoint variable is zero on the boundary by virtue of (3.33). Equation (3.49) is the weak form of the state problem with λ˙ playing the role of a weight function. Using (3.47) and (3.48) reduces (3.49) to: · ¸ Z £ ¤ ¡ s ¢ s inc 2 inc δξ A = Re vn ∇s λ · ∇s u − λ∇s · ∇s u − λk u + u dΓ.

(3.50)

Γ

Equation (3.50) vanishes only when the assumed boundary Γ coincides with the true boundary (or a local minimum). For an assumed boundary, the state and adjoint variables satisfying the first and second KKT conditions can be obtained by solving the state and adjoint problem (3.25)-(3.27) and (3.32)-(3.34), respectively. Therefore, the optimization process is equivalent to finding the boundary that forces (3.50) to vanish with the previously computed state and adjoint variables. This gives rise to the control problem: Control problem: · ¸ Z £ ¤ ¡ s ¢ s inc 2 inc Re vn ∇s λ · ∇s u − λ∇s · ∇s u − λk u + u dΓ = 0.

(3.51)

Γ

3.6

Inversion Process

In the previous sections, we defined the inverse misfit problem and constructed an augmented functional (3.17) by imposing the PDEs as side constraints. We 55

also derived the first-order optimality conditions that gave rise to the state (3.25)-(3.27), the adjoint (3.32)-(3.34), and the control (3.51) problems. In this section, we discuss the numerical implementation of the reduced-space method for the solution of the SDL problem. To solve the state and adjoint problems, any numerical method (finite differences, finite elements, boundary elements, etc) can be used; here we opted for the Boundary Element Method, for the benefit that it provides with the automatic satisfaction of the radiation condition, and for the dimensionality reduction that it affords for discretization purposes (only Γ needs to be discretized). Accordingly, the state problem (3.25)-(3.27) is tantamount (as discussed in section 3.1.1) to the following boundary integral equation: · inc ¸ ∂u 1 s s u + D[u ] = −S , on Γ. 2 ∂n

(3.52)

Similarly, the boundary integral equation for the adjoint problem (3.32)-(3.34) can be written as: ¶ µ Ns X 1 u¯s (x) usm (xm ) λ + D[λ] = − G(x, xm ) s , on Γ(x ∈ Γ).(3.53) 1− s 2 |um (xm )|2 u (x) m=1 ∂λ Notice that the single-layer term S[ ∂n ] vanishes for the adjoint variable due

to the boundary condition (3.33). Equations (3.52) and (3.53) are discretized per the standard procedures of the boundary element method [6]; here we use quadratic isoparametric elements. Notice further, that the left-hand-side operators of both integral equations are the same: only a single system matrix inversion is needed for both problems. 56

For every estimate (and description) of Γ, first (3.52) is solved to return us on Γ; notice that the forcing term on the right-hand-side of (3.52) depends on the incoming wave uinc . Once, us is obtained, the forcing term on the righthand-side of (3.53) is completely defined, and thus, (3.53) can be solved to return the adjoint variable or Lagrange multiplier, also on Γ. Both of these steps would satisfy the first two KKT optimality conditions. To satisfy the third KKT condition, or equivalently (3.51), we reason as follows: recall from the discussion in Chapter 2 that the variation of the augmented functional with respect to the parameters is equal to the gradient of the misfit term. That is: δpi A = ∇pi J,

(3.54)

where pi denotes the i-th model parameter (unknown). However, from (3.50), it follows that: ·

Z δpi A = Re

Γ

vni

£

s

∇s λ · ∇s u − λ∇s · ∇s u

inc

¤

2

¡

s

− λk u + u

inc

¢

¸ dΓ,

(3.55)

where vni denotes the transformation velocity corresponding to the i-th model parameter. Therefore, if we now seek to minimize the misfit J, using, for example, a conjugate-gradient scheme, the components of the gradient of the misfit are readily given by (3.55), while at the same time both the state and adjoint problems have been satisfied. All that remains is to define the components of the transformation velocity. To this end, let Ψ(p) denote the (vector) function describing the parameterization of the unknown boundary, in terms of a finite set of unknown parameters p. Then, the transformation velocity at a point 57

x ∈ Γ is defined as: vni (x)

¯ ∂Ψ(p) ¯¯ = · n(x). ∂pi ¯at x

(3.56)

Thus, the components of the gradient of the misfit functional, that are central to any gradient-based scheme, are given as: Z · ∇pi J = Re

Γ

¸ ∂Ψx ∂Ψy nx + ny × ∂pi ∂pi · ¸ £ ¤ ¡ s ¢ s inc 2 inc dΓ, ∇s λ · ∇s u − λ∇s · ∇s u − λk u + u

(3.57)

where Ψx and Ψy are the cartesian components of the parameterization function Ψ, and nx , ny are similarly the components of the normal vector. Schematically, the entire inversion process is summarized in Algorithm 4 shown below. Algorithm 4 Inversion Algorithm 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14:

Set IncidentWave (Frequency and Direction) Set Tolerance TOL Set InitialBoundaryParameterization Compute Misfit while (Misfit > TOL) do Solve State Problem - Integral Equation (3.52) Save StateVariables Solve Adjoint Problem - Integral Equation (3.53) Save AdjointVariables Solve Misfit Minimization using Algorithm 2 (CG) and Control Equation (3.57) Update Model Parameters Set NewBoundaryParameterization Compute Misfit end while

58

3.6.1

Frequency-continuation scheme

One of difficulties of the wavefield-based inversion is related to the presence of multiple local minima due to the highly oscillatory nature of the misfit functional. To improve the chances of the described procedure to converge we employ continuation schemes. To explain the continuation schemes, we discuss first how a single-frequency scheme works: given a single probing frequency and a set of initial guesses for the unknown parameters, our scheme for a single frequency is encapsulated in Algorithm 5 shown below: Algorithm 5 Single-Frequency Scheme 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18:

Compute Misfit Set Tolerances Tol1 and Tol2 Set MisfitNorm = 1 Set MaxIterations = 100 Set Iterations = 1 while (Misfit > Tol1) and (MisfitNorm > Tol2) and (Iterations < MaxIterations) do Set OldMisfit = Misfit Inversion Algorithm Compute Misfit Set NewMisfit = Misfit Set MisfitNorm = | (OldMisfit - NewMisfit) / OldMisfit) | Iterations = Iterations + 1 end while if Iterations == MaxIterations then Failed else Converged end if

Clearly, even if convergence is achieved for a single frequency, there is no guarantee that the converged parameters correspond to the global minimum. In addition, at high probing frequencies, the initial estimates have to 59

be quite close to the target ones, since there are multiple attraction basins that are quite narrow. The key idea behind continuation schemes is to employ a sequence of seemingly uncoupled problems, whereby the converged model parameters from one problem are fed as initial guesses to the next, and so forth, until all problems converge under the same set of model parameters. There are at least three forms the continuation scheme may take in the SDL problem: (i) continuation over multiple probing wave frequencies, (ii) continuation over multiple probing wave directions, and (iii) continuation over multiple probing wave frequencies and directions. In its simplest form, a frequency continuation scheme entails solving the inverse problem twice, for two distinct probing frequencies, where the converged model parameters from the first probing frequency problem are fed as initial guesses to the problem driven by the second probing frequency. Care, of course, should be taken so that the converged model parameters after the second inverse problem is solved, are also acceptable solutions to the first problem. Schematically, the frequency continuation scheme is encapsulated in Algorithm 6 below. There are at least two motivating factors that lend justification to the outlined continuation scheme concept. First, on physical and technological grounds: modern probing devices have considerable frequency and directionality agility. Second, the concept is akin to multigrid methods, a numerical approach whereby the same problem is solved over a sequence of increasingly finer spatial meshes, where the solution over the coarser mesh is projected onto the finer one, and so forth, until convergence. The frequency continua-

60

Algorithm 6 Frequency-Continuation Scheme 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17:

Single-Frequency Algorithm 5 Set PreviousGuess = ConvergedModelParameters for all Frequencies do Set ith-Frequency Set InitialGuess = PreviousGuess Single-Frequency Algorithm 5 Set PreviousGuess = ConvergedModelParameters Save ith-Misfit end for for all Frequencies do Compute Misfit (Use ConvergedModelParameters) if Misfit ≤ ith-Misfit then Converged else Failed end if end for

tion scheme we discuss herein could be loosely viewed as a “multigrid” method over the wavenumber or frequency space. Clearly, the algorithm as sketched, is concerned with a series of decoupled inverse problems, that is, one problem for each frequency (even at multiple wave incidence angles), even though the scatterer is the same. The coupling of all these problems, as sketched, is a loose one: it is achieved through the revisiting of the misfit functional values for all frequencies at the end of the process to ensure that the final set of converged model parameters satisfies all problems. Alternatively, one could redefine the J functional (3.16) to allow simultaneous optimization over all of the considered frequencies, as in: " M # N X A X 1 X ||us (xm , αi , kn )| − |usm (xm , αi , kn )||2 J2 (Γ) = , (3.58) s (x , α , k )|2 2 |u m i n m n=1 i=1 m=1 where kn denotes the n-th interrogating wavenumber (or frequency), N is the 61

total number of considered frequencies, αi denotes incidence angle, and A is the total number of probing incidence angles. The process implied by (3.58) is considerably more costly: we experimented with (3.58) and have found no appreciable difference to the detection process, and thus still favor the simplicity of (3.16) coupled with the continuation scheme. In practice, we apply the continuation scheme by starting with a low probing frequency and increase the frequency until convergence. As will be shown in the next chapter, our numerical results indicate that a few frequencies (three to four) are typically sufficient for resolving the SDL problem. It is noteworthy that low probing frequencies typically allow for the resolution of location, rather than the shape, whereas higher frequencies fine-tune the shape without affecting the location.

62

Chapter 4 Numerical Experiments - SDL

In the previous chapter, we presented a systematic methodology for detecting and localizing a sound-hard scatterer fully embedded in an acoustic host. To tackle the solution multiplicity difficulty we described the benefits of using an amplitude-based misfit functional coupled with a frequency- and/or directionality-continuation scheme. In this chapter we examine the performance of the proposed scheme via numerical experiments, and report on the results. In all example problems, the measured scattered pressures are synthesized numerically by solving the forward problem using the boundary element method with a mesh different from the one we use in the inversion process in order to avoid committing a classical “inverse crime”.

4.1

Example I - Circular Scatterer

In order to examine the robustness of the proposed scheme, we start with a simple example problem in which we have a-priori information that the unknown obstacle has a circular shape. Accordingly, the location and the size of the obstacle is defined by three unknown parameters: the center’s

63

coordinates x0 and y0 , and the radius r. The boundary parameterization function is cast as: ½ Ψ(p) =

x y

¾

½ =

x0 + r cos θ y0 + r sin θ

¾ ,

(4.1)

where θ ranges from 0 to 2π, and p = [x0 , y0 , r]T . The true obstacle is centered at (0, -10) and has unit radius. We start the search process with a circle of radius 3, centered at (15, −20). We use a plane wave with an incidence angle α = −45◦ . We measure the response at three observations stations located at (−10, 0), (0, 0) and (10, 0). The configuration of the problem is depicted in Fig. 4.1.

Stations

o

a=-45 True obstacle

Acoustic field Initial guess

Figure 4.1: Example I configuration; detection of a circular scatterer To resolve the SDL problem, we use both the single-frequency scheme, and the frequency-continuation scheme.

For the single-frequency scheme,

we use an incident wave with a wavenumber k = 0.1. For the frequencycontinuation scheme, we use three frequencies corresponding to k = 0.1, 64

k = 1.0, and k = 2.0. The convergence patterns of the unknown parameters using the single-frequency and the frequency-continuation scheme are shown in Fig. 4.2; the results are summarized in Table 4.1. The path to convergence when the frequency-continuation scheme is used is depicted in Fig. 4.3. Table 4.1: Estimated model parameters of Example I x0 y0 r Initial values 15.000 -20.000 3.000 5th iteration (k = 0.1) -37.187 -25.688 2.266 17th iteration (k = 1.0) -3.821 -10.852 1.129 34th iteration (k = 5.0) 0.002 -10.000 1.000 Target values 0.000 -10.000 1.000

As can be seen in Fig. 4.2, all three parameters converge to the target values under both the single-frequency scheme and the frequency-continuation scheme. However, for the frequency-continuation scheme the convergence is faster. We remark that convergence to the true obstacle failed when a single-frequency scheme was used with a higher frequency (e.g., k = 2.0). Furthermore, it is noteworthy that when, instead of using the amplitude-based misfit functional, we used J1 (defined in (3.15)), convergence to the true scatterer failed for the same initial guess and overall configuration; convergence was possible only when the initial guess came very close to the true one. Therefore, the results appear to support the claim that the combination of the amplitude-based misfit functional with the frequency-continuation scheme alleviate the difficulties associated with the solution multiplicity.

65

k=0.1 k=1.0

k=0.1 k=1.0

k=2.0

20

k=2.0

0

10 -10

0

y0

x0

-10 -20

-20 True x0

-30

True y0

Estimated x0 using single frequency Estimated x0 using frequency continuation

-40

Estimated y0 using single frequency Estimated y0 using frequency continuation

-30

-50

-40 0

5

10 15 20 25 30 35 40 45 50 55 60 65 70

0

5

10 15 20 25 30 35 40 45 50 55 60 65 70

Iteration

Iteration

(a) Convergence pattern of x0

k=0.1 k=1.0

(b) Convergence pattern of y0

k=2.0

3.0 True r Estimated r using single frequency Estimated r using frequency continuation

2.5

r

2.0

1.5

1.0

0.5 0

5 10 15 20 25 30 35 40 45 50 55 60 65 70

Iteration

(c) Convergence pattern of r

Figure 4.2: Convergence patterns of the model parameters of a circular scatterer using a single-frequency and a frequency-continuation scheme

66

0 -5 -10

16th

True, 34th

12nd

7th

-15

Initial 9th

y

-20 -25 -30 -35

15th

5th 4th 1st 2nd

-40 -45 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5

0

5

10 15 20 25

x Figure 4.3: Convergence path of a circular scatterer using the frequencycontinuation scheme

67

4.2

Example II - Penny-Shaped Scatterer

Next, we examine the performance of the proposed scheme using a pennyshaped scatterer. To realize the penny-shaped scatterer, we used an ellipse whose ratio of the minor axis length (b) to the major axis length (a) is 1/10. The center coordinate (x0 , y0 ) of the true scatterer is (0, −10) and the length of the major semi-axis (a) and minor semi-axis (b) are 1.5 and 0.15, respectively. Accordingly, the boundary parameterization function is cast as: ½ ¾ ½ ¾ x x0 + a cos θ Ψ(p) = = , y y0 + b sin θ

(4.2)

where θ ranges from 0 to 2π, and p = [x0 , y0 , a, b]T . We use two probing waves at incidence angles α = −45◦ , −135◦ ; the scattered pressure is measured at three stations located at (−10, 0), (0, 0) and (10, 0). The configuration of the problem is depicted in Fig. 4.4. We used a frequency-continuation scheme,

Stations

o

a=-135

o

a=-45 True obstacle

Acoustic field Initial guess

Figure 4.4: Example II configuration; detection of a penny-shaped scatterer where the insonification frequencies were set at k = 0.1, k = 1.0, and k = 2.0. 68

The convergence patterns of the four model parameters are shown in Fig. 4.5 and the results are summarized in Table 4.2. Table 4.2: Estimated model parameters of Example II Iteration No. x0 y0 a b Initial values 1.00e+1 -2.00e+1 2.00e+0 2.00e+0 13th (k = 0.1) 1.61e−1 -1.04e+1 1.52e+0 1.20e−1 19th (k = 1.0) 6.26e−6 -1.01e+1 1.51e+0 1.50e−1 96th (k = 2.0) 3.88e−9 -1.00e+1 1.50e+0 1.50e−1 Target values 0.00e+0 -1.00e+1 1.50e+0 1.50e−1

The final parameter values obtained using the frequency-continuation scheme are (3.876 × 10−9 , −9.9966, 1.5001, 0.1502), which are quite close to the true ones (0, -10, 1.5, 0.15). It can be seen that the frequency-continuation scheme yield more accurate estimates than the single-frequency scheme which, for the low frequency of k = 0.1 resulted in converged values of (0.16, -10.4, 1.52, 0.12). Clearly, as the frequency increases, the solutions come closer to the true values. However, if the process were to start at a higher frequency (e.g. k = 2.0), the solution would diverge due to the optimizer becoming trapped in a local minimum away from the global optimum. It appears important that the frequency initiating the continuation-scheme is a low one. Fig. 4.6 depicts the convergence path to the true penny-shaped scatterer.

69

k=0.1

k=1.0

k=2.0

Center coordinates, x0 and y0

20 10 0 -10 True x0

-20

Estimated x0 True y0

-30

Estimated y0 -40 0

10

20

30

40

50

60

Iteration

Length of major & minor semi-axes, a & b

(a) Convergence pattern of the center coordinates k=0.1

k=1.0

k=2.0

2.5 True a Estimated a True b Estimated b

2.0

1.5

1.0

0.5

0.0 0

10

20

30

40

50

60

Iteration

(b) Convergence patterns of the major and minor semi-axes

Figure 4.5: Convergence patterns of the model parameters of a penny-shaped scatterer using the frequency-continuation scheme

70

0

-8.5

k=0.1

k=1.0

-5

-9.0

True

-10

13th

8th

-9.5 True

-15 -20

y

y

Initial

-10.0

19th 14th

6th -10.5

1st

-25 4th

13th

2nd

-11.0

-30 -35 -20

-15

-10

-5

0

5

10

15

20

-11.5 -2.0

25

-1.5

-1.0

-0.5

x

0.0

0.5

1.0

1.5

2.0

x

(a) Convergence path of penny-shaped scatterer using k = 0.1

(b) Convergence path of penny-shaped scatterer using k = 1.0

-8.5 k=2.0 -9.0 -9.5 True,60th

y

30th -10.0

19th

-10.5 -11.0 -11.5 -2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

x

(c) Convergence path of penny-shaped scatterer using k = 2.0

Figure 4.6: Convergence path of a penny-shaped scatterer using the frequencycontinuation scheme

71

4.3

Example III - Potato-Shaped Scatterer

Next, we explore the proposed scheme using a scatterer with a non-convex shape. The target scatterer is defined as a star-shaped object using: r(θ) = a0 +

3 ½ X

¾ a2i−1 cos(iθ) + a2i sin(iθ) ,

(4.3)

i=1

x(θ) = a7 + r(θ)cos(θ),

(4.4)

y(θ) = a8 + r(θ)sin(θ).

(4.5)

Accordingly, the boundary parameterization function becomes:  " ½ ¾#  P  3    a7 + a0 + i=1 a2i−1 cos(iθ) + a2i sin(iθ) cos(θ) " Ψ(p) = ½ ¾#  P3     a8 + a0 + i=1 a2i−1 cos(iθ) + a2i sin(iθ) sin(θ)

          

,

(4.6)

where θ ranges from 0 to 2π, and p = [a0 , . . . , a8 ]T . As shown in (4.3)-(4.5) and (4.6), the shape is defined by nine parameters (ai , i = 0 . . . 8). In order to recover these unknown parameters, three probing waves are used at incidence angles of −45◦ ,−90◦ and−150◦ . We used three measurement stations located at (-15,0),(0,0) and (15,0). The search process is initiated with a circular shape centered at (5, −10) with a radius of 2. The configuration of the problem is depicted in Fig. 4.7. Using the frequency-continuation scheme, the probing frequencies are set at k = 0.1 (1st to 50th iteration), and k = 1.0 (51st to the 250th iteration). The results are summarized in Table 4.3 and the convergence path is shown in Fig. 4.8. 72

Stations

o

o

a=-45 a=-150 a=-90 o True obstacle

Initial guess

Acoustic field

Figure 4.7: Example III configuration; detection of a potato-shaped scatterer

Table 4.3: Estimated model parameters of Example III i Initial 50th iteration 250th iteration Target (k = 0.1) (k = 1.0) ai ai ai ai 0 2.0 1.021 1.001 1.000 1 0.0 -0.194 0.197 0.200 2 0.0 0.040 -0.297 -0.300 3 0.0 0.119 0.125 0.125 4 0.0 0.070 0.125 0.125 5 0.0 -0.123 -0.051 -0.050 6 0.0 -0.058 -0.050 -0.050 7 5.0 -1.621 -1.994 -2.000 8 -10.0 -5.251 -5.007 -5.000

73

-3

-3.0

k=0.1

True

-4

k=1.0

50th

-3.5

-5 -4.0

-6 14th 12th

-8

2nd

-9

True,200th

-4.5

y

y

-7

9th

-5.0 -5.5

1st

-10

Initial

6th

-11

100th

-6.0

130th 110th

-6.5

-12 -13 -7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-7.0 -4.0

7

x

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0 -0.5

0.0

0.5

1.0

x

(a) Convergence path of a potato-shaped scatterer using k = 0.1

(b) Convergence path of a potato-shaped scatterer using k = 1.0

Figure 4.8: Convergence path of a potato-shaped scatterer using the frequencycontinuation scheme As it can be seen from Table 4.3 and Fig. 4.8, the estimated boundary gradually approaches the true one. The same qualitative comments that were made in the previous two examples apply here as well.

74

4.4

Example IV - Kite-Shaped Scatterer

We discuss next one of the most severe problems treated in the literature [24, 26, 50]: a kite-shaped scatterer whose non-convex parts greatly complicate the wavefield (Fig. 4.9; see also Fig. 3.2). The true shape is defined by:

Stations o

o

a=-135 a=-45 o a=-90

True obstacle

Initial guess Acoustic field

Figure 4.9: Example IV configuration; detection of a kite-shaped scatterer

x(θ) = −10 + cos θ + 0.65 (cos 2θ − 1),

(4.7)

y(θ) = 1.5 sin θ,

(4.8)

where θ ranges from 0 to 2π. In order to resolve the true shape, we use three incident waves at angles of −45◦ ,−90◦ , and −135◦ . In addition, the scattered pressure is measured at five stations located at (-20,0), (-5,0), (0,0), (5,0), and (20,0), all in the backscattered region. We use k = 0.1 and k = 0.5 for the frequency-continuation scheme. In order to approximate the boundary, the

75

following boundary parameterization function is employed:  " ½ ¾#  P8     a17 + a0 + i=1 a2i−1 cos(iθ) + a2i sin(iθ) cos(θ) " Ψ(p) = ½ ¾#  P8     a18 + a0 + i=1 a2i−1 cos(iθ) + a2i sin(iθ) sin(θ)

          

,

(4.9)

where θ ranges from 0 to 2π, and p = [a0 . . . a18 ]T . Figure 4.10 depicts the convergence path. Figure 4.10 (a) is the convergence path for k = 0.1; in this case, the solution converged at the 14-th iteration, however, as shown in the figure, the shape is still far from the true one. In the next step, the wavenumber is increased to k = 0.5, and, per the continuation scheme, the minimization process started from the converged solution of the previous wavenumber. The results from the second wavenumber are shown in Fig. 4.10 (b). As it can be seen in the figure, convergence is attained.

76

-7 -8

14th

-9 True

-10

Y

-11 2nd

4th

-12 -13

1st

-14

3rd Initial

-15 -16 -17 -5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

X

(a) Convergence path of kite-shaped scatterer using k = 0.1

-7 15th

-8 -9 -10

69th 20th

Y

-11 -12 -13 -14 -15 -16 -17 -5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

X

(b) Convergence path of kite-shaped scatterer using k = 0.5

Figure 4.10: Convergence path of a kite-shaped scatterer using the frequencycontinuation scheme

77

Table 4.4: Estimated model parameters of Example IV i Initial 14th iteration 69th iteration ai (k = 0.1) (k = 0.5) 0 1.0 1.243 1.180 1 0.0 -0.038 -0.030 2 0.0 0.055 0.104 3 0.0 -0.215 -0.152 4 0.0 0.028 0.045 5 0.0 0.096 0.328 6 0.0 -0.035 -0.047 7 0.0 -0.175 -0.173 8 0.0 0.060 0.035 9 0.0 -0.038 0.002 10 0.0 -0.037 -0.024 11 0.0 0.014 0.078 12 0.0 -0.021 -0.021 13 0.0 0.023 -0.103 14 0.0 0.031 0.027 15 0.0 -0.001 0.038 16 0.0 -0.029 -0.028 17 5.0 -0.233 -0.215 18 -15.0 -9.617 -10.165

78

4.5

Example V - Arbitrarily-Shaped Scatterer

In the previous examples, we approximated the boundary by the parameterization function using a relatively small number of parameters. In this example, we seek the boundary directly by inverting for the nodal coordinates which are used in the boundary element discretization for solving the forward and adjoint problems. The potato-shaped scatterer, used originally in Example III, is used here again. We use five recording stations and three incident waves at angles of −45◦ ,−90◦ and−150◦ . The configuration of this problem is depicted in Fig. 4.11.

Stations

o

a=-150

o

a=-45

Acoustic field

Initial guess True obstacle

Figure 4.11: Example V configuration; detection of arbitrarily-shaped scatterer We resolve this problem using a single-frequency (k = 0.1) and two different boundary discretizations. First, we discretize the boundary using 10 quadratic boundary elements, which implies 20 nodes and a total of 40 parameters (for x and y). Notice that, in this case, the discretization is not fine enough 79

to obtain accurate solutions. Then, we also resolve the same problem using a finer mesh, with 100 nodes and a total of 200 inversion parameters. The results of these two cases are presented in Fig. 4.12. As shown in Fig. 4.12(a), the solution converges to the target one quite closely even though the discretization is not fine enough. However, in the case of using fine mesh (Fig. 4.12 (b)), it is observed that the estimated boundaries form a sawtooth shape and the final solution is less accurate than that of the previous case. When using nodal coordinates as inversion variables, we need to guarantee boundary smoothness in the form of an additional penalty in the augmented functional. A candidate form includes a Tikhonov-like penalty on the boundary curvature, as in: Z Rκ

∇κ · ∇κdΓ.

(4.10)

Γ

In addition, it is also required that self-intersecting shapes be rejected. These algorithmic features have not been implemented.

80

2 1

k=0.1

True

97th

0 -1 10th

y

-2 -3

7th

-4 -5

Initial

3rd

5th

-6 -7

1st

-8 -4

-3

-2

-1

0

1

2

3

4

5

6

7

8

x

(a) Convergence path of arbitrarily-shaped scatterer using 20 nodes

2 k=0.1

True

1 0 -1

5th

21st

y

-2

3rd

-3 -4 1st

-5

Initial

-6 -7 -8 -4

-3

-2

-1

0

1

2

3

4

5

6

7

8

x

(b) Convergence path of arbitrarily-shaped scatterer using 100 nodes

Figure 4.12: Convergence path of arbitrarily-shaped scatterer

81

4.6

Summary of SDL Numerical Experiments

We have discussed the performance of the proposed schemes for the shape detection and localization of a scatterer fully embedded in an acoustic medium via the described numerical experiments. Our main observations are summarized below: • While the misfit functional J1 can be used only with an initial guess that is quite close to the target one, the amplitude-based misfit functional J works well even when the initial guess is quite inaccurate. • A low probing frequency seems to enlarge the attraction basin, and as a result, the frequency-continuation scheme with a low initial frequency of the interrogating wave greatly improves performance. • The frequency-continuation scheme allows convergence and improves the convergence rate. Even when a low initial frequency is used to widen the attraction basin, the convergence rate in the region near the optimum is slow since the slope of the misfit functional is too flat at the low frequency. Therefore, gradual increase of the insonified wave’s frequency helps to accelerate the convergence rate. • The frequency-continuation scheme yields more accurate estimates than a single-frequency scheme, when both schemes converge. Increasing the frequency of the insonified wave results in a perceived increase of the misfit functional’s slope in the neighborhood of the optimum, thereby accelerating convergence and improving the solution exactness. 82

From the above observations, it can be concluded that the frequencycontinuation scheme coupled with the amplitude-based misfit functional improves the robustness and efficiency of the inversion process.

83

Chapter 5 Material Profile Reconstruction (MPR)

In Chapter 2, we presented a governing-equation-constrained optimization framework for the systematic treatment of wavefield-based inverse problems. In this chapter we discuss the application of the methodology to the material profile reconstruction problem, with an eye to geotechnical and geophysical applications. The problem of interest here is the identification of the material properties of interrogated soil deposits based on surficial measurements collected as either the response to dynamic loads imparted on the soil surface, or as seismic records. In this work, we consider scalar waves in both one- and in two-dimensional domains. To fix ideas, we start with the onedimensional problem where the boundaries are well described (free-surface and fixed-bottom). We, then, relax the requirement for a fixed bottom by allowing the truncation of the computational domain at an arbitrary depth and, thus, introduce absorbing boundary conditions at the truncation interface, and reformulate the inverse problem accordingly. We also discuss the incorporation of regularization schemes that improve the inversion process. The performance of the proposed schemes is studied via numerical experiments, and the results are reported in Chapter 6.

84

5.1

The Forward Problem

At first, we consider the response of a layered (heterogeneous) medium (soil deposits) overlaying a homogeneous halfspace (possibly made of rock) to surface excitation. We formally reduce the problem to a one-dimensional one by considering, for example, the case of compressional waves emanating from the surface of the soil due to a uniform excitation applied throughout the entire (two-dimensional) soil surface. Similar physical problems arise if one were to consider only shear waves in the same medium, or compressional waves in a rod. The latter problem was similarly treated by Tadi [71], albeit without the formalism lent in this work. Here, to fix ideas, we shall henceforth refer to compressional waves, which allow the reduction of the problem to one dimension; ultimately, our target application is the three-dimensional inversion of highly heterogeneous deposits. In principle, the approach we discuss herein, can be applied to this more complex problem with only minor modifications to account for the higher spatial dimensionality. Therefore, let u(x, t) denote the (scalar) displacement in the direction of the applied excitation (Fig. 5.1). Let l denote the depth of the soil deposits, and T the total observation period. Then, the strong form of the forward problem can be stated as: Forward Problem Find u(x, t), such that: µ ¶ ∂ 2 u(x, t) ∂ ∂u(x, t) ∂u(x, t) − = 0, (x, t) ∈ (0, l) × (0, T )(5.1) α(x) + β(x) 2 ∂t ∂x ∂x ∂t

85

f(t) (symbolic) Station x=0 x l Inversion variables: a(x), b(x) x=l

Figure 5.1: MPR problem configuration: soil deposits on bedrock with ∂u(0, t) = f (t), ∂x

(5.2)

u(l, t) = 0,

(5.3)

u(x, 0) = u(x, ˙ 0) = 0,

(5.4)

α(0)

where x denotes location and t denotes time (an overdot denotes derivative with respect to time1 ). In the above α(x) denotes the soil’s modulus (e.g. λ + 2µ for compressional waves, with λ, µ denoting the Lamé constants), or the square of the wave propagation velocity. Throughout we assume that the material density is constant (a reasonable assumption in geotechnical site investigations); in particular, in (5.1), we assume, without loss of generality, that the density ρ = 1. Furthermore, in (5.1), β(x) represents viscous damping (normalized with respect to the density too); the attenuation character of soil 1

We use both overdots and expanded forms to denote partial time derivatives.

86

deposits is much more complex than simple viscous damping could ever capture. However, here the emphasis is on the inversion and profile reconstruction process; we expect to tackle more realistic attenuation models in the future. We consider the bottom of the layered medium fixed (rigid) at x = l (condition (5.3)); the assumption is made for simplicity, but other conditions can be accommodated, including absorbing boundary conditions that, more realistically, account for the truncation of the infinite (or semi-infinite) medium. We assume further that the system is initially at rest (condition (5.4)), and that the source excitation is at the origin (condition (5.2)). Whereas in the forward problem the excitation and the material distributions α(x) and β(x) are known, in the inverse problem of interest herein, both α(x) and β(x) are unknown; known, however, is the response u(0, t), ∀t ∈ (0, T ).

5.2

The Inverse Problem

For the system defined by the forward problem given in (5.1)-(5.4), the inverse problem can be cast as a PDE-constrained optimization problem using a (leastsquares) misfit functional, as in: Minimize: 1 J= 2

Z

T

h

i2 u(0, t) − um (0, t) dt

(5.5)

0

subject to: µ ¶ ∂ ∂ 2 u(x, t) ∂u(x, t) ∂u(x, t) − = 0, (x, t) ∈ (0, l) × (0, T ), α(x) + β(x) 2 ∂t ∂x ∂x ∂t α(0)

∂u(0, t) = f (t), ∂x 87

u(l, t) = 0, u(x, 0) = u(x, ˙ 0) = 0.

5.3

Regularization Schemes

As discussed earlier and as is true for all inverse problems, the above minimization problem is inherently ill-posed due to, at a minimum, the incomplete data set. One way to overcome or alleviate this difficulty is to impose additional constraints in an attempt to “regularize” the solution, if it at all exists. The choice of the regularization scheme is of paramount importance. In the following section, we discuss two candidate regularization schemes. 5.3.1


One of most widely adopted regularization scheme is Tikhonov regularization [72]. We discussed Tikhonov regularization and its effects in Chapter 2 in the context of discrete systems; here we start with the continuous form. Again, a Tikhonov-type regularization enforces smoothness on the model parameters and may alleviate the ill-posedness of the system. We experiment first with the first-order Tikhonov regularization scheme; accordingly, let the Tikhonov regularization term RT k be defined as: R R (p) := 2

Z

Tk

∇p · ∇p dΩ.

(5.6)

Ω

where R is a regularization factor, p is a model parameter (e.g. elastic modulus), and Ω is the spatial domain of interest (0 to l in the present one88

dimensional case). We impose the regularization term RT k as a penalty term on the misfit functional given by (5.5). Clearly, the above Tikhonov scheme favors smooth profiles since the penalty term becomes smaller (modulo the regularization factor) for smooth p distributions, whereas it increases with high frequency perturbations of the model parameters. Therefore, the Tikhonov scheme works well for smooth target profiles, but is not applicable to sharply varying target profiles. In addition, the Tikhonov scheme requires initial estimates which are quite close to the target, since the scheme precludes large perturbations from the initial guess. Lastly, the regularization factor should be chosen with care in the Tikhonov scheme, since, as will be shown in Chapter 6, the solution is very sensitive to the choice of the regularization factor. 5.3.2

Time-dependent regularization

An alternative choice for the regularization term is to use time-derivatives of the design variables [71]. To this end, we assume that p ≡ p(x, t), thus violating the physical setting of the problem. Then, a possible form for the time-dependent regularization term RT D is: R

TD

R (p) := 2

Z TZ 0

p˙2 (x, t) dΩ dt.

(5.7)

Ω

Even though the model parameter is assumed to depend on both time and space, the minimization process enforces it to be independent of time: of all the possible trajectories p(x, t) for times t ∈ (0, T ), the time-independent p(x)

89

is the one minimizing (5.7). To accomplish that, we further impose that: p(x, 0) = p0 ,

(5.8)

p(x, ˙ T ) = 0.

(5.9)

In other words, we force at final time t = T , the material property distribution to be chosen to be time-independent, among all possible trajectories. Using either of the regularization schemes, the inverse problem defined in (5.5) can be now recast as: Minimize: 1 J= 2

Z

T

h

i2 u(0, t) − um (0, t) dt + Rα (α) + Rβ (β),

(5.10)

0

subject to: µ ¶ ∂ 2 u(x, t) ∂ ∂u(x, t) ∂u(x, t) − α(x) + β(x) = 0, (x, t) ∈ (0, l) × (0, T ), ∂t2 ∂x ∂x ∂t α(0)

∂u(0, t) = f (t), ∂x u(l, t) = 0,

u(x, 0) = u(x, ˙ 0) = 0, where Rα and Rβ are the regularization terms for the material/model parameters α and β, respectively. For the time-dependent regularization case they are given by: RTα D (α)

Rα = 2

Z TZ 0

90

l 0

α˙ 2 (x, t) dx dt,

(5.11)

RTβ D (β)

Rβ = 2

Z TZ 0

l

β˙ 2 (x, t) dx dt.

(5.12)

0

Similarly, for the Tikhonov regularization scheme, they take the form: RTα k (α)

Rα := 2

RTβ k (β)

Rβ := 2

Z

l

∇α(x) · ∇α(x) dx,

(5.13)

∇β(x) · ∇β(x) dx,

(5.14)

0

Z

l 0

in which Rα and Rβ are the regularization factors for α and β, respectively.

5.4

Augmented Functional

To reconstruct the material profile we seek to minimize (5.10) subject to the governing PDE and the boundary and initial conditions given by (5.1)-(5.4). To this end, we first recast the problem as an unconstrained optimization problem by defining an augmented functional based on (5.10), where now the governing PDE and the boundary/initial conditions have been imposed (added) via Lagrange multipliers as side constraints (notice, only Neumann-type conditions need to be added as part of the side constraints; essential conditions are explicitly enforced). We then seek to satisfy the first-order optimality conditions. We discuss first the case of the time-dependent regularization; the Tikhonov scheme is simpler. The details follow: we define the augmented

91

functional as: A(u, λ, α, β) = + + − +

Z i2 1 Th u(0, t) − um (0, t) dt 2 0 Z Z Z Z Rα T l 2 Rβ T l ˙ 2 α˙ dx dt + β dx dt 2 0 0 2 0 0 µ ¶ ¾ Z TZ l ½ 2 ∂ ∂u ∂u ∂ u − α +β dx dt λ ∂t2 ∂x ∂x ∂t 0 0 · ¸ Z T ∂u(0, t) λ(0, t) α(0, t) − f (t) dt ∂x 0 Z l ∂u(x, 0) λ(x, 0) dx, ∂t 0

(5.15)

where λ(x, t) is the Lagrange multiplier, and, wherever appropriate, functional dependence has been dropped for brevity. Notice that the originally spatiallydependent variables α(x) and β(x) have been modified to α(x, t) and β(x, t) to account for the temporal dependence per the time-dependent regularization scheme. For the Tikhonov scheme, the augmented functional may be defined in a similar way. Next the first-order optimality conditions are obtained from the variation of the augmented functional with respect to the state variable u, the Lagrange multiplier or adjoint variable λ, and the design variables α and β:

 δλ A    δu A δα A    δβ A

      

= 0.

We derive next the KKT conditions postulated in (5.16).

92

(5.16)

5.5 5.5.1

The First-Order Optimality Conditions First KKT condition

At the optimum, the gradients of the augmented functional must vanish. Accordingly: µ ¶ ¾ ∂u ∂u ∂ ∂ 2u δλ A = α +β dx dt δλ − ∂t2 ∂x ∂x ∂t 0 0 · ¸ Z T ∂u(0, t) − δλ(0, t) α(0, t) − f (t) dt ∂x 0 Z l ∂u(x, 0) dx = 0, + δλ(x, 0) ∂t 0 Z TZ

l

½

(5.17)

where δλ denotes an arbitrary variation of λ. By taking into account the explicitly imposed homogeneous essential boundary u(l, t) = 0, and initial condition u(x, 0) = 0 we recover the state problem: State problem µ ¶ ∂2u ∂ ∂u ∂u − α +β = 0, (x, t) ∈ (0, l) × (0, T ), 2 ∂t ∂x ∂x ∂t α(0, t)

∂u(0, t) = f (t), ∂x

(5.18) (5.19)

u(l, t) = 0,

(5.20)

u(x, 0) = u(x, ˙ 0) = 0.

(5.21)

Clearly the state problem is identical to the forward problem given by (5.1)(5.4).

93

5.5.2

Second KKT condition

The variation of the augmented functional with respect to the state variable u yields the second KKT condition. Accordingly: Z

i u(0, t) − um (0, t) δu(0, t) dt 0 µ ¶ ¾ Z TZ l ½ 2 ∂ ∂δu ∂δu ∂ δu − α +β dx dt + λ ∂t2 ∂x ∂x ∂t 0 0 Z T ∂δu(0, t) − λ(0, t)α(0, t) dt ∂x 0 Z l ∂δu(x, 0) + dx. λ(x, 0) ∂t 0 T

h

δu A =

(5.22)

By integrating by parts and taking into account the boundary and initial conditions (δu(l, t) = 0, δu(x, 0) =

+ − + −

= 0) there results:

µ ¶ ¾ ∂ 2λ ∂ ∂λ ∂¡ ¢ δu − α − βλ dx dt ∂t2 ∂x ∂x ∂t 0 0 ½h ¾ Z T i ∂λ(0, t) δu(0, t) u(0, t) − um (0, t) − α(0, t) dt ∂x 0 Z T ∂δu(l, t) λ(l, t)α(l, t) dt ∂x 0 ½ ¾ Z l ∂δu(x, T ) λ(x, T ) + β(x, T )δu(x, T ) dx ∂t 0 Z l ∂λ(x, T ) δu(x, T ) dx. (5.23) ∂t 0 Z TZ

δu A =

∂δu(x,0) ∂t

l

½

Since δu is arbitrary, by setting δu A = 0 the following adjoint problem results: Adjoint problem µ ¶ ∂2λ ∂ ∂λ ∂¡ ¢ − βλ = 0, (x, t) ∈ (0, l) × (0, T ), α − 2 ∂t ∂x ∂x ∂t 94

(5.24)

α(0, t)

¤ ∂λ(0, t) £ = u(0, t) − um (0, t) , ∂x

(5.25)

λ(l, t) = 0,

(5.26)

˙ λ(x, T ) = λ(x, T ) = 0.

(5.27)

We remark that the adjoint problem is similar to the state problem with two important differences: first, the right-hand-side of (5.25), i.e., the source term, differs from the right-hand-side of (5.19); notice that the source in the adjoint problem is provided by the misfit (this was exactly the case with the SDL adjoint problem discussed in Chapter 3). Secondly, by virtue of (5.27), the adjoint problem is a final value problem; in addition, the damping term’s sign in (5.24) has changed. If the Tikhonov scheme were used, the same adjoint problem would have been obtained, with, naturally, α and β being only spatially dependent. 5.5.3

Third KKT condition

We obtain the KKT condition as the variation of the augmented functional with respect to α; there results: Z TZ

Z TZ

l

l

δα A = Rα αδ ˙ α˙ dx dt − 0 0 0 0 Z T ∂u(0, t) − λ(0, t) δα(0, t) dt ∂x 0

95

µ ¶ ∂ ∂u λ δα dx dt ∂x ∂x

Z TZ l ·

= − − = − − + −

¸ ¢ ∂¡ Rα αδα ˙ −α ¨ δα dx dt ∂t 0 0 ¶ ¸ Z TZ l · µ ∂ ∂u ∂λ ∂u λδα − δα dx dt ∂x ∂x ∂x ∂x 0 0 Z T ∂u(0, t) λ(0, t) δα(0, t) dt ∂x 0 Z l Rα [α(x, ˙ T )δα(x, T ) − α(x, ˙ 0)δα(x, 0)] dx 0 Z TZ l Rα α ¨ δα dx dt 0 0 ¸ Z T· ∂u(l, t) ∂u(0, t) − λ(0, t)δα(0, t) dt λ(l, t)δα(l, t) ∂x ∂x 0 Z TZ l ∂λ ∂u δα dx dt ∂x ∂x 0 0 Z T ∂u(0, t) λ(0, t) δα(0, t) dt = 0. (5.28) ∂x 0

By taking into account that δα(x, 0) = 0, α(x, ˙ T ) = 0 and λ(l, t) = 0, (5.28) reduces to:

Z TZ l ½ δα A = 0

0

¾ ∂λ ∂u − Rα α ¨+ δα dx dt = 0. ∂x ∂x

(5.29)

Similarly, the variation of the augmented functional with respect to α in the case of Tikhonov regularization results in: Z l½ δα A =

Z − Rα αxx +

0

0

96

T

¾ ∂λ ∂u dt δα dx = 0. ∂x ∂x

(5.30)

5.5.4

Fourth KKT condition

The last KKT condition can be obtained similarly to the third one, by taking the variation of the augmented functional with respect to β and there results: Z TZ l Z TZ l ∂u ˙ ˙ δβ A = Rβ βδ β dx dt + dx dt λδβ ∂t 0 0 0 0 ¸ Z TZ l Z TZ l · ³ ∂u ∂ ˙ ´ ¨ βδβ − βδβ dx dt + λ δβ dx dt = Rβ ∂t ∂t 0 0 0 0 Z lh i ˙ T )δβ(x, T ) − β(x, ˙ 0)δβ(x, 0) dx = Rβ β(x, 0 Z TZ l Z TZ l ∂u ¨ − Rβ βδβ dx dt + λ δβ dx dt = 0. (5.31) ∂t 0 0 0 0 ˙ T ) = 0, (5.31) yields: By taking into account that δβ(x, 0) = 0, β(x, ¾ Z TZ l ½ ∂u δβ A = δβ dx dt = 0 − Rβ β¨ + λ ∂t 0 0

(5.32)

Lastly, if Tikhonov regularization were used, instead of (5.32), the last KKT condition would have read: Z l½ Z δβ A = − Rβ βxx + 0

T 0

¾ ∂u dt δβ dx. λ ∂t

(5.33)

Conditions (5.29) and (5.32) constitute the control problem for the timedependent regularization case, whereas (5.30) and (5.33) are the corresponding ones for the Tikhonov regularization case.

5.6

Inversion Process

In order to seek the minimum satisfying the KKT conditions, we adopt a reduced-space method, similarly to the SDL problem. Again, notice that, in 97

principle, the state problem (5.18)-(5.21), the adjoint problem (5.24)-(5.27), and the control conditions (5.29) and (5.32) (or (5.30) and (5.33)) can be solved as a coupled problem. However, the computational cost per iteration increases, given the resulting sizes. We remark that for the solution of the state and the adjoint problem (either as a coupled system or individually) any numerical scheme may be used (finite differences, finite elements, etc.). By contrast to a full-space method, here we opt for a reduced-space method that maps the optimization problem to the space of the design variables (α and β), thereby eliminating the state and adjoint variables. We start by solving the state problem (5.18)-(5.21) to obtain the state variables u, for an estimate of the model parameters, thereby satisfying the first KKT condition δλ A = 0. Then, we solve the adjoint problem using the state variables computed in the first step, to obtain the Lagrange multipliers λ that satisfy the second KKT condition δu A = 0. To solve both the state and adjoint problems, we employ conventional finite elements. Notice that in this case, and owing to a self-adjoint operator for the original problem, we expect that the system matrices of both the state and adjoint problem to be identical (there is a slight difference with the damping term). Then, there remains to seek to update the design variables, α and β, so that the third and fourth KKT conditions (5.29) and (5.32) (or (5.30) and (5.32) for the Tikhonov scheme), respectively, be satisfied. We use the control equations to iteratively provide updates to the model parameters. The details of the inversion process follows.

98

5.6.1

State and adjoint semi-discrete forms

In order to satisfy the first KKT system for the assumed inversion variables, we have to solve the state problem given in (5.18)-(5.21). We use a standard Galerkin approach to solve the state problem. Accordingly, the weak form can be obtained by multiplying the state equation (5.18) by an appropriate test function υ(x) (with υ(l) = 0) and integrating over the entire domain. Using integration by parts, there results: Z l· 0

∂ 2 u(x, t) ∂u(x, t) ∂υ(x) υ(x) + α(x, t) 2 ∂t ∂x ∂x ¸ ∂u(x, t) +β(x, t) υ(x) dx = −υ(0)f (t), ∂t

(5.34)

where the boundary conditions have been taken into account. With a similar process, where q(x) is now used as a test function, we obtain the weak form of the adjoint problem: Z l· 0

∂ 2 λ(x, t) ∂λ(x, t) ∂q(x) q(x) + α(x, t) 2 ∂t ∂x ∂x ¸ ∂ −q(x) (β(x, t)λ(x, t)) dx = q(0) [um (0, t) − u(0, t)] . ∂t

(5.35)

We introduce next standard polynomial approximations for the state u(x, t), the adjoint λ(x, t), and the corresponding test functions υ(x) and q(x); let: u(x, t) =

N X

ui (t)φi (x),

υ(x) =

i=1

λ(x, t) =

N X

N X

υi φi (x),

(5.36)

qi φi (x),

(5.37)

i=1

λi (t)φi (x), q(x) =

i=1

N X i=1

99

where N is the number of nodal points, φ are basis functions, and ui , λi , υi , qi denote nodal quantities . Then the semi-discrete forms of the state and adjoint problems can be cast as: ¨ (t) + K(t) u(t) + C(t) u(t) ˙ Mu = F(t),

(5.38)

¨ + [K(t) + Q(t)] λ(t) − C(t) λ(t) ˙ M λ(t) = G(t),

(5.39)

where:

Z

Z

l

Mij = 0

Z

l

φi φj dx, Kij = 0

α(x, t)φ0i φ0j dx,

l

Cij =

Z

l

β(x, t)φi φj dx, Qij = − 0

˙ t)φi φj dx, β(x,

(5.40) (5.41)

0

Fi = −f (t)δi1 , Gi = [um (0, t) − u(0, t)]δi1 .

(5.42)

In the above, δi1 denotes the Kronecker delta, u and λ are the vectors of the nodal state and adjoint variables, respectively, and customary notation has been used for the matrices. Notice that, whereas the mass matrix M is independent of time, both the stiffness K and damping C matrices depend on time, due to the presence of the (assumed) time-dependent moduli. 5.6.2

Temporal discretization

To arrive at a solution first for the state variables and then for the adjoint variables, the semi-discrete forms (5.38) and (5.39) need next be discretized in time. We note that, whereas (5.38) is an initial value problem for which ˙ ) = 0. ˙ u(0) = u(0) = 0, (5.39) is a final value problem for which λ(T ) = λ(T In addition, the time-dependent matrices K(t), C(t), and Q(t) need to be 100

appropriately treated. Their temporal dependence stems from the moduli, which in turn, need also be discretized in both space and time. Accordingly, let: α(x, t) =

N X

aj (t)ϕj (x),

(5.43)

bj (t)ϕj (x),

(5.44)

j=1

β(x, t) =

N X j=1

in which ϕj are basis functions, and aj and bj denote nodal values of α and β, respectively. Whereas, one could borrow numerical integration schemes from plasticity in order to tackle the temporal discretization of the matrices, here, for implementation purposes, we opted to simplify by letting both moduli be constant in time and, piecewise constant in space. Choices for the constant values in time include: aj (t) '< aj (t) >, or aj (t) ' aj (T ), ∀j = 1, . . . , N,

(5.45)

where the former expression refers to the mean value of α over the period (0, T ), and the latter expression refers to its final value. Similarly, for the coefficients of β, bj . We opted for the second of (5.45). Thus, effectively, over an element e, (5.43) and (5.44) can be rewritten as: α(x, t)|e ' a(T )ϕe (x),

(5.46)

β(x, t)|e ' b(T )ϕe (x).

(5.47)

Consequently, the matrices in (5.40)-(5.41) are modified to now read: Z l Z l Mij = φi φj dx, Kij = a(T )ϕe φ0i φ0j dx, (5.48) 0

0

101

Z

l

Cij =

b(T )ϕe φi φj dx,

Qij = 0,

(5.49)

0

where ϕe denotes the element-per-element constant, derived from the spatial discretization of the moduli. With these simplifications in mind, standard temporal integration schemes can be used: here we opted for Newmark’s averageacceleration scheme. The algorithms are summarized below for both the state and adjoint problems (Algorithms 7 and 8). Algorithm 7 State Problem FEM 1: Construct the stiffness K, mass M , and damping C ˙ 0 and u ¨0 2: Initialize u0 , u 3: Select time step ∆t ˆ K ˆ = K + 42 M + 2 C 4: Form effective stiffness matrix K: ∆t ∆t ˆ K ˆ = LDLT 5: Triangularize K: 6: for all time steps: do 7: Calculate effective load vector at t + ∆t: 2 ˆ t+∆t = F t+∆t + M ( 4 2 ut + 4 u˙ t + u ¨ t ) + C( ∆t ut + u˙ t ) F ∆t ∆t 8: Solve for displacement at t + ∆t: ˆ t+∆t LDLT ut+∆t = F 9: Calculate accelerations and velocities at t + ∆t: 4 ¨ t+∆t = ∆t4 2 (ut+∆t − ut ) − ∆t ¨t u u˙ t − u ¨ t+∆t ) u˙ t+∆t = u˙ t + ∆t (¨ ut + u 2 10: end for ˆ is the same for both the state and Notice that the effective stiffness matrix K the adjoint problems, and thus we need to invert (or triangularize) it only once. Notice further the forward and backward traversing of the time line for the two problems.

102

Algorithm 8 Adjoint Problem FEM 1: Construct the stiffness K, mass M , and damping C T ˙T ¨T 2: Initialize λ , λ and λ 3: Select time step ∆t ˆ K ˆ = K + 42 M + 2 C 4: Form effective stiffness matrix K: ∆t ∆t ˆ K ˆ = LDLT 5: Triangularize K: 6: for all time steps: do 7: Calculate effective load vector at t − ∆t: ¨ t ) + C( 2 λt − λ˙ t ) ˆ t−∆t = Gt−∆t + M ( 4 2 λt − 4 λ˙ t + λ G ∆t ∆t ∆t 8: Solve for displacement at t − ∆t: ˆ t−∆t LDLT λt−∆t = G 9: Calculate accelerations and velocities at t − ∆t: ¨ t−∆t = 4 2 (λt−∆t − λt ) + 4 λ˙ t − λ ¨t λ ∆t ∆t t−∆t t ¨t + λ ¨ t−∆t ) λ˙ = λ˙ − ∆t (λ 2 10: end for 5.6.3

Model parameter updates

By solving the state and adjoint problem, the state variable u and the adjoint variable λ satisfying the first and the second KKT conditions, respectively, are obtained. Then, the problem is reduced to a minimization problem with respect to the model parameters, α and β. Again, similarly to the SDL problem, the variations with respect to the model parameters of the augmented functional are tantamount to the gradient components of the misfit. What remains to be done is to provide the mechanism for updating the model parameters: this can be directly accomplished via the control equations derived for the time-dependent and Tikhonov regularizations. We outline the details below.

103

5.6.3.1

Time-dependent regularization

The control equation (5.29) yields: α ¨ (x, t) =

1 ∂λ(x, t) ∂u(x, t) . Rα ∂x ∂x

(5.50)

The right-hand-side of (5.50) can be readily computed, once u and λ have been obtained. Then, the update to α can be obtained using (5.50), and the approximations (5.36), (5.37), and (5.46), by integrating (5.50) twice in time, while taking into account the conditions shown below (superscripts to a indicate new and previous values between inversion iterations): ϕe anew e (t)

1 = Rα

ZZ T

T

φ0 (x) λ(t) φ0 (x) u(t) dt dt,

(5.51)

previous anew (T ), e (0) = ae

(5.52)

a˙ new e (0) = 0.

(5.53)

There results: anew e (T )

=

aprevious (T ) e

1 + ϕe Rα

·Z Z

¸ 0T

0T

φ (x) λ(t) φ (x) u(t) dt dt

. t=T

(5.54)

Similarly, for the β updates we use the control equation (5.32) to obtain: ¨ t) = 1 λ(x, t) ∂u(x, t) , β(x, Rβ ∂t

(5.55)

and therefore: bnew e (T )

=

(T ) bprevious e

1 + ϕ e Rβ

·Z Z

¸ T

T

˙ φ (x) λ(t) φ (x) u(t) dt dt

. (5.56) t=T

104

Notice that, as evidenced by the above relations, the use of high values for the regularization factors does not distort the misfit information, since the regularization terms vanish as the time-derivatives of the inversion variables become or approach zero. However, as shown in (5.50) and (5.55), the accelerations of the time-dependent coefficients of the design variables are inversely proportional to the regularization factors Rα and Rβ . Therefore, the use of higher values of regularization factors will force the convergence rate to be slower. It is, thus, beneficial to use lower regularization factor values (as long as the resulting optimization problem converges). The entire inversion process with the time-dependent regularization scheme is summarized in Algorithm 9 below. To select an optimal value for the regularization factor, one could, alternatively, use a line search scheme. In (5.54) and (5.56), the integration part plays the role of the search direction, and the reciprocal of the regularization factor plays the role of the step length. Therefore, by combining any gradientbased scheme with the line search scheme, we could recover the optimal value for the regularization, and, thus, accelerate the convergence rate. We use the steepest descent method with the inexact line search scheme; the algorithm is summarized in Algorithm 10 below.

105

Algorithm 9 Inversion Algorithm using Time-Dependent Regularization 1: Choose Rα , Rβ 2: Set k=0 3: Set initial guess of inversion variables, ake and bke 4: Set convergence tolerance TOL 5: Set misfit = TOL + 1 6: while (misfit > TOL) do 7: Solve the state problem and obtain u (Algorithm 7) 8: Solve the adjoint problem and obtain λ (Algorithm 8) ¨ using: Compute a 9: ¯ 1 ∂λ ∂u ¯¯ a ë = Rα ∂x ∂x ¯e 10:

¨ using: Compute b

11:

Update the model parameters a using: ZZ k+1 k ¨ dt dt a =a + a

12:

Update the model parameters b using: ZZ k+1 k ¨ dt dt b =b + b

¯ ¯ 1 ∂u ¨be = λ ¯¯ Rβ ∂t e

k=k+1 13: 14: end while

106

Algorithm 10 Inversion Algorithm using Time-Dependent Regularization with Optimal Regularization Factor 1: Choose θ, ρ, µ, Rα , Rβ 2: Set k=0 3: Set initial guess of inversion variables, ake and bke , Set p = {a b}T 4: Set convergence tolerance TOL 5: Set misfit = TOL + 1 6: while (misfit > TOL) do 7: Solve the state problem and obtain u (Algorithm 7) 8: Solve the adjoint problem and obtain λ (Algorithm 8) ¨ using: 9: Compute a ¯ 1 ∂λ ∂u ¯¯ a ë = Rα ∂x ∂x ¯e 10:

11:

¨ using: Compute b

¯ ¯ ∂u 1 ¨be = λ ¯¯ Rβ ∂t e ½ ¾ dα Compute the search direction dk = dβ where ZZ ZZ ¨ dt dt ¨ dt dt and dβ = dα = a b

while (J(pk + θk dk ) ≤ J(pk ) + µθk pk · ∇J(pk ) ) do 12: 13: θ ← ρθ 14: end while 15: Update the estimates pk+1 = pk + θk dk 16: k=k+1 17: end while

107

5.6.3.2


For the Tikhonov regularization, the inversion process is similar to Algorithm 10; the primary difference is in the computation of the search direction. Again, by solving the state and adjoint problems, we obtain the state variables u and the Lagrange multipliers λ, which satisfy the first and the second KKT condition under given estimates of the model parameters. Then, the remaining third and the fourth KKT conditions given by (5.30) and (5.33), respectively, essentially provide the first-order variations (or components of the gradient) of the misfit functional with respect to the model parameters. Using these firstorder variations, we can adapt appropriately any gradient-based optimization scheme, similar to what we did with the SDL problem. Here we opt for the steepest descent method combined with an inexact line search. In other words, we take the search direction as the negative of the first-order variations. The entire inversion process using Tikhonov regularization and a classical optimization scheme is summarized in Algorithm 11.

108

Algorithm 11 Inversion using Tikhonov regularization 1: Choose θ, ρ, µ, Rα , Rβ 2: Set k=0 3: Set initial guess of inversion variables, ake and bke , Set p = {a b}T 4: while (misfit > TOL) do 5: Solve the state problem and obtain u (Algorithm 7) 6: Solve the adjoint problem and obtain 8) ½ λ (Algorithm ¾ dα Compute the search direction dk = 7: dβ where ¯ ¯ Z T ∂ 2 α ¯¯ ∂λ ∂u ¯¯ dα,e = Rα 2 ¯ − ¯ dt ∂x e 0 ∂x ∂x e ¯ ¯ Z T ∂ 2 β ¯¯ ∂u ¯¯ dβ,e = Rβ 2 ¯ − λ ¯ dt ∂x e ∂t e 0 8: while (J(pk + θk dk ) ≤ J(pk ) + µθk pk · ∇J(pk ) ) do θ ← ρθ 9: 10: end while 11: Update the estimates pk+1 = pk + θk dk 12: k=k+1 13: end while

109

5.7

Inversion in 1D Truncated Domains

In the previous sections, we discussed the mathematical formulation and numerical implementation of inverse problems arising in soil deposits (one-dimensional assumptions) whose depth-to-bottom is a-priori known. We discuss next, the case when the depth is not known, and perforce, the computational domain is truncated, e.g. at depth x = l, which does not correspond to a rigid bottom. The goal is again to reconstruct the material profile of the soil deposits between x = 0 and the truncated interface at x = l. From a mathematical perspective, and to account for the part of the physical domain that will be left out of the computations, we introduce an absorbing boundary condition at the truncation boundary. The strong form of the ensuing forward problem is: Forward Problem Find u(x, t), such that: µ ¶ ∂ 2 u(x, t) ∂ ∂u(x, t) − α(x) = 0, (x, t) ∈ (0, l) × (0, T ), ∂t2 ∂x ∂x

(5.57)

with ∂u(0, t) = f (t), ∂x ∂u(l, t) 1 ∂u(l, t) = −p , ∂x α(l) ∂t α(0)

u(x, 0) = u(x, ˙ 0) = 0.

(5.58) (5.59) (5.60)

We remark that for homogeneous deposits, the absorbing condition (5.59) is exact; for inhomogeneous deposits (of interest here) it is only approximate. 110

From a physical point of view, the approximate character of (5.59) will introduce reflections at the truncation boundary. The above problem is similar to one we defined in section 5.1; we have dropped the damping term for simplicity2 . We modify next the main steps we followed in previous sections, to illustrate the changes needed to accommodate the truncated boundary; we use again the time-dependent regularization: Minimize: 1 J= 2

Z

T 0

Z Z h i2 Rα T l 2 u(0, t) − um (0, t) dt + α˙ (x, t) dx dt, 2 0 0

(5.61)

subject to: µ ¶ ∂ 2 u(x, t) ∂ ∂u(x, t) − α(x, t) = 0, (x, t) ∈ (0, l) × (0, T ), ∂t2 ∂x ∂x ∂u(0, t) = f (t), ∂x ∂u(l, t) 1 ∂u(l, t) = −p , ∂x α(l, t) ∂t α(0, t)

u(x, 0) = u(x, ˙ 0) = 0. We introduce next the augmented functional, imposing the PDE and the 2

The presence of damping, or in general, of viscoelastic behavior, greatly complicates the absorbing boundary condition; indeed, to date, and for, for example, the linear standard solid model, there are no absorbing boundary conditions developed capable of adequately modeling inhomogeneous truncated 1D domains.

111

boundary/initial conditions as side constraints: Z i2 1 Th u(0, t) − um (0, t) dt 2 0 Z Z Rα T l 2 α˙ dx dt 2 0 0 µ ¶¾ Z TZ l ½ 2 ∂ ∂u ∂ u − α dx dt λ ∂t2 ∂x ∂x 0 0 ½ ¾ Z T ∂u(0, t) λ(0, t) α(0, t) − f (t) dt ∂x 0 ¾ ½ Z T ∂u(l, t) p ∂u(l, t) λ(l, t) α(l, t) + α(l, t) dt ∂x ∂t 0 Z l ∂u(x, 0) λ(x, 0) dx. (5.62) ∂t 0

A(u, λ, α) = + + − + +

We derive next the first-order optimality conditions:    δλ A  δu A = 0.   δα A

(5.63)

From the first KKT condition, δλ A = 0, we obtain: µ ¶¾ ∂ 2u ∂ ∂u − α dx dt δλ ∂t2 ∂x ∂x 0 0 · ¸ Z T ∂u(0, t) δλ(0, t) α(0, t) − f (t) dt ∂x 0 ½ ¾ Z T ∂u(l, t) p ∂u(l, t) δλ(l, t) α(l, t) + α(l, t) dt ∂x ∂t 0 Z l ∂u(x, 0) dx = 0, δλ(x, 0) ∂t 0 Z

δλ A = − + +

T

Z

l

½

(5.64)

and by taking into account the explicitly imposed initial condition of u(x, 0) = 0, there results: 112

State problem µ ¶ ∂ 2 u(x, t) ∂ ∂u(x, t) − α(x, t) = 0, (x, t) ∈ (0, l) × (0, T ), ∂t2 ∂x ∂x

(5.65)

∂u(l, t) 1 ∂u(l, t) = −p , ∂x α(l, t) ∂t

(5.66)

u(x, 0) = u(x, ˙ 0) = 0,

(5.67)

∂u(0, t) = f (t). ∂x

(5.68)

α(0, t)

The above state problem is identical to the forward problem given by (5.57)(5.60). Similarly, from the variation of the augmented functional with respect to the state variable, we obtain: Z Th i u(0, t) − um (0, t) δu(0, t) dt δu A = 0 µ ¶¾ Z TZ l ½ 2 ∂ δu ∂ ∂δu + λ − α dx dt ∂t2 ∂x ∂x 0 0 Z T ∂δu(0, t) − λ(0, t)α(0, t) dt ∂x 0 ½ ¾ Z T ∂δu(l, t) p ∂δu(l, t) + λ(l, t) α(l, t) + α(l, t) dt ∂x ∂t 0 Z l ∂δu(x, 0) + λ(x, 0) dx. ∂t 0

(5.69)

Integrating by parts while taking into account any homogeneous essential boundary and initial conditions, there results: µ ¶¾ Z TZ l ½ 2 ∂ λ ∂ ∂λ − δu α dx dt δu A = ∂t2 ∂x ∂x 0 0 ½ ¾ Z T £ ¤ ∂λ(0, t) + δu(0, t) u(0, t) − um (0, t) − α(0, t) dt ∂x 0 ½ Z T ´¾ ∂ ³p ∂λ(l, t) − α(l, t)λ(l, t) dt + δu(l, t) α(l, t) ∂x ∂t 0 113

Z

p ∂δu(x, T ) dx + α(l, T )λ(l, T )δu(l, T ) ∂t 0 Z l ∂λ(x, T ) − δu(x, T ) dx. ∂t 0 T

+

λ(x, T )

(5.70)

Since δu is arbitrary, by setting δu A = 0 the following adjoint problem results: Adjoint problem µ ¶ ∂ 2 λ(x, t) ∂ ∂λ(x, t) − α(x, t) = 0, (x, t) ∈ (0, l) × (0, T ), ∂t2 ∂x ∂x α(l, t)

´ ∂λ(l, t) ∂ ³p = α(l, t)λ(l, t) , ∂x ∂t

α(0, t)

(5.71) (5.72)

˙ λ(x, T ) = λ(x, T ) = 0,

(5.73)

¤ ∂λ(0, t) £ = u(0, t) − um (0, t) . ∂x

(5.74)

Lastly, the variation of the augmented functional with respect to α yields: Z TZ

Z TZ

l

l

µ ¶ ∂u ∂ δα dx dt λ ∂x ∂x

δ α A = Rα αδ ˙ α˙ dx dt − 0 0 0 0 Z T ∂u(0, t) λ(0, t)δα(0, t) − dt ∂x 0 ½ ¾ Z T ∂u(l, t) δα(l, t) ∂u(l, t) + λ(l, t) δα(l, t) + p dt. ∂x 2 α(l, t) ∂t 0

(5.75)

By integrating by parts and taking into account that α(x, ˙ T ) = 0, δα(x, 0) = 0, while canceling like-terms, (5.75) yields: Z TZ l ½ δα A =

∂λ(x, t) ∂u(x, t) ∂x ∂x ¾ λ(x, t) ∂u(x, t) + p δ(x − l) δα(x, t) dx dt = 0. (5.76) 2 α(x, t) ∂t − Rα α ¨ (x, t) +

0

0

114

Equation (5.76) is nearly identical to (5.29) derived before for a fixed bottom; only the last term differs to reflect the contribution of the absorbing boundary. Therefore, the problem can be treated similarly to the process we described in Algorithms 9 and 10.

5.8


Next, we extend the formulation to the two-dimensional heterogeneous semiinfinite medium case. The strong form of the forward problem can be cast as: Forward Problem Find u(x, t), such that: ∂ 2 u(x, t) − ∇ · [α(x)∇u(x, t)] = 0, (x, t) ∈ Ω × (0, T ), ∂t2

(5.77)

with u(x, 0) = u(x, ˙ 0) = 0, x ∈ Ω, α(x)

∂u(x, t) = f (x, t), (x, t) ∈ Γf × (0, T ), ∂n µ ¶ √ 1 lim r ur + p u˙ = 0. r→∞ α(x)

(5.78) (5.79) (5.80)

In the above, x denotes the position vector, and n the normal vector to the bounding surfaces of Ω. Equation (5.80), in which r denotes the radial distance and ur the derivative of u along the radial direction, is the Sommerfeld radiation condition. To resolve this problem, the semi-infinite domain Ω is truncated via the introduction of an artificial boundary Γa (Fig. 5.2). The 115

f(x,t)

f(x,t) Free surface G

f

Measuring stations Finite domain W

f

Inversion variable: a(x) Semi-infinite domain W

Absorbing boundary G

(a)

a

(b)

Figure 5.2: (a)Semi-infinite domain; and (b)truncated finite domain using absorbing boundary process gives rise to the finite computational domain Ωf (Fig. 5.2(b)). In this case, it is necessary to specify a boundary condition on Γa that will ensure that the outgoing waves crossing Γa are undisturbed by the presence of this boundary. This boundary condition can be determined in terms of the actual solution u on Γa . We use again local absorbing boundaries, and in particular a first-order approximation, which reads: 1 1 ∂u(x, t) u(x, ˙ t) + κ(x)u(x, t), (x, t) ∈ Γa × (0, T ), = −p ∂n 2 α(x)

(5.81)

in which κ denotes the curvature of Γa . We truncate the semi-infinite domain as a semi-circular domain with radius R, and in this case κ(x) = − R1 , ∀x ∈ Γa . Then (5.81) becomes: 1 ∂u(x, t) ∂u(x, t) 1 u(x, ˙ t) − ≡ = −p u(x, t), (x, t) ∈ Γa × (0, T ). ∂n ∂r 2R α(x) (5.82) 116

Then, following the same process as before, the inverse problem can be cast as follows: Minimize: Ns Z TZ h i2 1X J= u(x, t) − um (x, t) δ(x − xm,j ) dt dΓf 2 j=1 0 Γf Z Z Rα T + α˙ 2 (x, t) dΩf dt, 2 0 Ωf

(5.83)

subject to: ∂ 2 u(x, t) − ∇ · [α(x, t)∇u(x, t)] = 0, ∂t2

(x, t) ∈ Ωf × (0, T ),

u(x, 0) = u(x, ˙ 0) = 0, x ∈ Ωf , ∂u(x, t) = f (x, t), (x, t) ∈ Γf × (0, T ), ∂n ∂u(x, t) 1 1 = −p u(x, ˙ t) − u(x, t), (x, t) ∈ Γa × (0, T ). ∂r 2R α(x, t) α(x, t)

(5.84) (5.85) (5.86) (5.87)

We again recast the above PDE-constrained minimization problem as an unconstrained optimization problem by defining an augmented functional as: M Z Z i2 1X T h u(x, t) − um (x, t) δ(x − xm,j ) dt dΓf A(λ, u, α) = 2 j=1 0 Γf Z Z Rα T + α˙ 2 dΩf dt 2 0 Ωf ¾ ½ 2 Z TZ ∂ u + − ∇ · α∇u dΩf dt λ 2 ∂t 0 Ωf ½ ¾ Z TZ ∂u − f dΓf dt + λ α ∂n f 0 Γ ½ ¾ Z TZ ∂u √ α + λ α + αu˙ + u dΓa dt ∂n 2R a Z0 Γ ∂u(x, 0) + λ(x, 0) dΩf . (5.88) ∂t Ωf 117

We then seek to satisfy the following first-order optimality conditions:    δλ A  δu A = 0. (5.89)   δα A From the first KKT condition, δλ A = 0, we obtain: ¾ ∂2u − ∇ · α∇u dΩf dt δλ 2 ∂t 0 Ωf ½ ¾ Z TZ ∂u δλ α − f dΓf dt ∂n f 0 Γ ½ ¾ Z TZ ∂u √ α δλ α + αu˙ + u dΓa dt ∂n 2R a Z0 Γ ∂u(x, 0) dΩf dt, δλ(x, 0) ∂t Ωf Z TZ

δλ A = + + +

½

(5.90)

and by taking into account the explicitly imposed initial condition u(x, 0) = 0, there results: State problem ∂ 2 u(x, t) − ∇ · [α(x, t)∇u(x, t)] = 0, (x, t) ∈ Ωf × (0, T ), ∂t2

(5.91)

with u(x, 0) = u(x, ˙ 0) = 0, x ∈ Ωf , ∂u(x, t) = f (x, t), (x, t) ∈ Γf × (0, T ), ∂n ∂u(x, t) 1 1 = −p u(x, ˙ t) − u(x, t), (x, t) ∈ Γa × (0, T ). ∂n 2R α(x, t) α(x, t)

(5.92) (5.93) (5.94)

Similarly, from the variation of the augmented functional with respect to the

118

state variable, we obtain: δu A =

M Z TZ X j=1

+ + +

Γf

¾ ∂ 2 δu λ − ∇ · α∇δu dΩf dt 2 ∂t f 0 Ω Z TZ ∂δu λα dΓf dt ∂n 0 Γf ½ ¾ Z TZ ∂δu √ α λ α + αδ u˙ + δu dΓa dt ∂n 2R a Z0 Γ ∂δu(x, 0) λ(x, 0) dΩf dt. ∂t f Ω

Z TZ +

0

h i δu u − um δ(x − xm,j ) dΓf dt ½

(5.95)

By integrating by parts and taking into account the explicitly imposed homogeneous essential boundary and initial conditions, there results: ¾ ½ 2 Z TZ ∂ λ δu A = − ∇ · α∇λ dΩf dt δu 2 ∂t f Z Z0 Ω p ∂δu(x, T ) f λ(x, T ) dΩ + + δu(x, T )λ(x, T ) α(x, T ) dΓa ∂t f Γa ZΩ ∂λ(x, T ) − δu(x, T ) dΩf ∂t f Ω Z TZ + δuα∇λ · n d(Γf ∪ Γa ) dt +

0 Γf ∪Γa M Z TZ X j=1

Z TZ − 0

0

h i δu u − um δ(x − xm,j ) dΓf dt

Γf

∂¡ √ ¢ a λ α dΓ dt + δu ∂t Γa

Z TZ δu 0

Γa

α λ dΓa dt. 2R

(5.96)

Since δu is arbitrary, by setting δu A = 0 the following adjoint problem results: Adjoint problem ∂ 2 λ(x, t) − ∇ · α(x, t)∇λ(x, t) = 0, (x, t) ∈ Ωf × (0, T ), ∂t2 119

(5.97)

˙ λ(x, T ) = λ(x, T ) = 0, x ∈ Ωf ,

(5.98)

M Z TZ h i X ∂λ(x, t) =− u − um δ(x − xm,j ) dΓf dt, α(x, t) ∂n f 0 Γ j=1

with (x, t) ∈ Γf × (0, T ),

α(x, t)

(5.99)

´ ∂λ(x, t) 1 ∂ ³p α(x, t)λ(x, t) − = α(x, t)λ(x, t), ∂n ∂t 2R with (x, t) ∈ Γa × (0, T ).

(5.100)

Finally, the variation of the augmented functional with respect to α results in: Z TZ

Z TZ

δα A = Rα αδ ˙ α˙ dΩ dt − λ∇ · δα∇u dΩ dt 0 Ω 0 Ω Z TZ ∂u + dΓf dt λδα ∂n f 0 Γ ½ ¾ Z TZ δα ∂u δα + λ δα + √ u˙ + u dΓa dt ∂n 2R 2 α a 0 Γ

(5.101)

By integrating by parts and taking into account that δ α(x, ˙ 0) = δα(x, T ) = 0, (5.101) yields: ½

Z TZ

¾

δα − Rα α ¨ + ∇λ · ∇u dΩf dt 0 ¾ ½ Z TZ u˙ u + dΓa dt = 0. δα λ √ + 2R 2 α a 0 Γ

δα A =

Ωf

(5.102)

Equation (5.102) is similar to the one-dimensional control equations (5.29) and (5.76). Therefore, the two-dimensional problem can be solved in a way similar to that described in Algorithms 9 and 10.

120

Chapter 6 Numerical Experiments - MPR

We investigate the efficiency and robustness of the algorithms described in the previous sections via a series of numerical experiments. At first, we start with one-dimensional problems in which several layers overlay the rigid bottom. Through the numerical experiments, we test the performance of the proposed algorithms for both smooth and sharp target profiles. In addition, we study the effect of the regularization factor and the performance against noisy data. We also test the algorithmic performance for different initial estimates. We also report on both the one- and two-dimensional truncated domain cases. For the two-dimensional problems, we examine the performance with various configurations of sources and receivers, different mesh densities, and sources whose frequency content differs. In all cases, we use synthetic data produced in a manner that avoids committing classical inverse crimes. That is, we obtain the measured response via means that are not based on the same numerical scheme (or mesh density) used for the solution of the state problem as part of the inversion process.

121

6.1

On Smooth Profiles

For the first example problem, we consider a simple one-dimensional case with a fixed bottom, in which only the modulus (or wave velocity) is the unknown. We consider l = 1, and thus x ∈ [0, 1]. The target modulus profile is a Gaussian bell-like distribution, given as: "

µ

α(x) = 1 + exp

(x − 0.5)2 − 0.04

¶# ,

(6.1)

and it is depicted in Fig. 6.1. The source excitation is a rapidly-decaying

2.5

α(x)

2.0

1.5

1.0

0.5 0.0

0.2

0.4

0.6

0.8

1.0

x

Figure 6.1: Smooth target profile of α pulse-like signal given by (6.2). Both the signal and its Fourier transform are depicted in Fig. 6.2.

·

¸ (t − 0.1)2 f (t) = exp − . 0.0025

(6.2)

For the given target profile and source excitation, the measured data um (0, t) is synthesized and it is shown in Fig. 6.3. Based on the given measured data, we

122

0.10

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Amplitude

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5

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10

15

20

Frequency (Hz)

(a) Source signal

(b) Frequency content of source signal

Figure 6.2: Source signal and its frequency content

0.15 0.10

m

0.0

Measured data, u (0,t)

Applied force, f(t)

1.0

0.05 0.00 -0.05 -0.10 -0.15 0

1

2

3

4

Time, t

Figure 6.3: Measured data, um (0, t)

123

5

inverted for the modulus profile via the time-dependent regularization scheme (Algorithm 9) and via the Tikhonov regularization (Algorithm 11). We examined the performance of the schemes against: (i) different choices of the regularization factor; (ii) different initial estimates; and (iii) noisy data. The results are reported in the following sections. 6.1.1

Regularization factor effects

We begin by choosing an initial distribution for the modulus that is constant throughout the entire domain, that is, we set the initial guess to α(x) = 1.0. We investigate the effect the magnitude of the regularization factors has on the performance of the two regularization schemes we considered. Figure 6.4 summarizes the α profiles obtained using the time-dependent regularization scheme for three values of Rα = 1.0, 0.5, 0.1, respectively (the results are reported for the same level of misfit error tolerance, set at 10−6 ). As it can be seen from Fig. 6.4, the solution converges to the target profile, regardless of the value of the regularization factor, albeit at higher computational cost associated with higher factor choices (notice that, here, we employ Algorithm 9 in which the line search scheme is not implemented, and, as a result, the regularization factor remains constant throughout the process). Naturally, smaller factor values fail to achieve the penalty intent of the regularization term: for example in this first problem, whereas for Rα = 0.1 satisfactory results were obtained, for Rα = 0.01 we observed divergence and increased misfit errors. Figure 6.5 summarizes the convergence patterns of the misfit error for the

124

2.5

2.5 Target Initial 10000th 100300th

2.0

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(a) Rα = 1.0

(b) Rα = 0.5

2.5 Target Initial 10000th

α(x)

2.0

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1.0

0.5 0.0

0.2

0.4

0.6

0.8

1.0

x

(c) Rα = 0.1

Figure 6.4: Target, initial, and estimated profile of α(x) using the timedependent regularization scheme

125

considered values of the regularization factors, whereas Fig. 6.6 depicts the convergence patterns for the modulus.

1e-1 Rα=1.0

1e-2

Rα=0.5 Rα=0.1

Misfit error

1e-3 1e-4 1e-5 1e-6 1e-7 1e-8 0

20000

40000

60000

80000

100000

Number of iteration

Figure 6.5: Misfit error using the time-dependent regularization scheme Next, we consider a Tikhonov regularization scheme using different values for the regularization factors; the resulting estimated profiles are shown in Fig. 6.7. We continue the iteration process until the misfit error converges. As it can be seen in Fig. 6.7(a), when Rα = 0.01, the solution does not converge to the target profile. However, as smaller values of the regularization factor are adopted, the solution tends closer to the target profile (Fig. 6.7(b), (c)). However, we failed to obtain a solution as close to the target profile as we obtained in the case of the time-dependent regularization, for any value of the regularization factor. Figure 6.8 depicts the convergence patterns for the modulus, whereas Fig. 6.9 depicts the convergence patterns of the misfit error for the Tikhonov regularization scheme.

126

2

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1.5 α (x)

α (x)

2

1

1 1

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x

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0

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(a) Rα = 1.0

(b) Rα = 0.5

2

α (x)

1.5

1 1 0.5 0

2000

0.5 4000

6000

8000

10000

0

x

Iteration

(c) Rα = 0.1

Figure 6.6: Convergence patterns of α(x) using the time-dependent regularization scheme

127

2.5


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α(x)

2.0

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x

(c) Rα = 10−11

Figure 6.7: Target, initial and estimated profile of α(x) using the Tikhonov regularization scheme

128

1.5

1.5 α (x)

2

α (x)

2

1

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1

1 100

200

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0

1

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0.5

100

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x

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(a) Rα = 0.001

(b) Rα = 0.00001

2

α (x)

1.5

1

0.5 0

1 100

200

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400

0

x

Iteration

(c) Rα = 10−11

Figure 6.8: Convergence pattern of estimated profile of α(x) using the Tikhonov regularization scheme

129

1e-1 Rα=0.001 Rα=0.00001

Misfit error

1e-2

Rα=10-11

1e-3

1e-4

1e-5 0

50

100

150

200

250

300

Number of iteration

Figure 6.9: Misfit error using the Tikhonov regularization scheme 6.1.2

Initial estimates effects

Even though in the preceding problem, our initial guess for the modulus distribution is a constant one throughout the domain, it is still close to the modulus values at the ends of the domain. To explore the algorithmic performance when the initial guess, while still constant, is not close to the end values, we seek to reconstruct the profile starting with α(x) = 0.7. We test again using different regularization factors, Rα = 1.0, 0.5, 0.1. We set the tolerance to 10−7 . As it can be seen in Fig. 6.10, the estimated solutions converged to the target one when the time-dependent regularization was used. Similar to the earlier case, the solutions converged to the target one regardless of the values of the regularization factor Rα . However, the effect of the regularization factor on the rate of convergence remains, as shown in Fig. 6.11. Next, we used the Tikhonov regularization scheme, exploring various

130

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(a) Rα = 1.0

(b) Rα = 0.5


α(x)

2.0

1.5

1.0

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0.2

0.4

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0.8

1.0

x

(c) Rα = 0.1

Figure 6.10: Target, initial and estimated profile of α(x) using the timedependent regularization scheme

131

1e-1 Rα=1.0 Rα=0.5

1e-2

Rα=0.1 Misfit error

1e-3 1e-4 1e-5 1e-6 1e-7 0

10000

20000

30000

40000

50000

60000

70000

Number of iteration

Figure 6.11: Misfit error using the time-dependent regularization scheme values for the regularization factor, in an attempt to reconstruct the profile when the initial guess was again set at α(x) = 0.7. By contrast to the case where the initial guess was closer to the target profile, here all attempts failed to converge as depicted in Fig. 6.12. The reason of the failure is that the regularization prevents the estimates from rapidly changing from the initial guess to estimates closer to the target profile.

132

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Target Initial 27th

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133

6.1.3

Noise effects

One of the practical difficulties arising in inversion is associated with the presence of noise in the measured response (and/or on the source) that typically result in contaminated data. It is thus of interest to study the performance of the algorithms in the presence of noisy data. The problem parameters are the same as those of the first test case, that is a smooth target profile with an initial guess at α(x) = 1. However, now noise-contaminated data are used. To artificially inject noise in the measured data, we use Gaussian noise having standard deviation of 1%, 5%, and 10% with respect to the maximum amplitude of the measured data. The noise-contaminated data are shown in Fig. 6.13. The reconstructed profiles obtained using the time-dependent regularization scheme are shown in Fig. 6.14. For all cases, Rα = 0.1 is employed. As can be seen in the figure, the solution converged satisfactorily to the target one in all cases. However, as the level of the noise in the signal increases, the estimations show higher level of fluctuations due to the effects of the noise. If the regularization factor increases, similar results to the above can be obtained, albeit at a slower convergence rate: figure 6.15 summarizes the convergence of the misfit error for the data containing different noise levels. Next, we consider again a Tikhonov scheme and study the performance for a single case with 10% Gaussian noise. We adopt different values of the regularization factor. The inverted profiles are shown in Fig. 6.16. Figure 6.16(a) depicts the reconstructed profile when no regularization is used (Rα = 0); inter134

0.10

0.10 Measured data, u (0,t)

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(a) 1%-Gaussian noise

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m

0


m


0.15

0.05 0.00 -0.05 -0.10 -0.15 0

1

2

3

4

5

Time, t

(c) 10%-Gaussian noise

Figure 6.13: Measured data contaminated by noise

135

4

5

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α(x)

α(x)

2.0


1.5

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α(x)

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x


Figure 6.14: Target, initial and estimated profile of α(x) in case of timedependent regularization scheme

136

1e-1 1% Gaussian noise 5% Gaussian noise 10% Gaussian noise

Misfit error

1e-2

1e-3

1e-4

1e-5

1e-6 0

2000

4000

6000

8000

10000

Number of iteration

Figure 6.15: Misfit error using the time-dependent regularization scheme estingly, despite the mild oscillations, the profile is competitive to the one produced by the time-dependent regularization (Fig. 6.14(c)). With Rα = 10−5 , the reconstructed profile becomes smoother, albeit somewhat deviating from the target (Fig. 6.16(b)) at the neighborhood of the peak. As the regularization factor increases, further deviation (or failure) is observed (e.g. Fig. 6.16(c)), since the Tikhonov regularization begins to weigh heavily on the inversion process.

137

2.5


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α(x)

α(x)

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2.5 Target Initial 153rd

α(x)

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(c) 10%-Gaussian noise, Rα = 0.001


138

6.2

On Sharp Profiles

In the next series of numerical experiments we study the performance of the algorithms on sharply changing profiles. We start again by considering α(x) as the sole profile to invert for, and adopt the following step form for it:  0.0 ≤ x < 0.3  1.0 2.0 0.3 ≤ x < 0.7 . (6.3) α(x) =  1.0 0.7 ≤ x < 1.0 We choose α(x) = 1.0 as the initial guess and reconstruct the profile using both a time-dependent and a Tikhonov regularization scheme. We also investigate the performance of each scheme with noise-contaminated data. For each scheme, we tested the performance using 0%, 1%, 5% and 10% Gaussian noise. In Fig. 6.17, the measured noise-contaminated data are shown. The estimated profiles of α(x) using the time-dependent regularization scheme are shown together with the target and initial profiles in Fig. 6.18. In particular, Figs. 6.18(a),(b), and (c) depict the reconstructed profiles for different regularization factors and noise-free data. In all cases the results appear satisfactory; however, depending on the choice of the regularization factor, the convergence rate changes. Figures 6.18(d), (e), and (f) depict the reconstructed profiles obtained using the noise-contaminated data; here we chose to report results only for Rα = 0.1. The results are deemed satisfactory in all cases. The estimated profiles of α(x) using the Tikhonov regularization scheme are shown together with the target and initial profiles in Fig. 6.19. By con-

139

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0.15 0.10

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m


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Figure 6.17: Measured data contaminated by different levels of noise

140

5

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141

trast to the time-dependent regularization scheme, we observe here that the solutions depend on the choice of the regularization factor.

2.5


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Figure 6.19: Target, initial and estimated profile of α(x) using the Tikhonov regularization scheme Next, we seek to reconstruct the same profile starting with α(x) = 0.7 to explore the algorithmic performance when the initial guess is not close to the end value. We again invert the target profile using both a time-dependent and a Tikhonov regularization scheme and, for each scheme, we tested the performance using data contaminated with 0%, 1%, 5% and 10% Gaussian 142

noise. The estimated profiles of α(x) using the time-dependent regularization scheme are shown together with the target and initial profiles in Fig. 6.20. Similarly to the previous case, it is observed that the profiles converge to the target profile in all cases, but the convergence rate depends on the magnitude of the regularization factor (Fig. 6.20(a), (b), (c)). In addition, the reconstructed profiles using noisy data are depicted in Fig. 6.20(d), (e), and (f); here we chose to report results only for Rα = 0.2. The results are again deemed satisfactory in all cases. By contrast, Fig. 6.21 depicts attempts to reconstruct the profile using the Tikhonov scheme, which failed in all cases. Theses results show that the Tikhonov scheme is inappropriate for inexact initial estimates.

143

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0.6 x

(a) 0%-Gaussian noise, Rα =1.0

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144

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Target Initial 28th

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(c) 0%-Gaussian noise, Rα =10−6


6.3

Simultaneous Inversion of Modulus and Damping

Next, we report on the performance of the algorithm using the time-dependent regularization in the case of two inversion profiles, the modulus α and the attenuation metric β. For the modulus we choose a sharp profile exhibiting multiple layers of different moduli, whereas for the damping we use a smooth linear-varying profile. The target profiles are:  1.5 for 0.0 ≤ x < 0.25    2.0 for 0.25 ≤ x < 0.5 α(x) = , 1.0 for 0.5 ≤ x < 0.75    2.5 for 0.75 ≤ x < 1.0 β(x) = 1.0 − 0.5x.

(6.4)

(6.5)

For the initial guesses, constant α(x) = 1.2 and β(x) = 0.5 are chosen, and the profiles are reconstructed using both noise-free data, and data contaminated with 10% Gaussian noise. The data for each case are shown in Fig. 6.22; the

145

regularization factors were set at Rα = 0.1 and Rβ = 0.1 using the timedependent scheme.

0.10

0.08 0.06 Measured data, u (0,t)

0.05 m

0.02

m


0.04

0.00 -0.02 -0.04 -0.06 -0.08

0.00

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-0.10 -0.12

-0.15 0

1

2

3

4

5

0

1

Time, t

2

3

4

5

Time, t



Figure 6.22: Measured data The estimated profiles are shown together with the target and initial profiles in Fig. 6.23. The performance as exhibited in these figures is quite satisfactory; notice that the described process allows, in essence, the recovery of the number of layers, the material composition of each layer, and the thickness (or depth) of each layer, and without having to explicitly declare them as model parameters.

146

3.0

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β(x)

α(x)

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(a) α(x) for 0%-Gaussian noise

(b) β(x) for 0%-Gaussian noise

3.0

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α(x)

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1.5 0.50

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x

(c) α(x) for 10%-Gaussian noise

(d) β(x) for 10%-Gaussian noise

Figure 6.23: Target, initial and estimated profiles of α(x) and β(x) using the time-dependent regularization scheme

147

6.4


In the previous chapter, we also formulated the one-dimensional case in which the layered soil structure overlays a homogeneous halfspace. Next, we study the performance of the algorithms in this case. We consider the problem of inverting for the modulus having a sharp target profile defined as follows:  1.0 0.00 ≤ x < 0.25     0.25 ≤ x < 0.50  2.0 1.0 0.50 ≤ x < 0.75 . α(x) = (6.6)   2.5 0.75 ≤ x < 1.00    3.0 1.00 ≤ x We truncate the domain at x = 1 and impose the absorbing boundary condition given by (5.59) (Fig. 6.24). For the initial estimate, we choose a constant modulus, that is, we set the initial guess to α(x) = 1.2 and the target profile and the initial estimate are depicted in Fig. 6.25. We synthesize the measured data for the source described in Fig. 6.2 and invert for the modulus profile using both noise-free and noisy data. For the noisy data, we add Gaussian noise having a standard deviation of 10% with respect to the maximum amplitude of the measured data. The noise-free and noisy data are depicted in Fig. 6.26.

We invert for the modulus profile using a modified version of Algorithm 9 that accommodates the absorbing boundary condition. The results shown in Fig. 6.27 exhibit quite satisfactory performance for both the noise-free data and noisy data cases.

148

f(t) (symbolic) Station 0.25

a=1.5 x

a=2.0

0.25

a=1.0

0.25

a=2.5

0.25 Truncated at x=1 and ABC is imposed

a=3.0

Semi-infinite domain

Figure 6.24: Problem configuration: layered medium with truncation boundary

3.5 Target Initial

3.0

α(x)

2.5

Truncated at x=1 and ABC is imposed

2.0 1.5 1.0 0.5 0.0

0.5

1.0

1.5

2.0

x

Figure 6.25: Target and initial profile of α(x)

149

0.02

0.00


-0.02

-0.02

m

m


0.02

-0.04 -0.06 -0.08

-0.04 -0.06 -0.08 -0.10

-0.10

-0.12 0

1

2

3

4

5

0

1

Time, t

2

3

4

5

Time, t



Figure 6.26: Measured data

3.5


3.0 2.5

2.5 ABC

α(x)

α(x)


3.0

2.0

ABC 2.0

1.5

1.5

1.0

1.0

0.5

0.5 0.0

0.5

1.0

1.5

2.0

x

0.0

0.5

1.0

1.5

2.0

x




150

6.5


In the previous sections, we examined the performance of the proposed algorithms for one-dimensional problems through numerical experiments. Next, we apply the algorithms to the reconstruction of the modulus profile of a two-dimensional semi-infinite domain. Here, we consider two different target profiles: first, a homogeneous host containing an inclusion of different modulus, and, second, a layered medium also containing an inclusion. We solve both problems with varying number and location of sources and receivers, various mesh densities, and with sources having varying frequency content. In addition, we examine the performance against noisy data. For all cases that follow, we employ Algorithm 10, that is, the time-dependent regularization combined with a line search scheme. With the line search scheme, the regularization factor is modified at every iteration and, as a result, the convergence rate can be improved. 6.5.1

Embedded inclusion in homogeneous medium

To test the performance of the proposed algorithm for two-dimensional case, first, we consider the problem of detecting a homogeneous inclusion embedded in an also homogeneous semi-infinite medium. As discussed in Chapter 5, we truncate the domain via the introduction of a semi-circular boundary and impose an absorbing boundary condition on the truncation interface. The truncated finite domain and the target profile are depicted in Fig. 6.28. To examine the effects of the number and location of the sources and receivers,

151

R=0.03 0.0075

0.0075 a=3.0 0.00375 a=1.5

Truncated absorbing boundary unit: km, sec

Figure 6.28: Target profile: an inclusion embedded in homogeneous medium we invert for the target profile based on different configurations of sources and receivers; these are depicted in Fig. 6.29. In addition, we solve the problem using different mesh densities, and source signals containing different frequency content. The mesh discretizations employed are depicted in Fig. 6.30, and the source signals are shown in Fig. 6.31 together with their Fourier transforms. For every case, we start with an initially homogeneous profile, that is, with α(x) = 1.5 for the entire domain. First, we solve the problem using different number and locations of sources and receivers. We use the dense mesh shown in Fig. 6.30(d), and Source 1, as shown in Fig. 6.31(a), is applied at each source point. In Fig. 6.32, the convergence pattern in the case of 119 sources and receivers (Fig. 6.29 (a)) is shown; as it can be seen the estimates converge to the target profile quite satisfactorily. The final results obtained from the various configurations of sources

152

[email protected]

[email protected]

(a) 119 sources and receivers

(b) 16 sources and receivers

[email protected]

[email protected]

(c) 11 sources and receivers

(d) 7 sources and receivers

[email protected]

[email protected]

(e) 5 sources and receivers

(f) 3 sources and 5 receivers

Figure 6.29: Configurations of sources and receivers: the locations of sources and receivers are marked with H and , respectively

153

(a) 1350 elements

(b) 2400 elements

(c) 3750 elements

(d) 5400 elements

Figure 6.30: Finite element discretizations

154

0.020

1.2

0.016

0.8

Amplitude

Applied force, f(t)

1.0

0.6

0.012

0.008

0.4

0.004

0.2

0.000

0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

0

50

100

(a) Source 1

250

300

0.020

1.0

0.016

0.8

Amplitude

Applied force, f(t)

200

(b) Frequency components of Source 1

1.2

0.6

0.012

0.008

0.4

0.004

0.2

0.000

0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

0

50

100

150

200

250

300

Frequency (Hz)

Time, t

(c) Source 2

(d) Frequency components of Source 2

0.020

1.2 1.0

0.016

0.8

Amplitude

Applied force, f(t)

150 Frequency (Hz)

Time, t

0.6

0.012

0.008

0.4

0.004

0.2 0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

0.000 0

50

100

150

200

250

300

Frequency (Hz)

Time, t

(e) Source 3

(f) Frequency components of Source 3

Figure 6.31: Source signals and their frequency components

155

and receivers are shown in Fig. 6.33. In every case, the inclusion is detected. However, we observe that the accuracy of the estimated material property becomes worse as less sources and receivers are used. In addition, fewer sources and receivers require more iterations to render a converged solution. Then, we examine the effect the frequency content of the source signal has on the inversion process. To this end, we solve the same problem using different sources, as shown in Fig. 6.31. As it can be seen, Source 3 contains frequency components ranging from 0 to 50Hz, while Source 2 and Source 1 contain higher frequency components, up to 100Hz and 150Hz, respectively. For each case, we use the configuration of the sources and receivers shown in Fig. 6.29(c) in which 11 sources and receivers are used. For the finite element discretization, we use the mesh shown in Fig. 6.30(d). The results obtained using the different source signals are depicted in Fig. 6.34. From Fig. 6.34, it can be seen that the source containing the higher frequency components yields more accurate estimates (Fig. 6.34(c)), compared with the results obtained using the source signal containing lower frequency components (Fig. 6.34(a)). Next, we examine the effect of the mesh density by solving the same problem using different meshes as shown in Fig. 6.30. For each case, we employ Source 1 (Fig. 6.31(a)), and the source-receiver configuration shown in Fig. 6.29(c). The obtained results are shown in Fig. 6.35. As shown in Fig. 6.35, satisfactory results are obtained in all cases.

156

(a) Initial estimates

(b) 1st iteration

(c) 50th iteration

(d) 100th iteration

(e) 150th iteration

(f) 225th iteration

Figure 6.32: Results using 119 sources and receivers 157

(a) 121 sources and receivers: 225 iterations

(b) 16 sources and receivers: 232 iterations

(c) 11 sources and receivers: 508 iterations

(d) 7 sources and receivers: 1215 iterations

(e) 5 sources and receivers: 2177 iterations

(f) 3 sources and 5 receivers: 863 iterations

Figure 6.33: Results using different configurations of sources and receivers

158

(a) Source 3: 67 iterations

(b) Source 2: 644 iterations

(c) Source 1: 508 iterations

Figure 6.34: Results using sources with different frequency content

159

(a) 1350 elements: 174 iterations

(b) 2400 elements: 254 iterations

(c) 3750 elements: 402 iterations

(d) 5400 elements: 508 iterations

Figure 6.35: Results using different mesh densities

160

6.5.2

Embedded inclusion in layered medium

We consider next the problem of identifying simultaneously both the layered structure as well as an embedded inclusion within a semi-infinite domain. Figure 6.36 depicts the problem configuration and the target profile. R=0.03 0.0075

0.0075 a=1.5

0.01

a=3.0 0.00375

a=2.0

Truncated absorbing boundary unit: km, sec

Figure 6.36: Target profile: an inclusion embedded in a layered medium We solve this problem using both noise-free and noisy data and, for every case, we start with an initially homogeneous profile, that is, with α(x) = 1.5 for the entire domain. For the noisy data, we synthesize the data using a source signal containing Gaussian noise whose standard deviation is 10% of the maximum amplitude of the source. The sources used to synthesize the noise-free and noisy data are depicted in Figs. 6.37(a) and (c), respectively, and the corresponding frequency content is shown in Figs. 6.37 (b) and (d). The noisy source shown in Fig. 6.37(c) is for one location only; due to the randomness of the noise generation process, there is different noise associated 161

0.006

1.0

0.005

0.8

0.004 Amplitude

Applied force, f(t)

1.2

0.6

0.003

0.4

0.002

0.2

0.001

0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

0.000 0

100

200

300

Time, t

500

600

700

800

900 1000

Frequency (Hz)

(a) Noise-free source

(b) Frequency components of noise-free source

1.2

0.006

1.0

0.005

0.8

0.004 Amplitude

Applied force, f(t)

400

0.6

0.003

0.4

0.002

0.2

0.001

0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

0.000

Time, t

0

100

200

300

400

500

600

700

800

900 1000

Frequency (Hz)

(c) Noisy source

(d) Frequency components of noisy source

Figure 6.37: Sources and their frequency components

162

with the various source points, albeit at the same noise level. A total of 19 sources and receivers are used and the configuration of the sources and receivers is depicted in Fig. 6.38. [email protected]

Figure 6.38: Configurations of sources and receivers: the locations of sources and receivers are marked with H and , respectively In Fig. 6.39, the results for both noise-free data and noisy data cases are shown. As it can be seen in Fig. 6.39, the solutions converge to the target profile quite satisfactorily for both cases even though the case of noisy data yields less accurate results.

163

(a) Noise-free data: 1006 iterations

(b) Noisy data: 131 iterations

Figure 6.39: Results using sources containing different frequency contents

164

6.6

Summary of MPR Numerical Experiments

We examined the performance of various algorithms for the reconstruction of material profiles via numerical experiments. We used a time-dependent regularization approach and applied it to both one-dimensional and two-dimensional problems. First, we considered one-dimensional problems with a fixed boundary and solved them using both a time-dependent regularization scheme and a Tikhonov regularization scheme. For both of these schemes, we tested the performance for smooth and sharp profiles. In addition, we examined the effect of the regularization factor and the algorithmic performance against noisy data. We also tested the algorithmic performance for different initial estimates. Our observations: • The proposed scheme can capture both sharp and smooth profiles while the Tikhonov scheme exhibits difficulties when confronted with sharp profiles. • The solutions obtained using the time-dependent scheme appear insensitive to the regularization factor, which is typically a source of difficulties in inversion algorithms. In addition, by using a line search scheme in the optimization process, the regularization factor can be modified to approach an optimal value. • The proposed scheme appears more robust under inexact initial guesses when compared to Tikhonov-type schemes.

165

We also considered the more realistic cases of one- and two-dimensional truncated domains. Through the numerical experiments, it is observed that the proposed scheme works well even in the presence of local absorbing boundary conditions, that typically add noise to the solution (even for forward problems). In addition, via the numerical experiments for two-dimensional problems, it is observed that the accuracy of the solution depends, as expected, on the frequency content of the source as well as on the number and location of sources and receivers.

166

Chapter 7 Conclusions 7.1

Summary

In this thesis we presented a general framework for solving wavefield-based inverse problems using a governing-equation-constrained optimization approach. We have applied the developed methodology to two classical inverse problems: the shape detection and localization problem, and the material profile reconstruction problem. The discussed framework provides a systematic way for treating inverse problems, while allowing the ready incorporation of different misfit norms and regularization schemes. In the shape detection and localization problem, we used integral equations and boundary elements to realize the modeling framework. In order to account for the evolution of the boundary shapes during the iterative detection process, we used the apparatus of total derivatives that allowed in an elegant manner the computation of the misfit gradients, without resorting to the computationally intensive and failure-prone updates based on finite difference schemes. To alleviate the difficulties associated with solution multiplicity and improve on the chances of the optimizer to capture the global

167

optimum, and thus resolve the location and shape of the scatterer, we also adopted an amplitude-based misfit functional, and, more importantly, introduced frequency- and directionality-continuation schemes. Through the numerical experiments, it appears that the above combination results in robust algorithmic behavior. This is so, especially in view of the fact that all numerical experiments were conducted with data collected solely in the backscattered region of the sought scatterer. In the material profile reconstruction problem, we also followed the framework to invert for the profile of one- and two-dimensional finite and semi-infinite domains. Here, we employed a time-dependent regularization scheme and examined the performance of the proposed schemes via numerical experiments. From the experiments, it is observed that the time-dependent regularization scheme is superior to classical Tikhonov-type schemes. The time-dependent regularization scheme works satisfactorily for both sharp and smooth profiles; in addition the solution appears insensitive to the choice of the regularization factor, which is often a source of difficulties in regularization schemes for inverse problems. It was also observed that the time-dependent regularization scheme works better for inexact initial guess when compared to Tikhonov-type schemes. In addition, when the time-dependent regularization is coupled with a line search scheme, the regularization factor can be modified at every iteration to approach an optimal value. Finally, we also examined the performance of the proposed algorithm against noisy data via the numerical experiments and satisfactory results were again obtained.

168

7.2

Pointers to Future Research

The following constitute possible extensions of the present work: • A straightforward extension of this work is the more realistic threedimensional half-space case for both the SDL and MPR problems using, however, elastodynamic wavefields. • For the shape detection and localization problem, we considered an inclusion embedded in homogeneous medium using an integral-equationconstrained optimization approach. One possible extension of this work is to considering multiple inclusions and a layered host medium, for which the Green’s functions are readily available. • For the material property reconstruction problem, forward models that take into account more realistic soil attenuation models are highly desirable. The discussed framework can readily accommodate such models, provided that parallel advances are realized in the area of absorbing boundary conditions for such viscoelastic models. • In the shape detection and localization problem we improved the convergence rate as well as the solution exactness by using the frequencycontinuation scheme. Also, through the numerical experiments for the two-dimensional material profile reconstruction problem, we observed that the solution exactness and convergence rate depend on the frequency content of the source. Based on these observations, a scheme 169

sharing the same concept with the frequency-continuation scheme can be tried out in the material profile reconstruction problem.

170

Appendix

171

On the proof of (3.22) We wish to prove that: ·

D ∂f Dξ ∂n

¸ ξ=0

µ ∗ ¶ ∗ ∗ ∂f ∂f ∂vn = . =(∇f ) ·n = ∇ f ·n − ∂n ∂n ∂n

(1)

Proof: For any scalar field a, the following gradient decomposition holds: ∇a = ∇s a +

∂a n, ∂n

(2)

where ∇s denotes surface gradient operator, and n is the normal vector. Then, for any two scalar fields a and b, there holds: ¶ µ ¶ µ ∂b ∂a n · ∇s b + n ∇a · ∇b = ∇s + ∂n ∂n ∂a ∂b = ∇s a · ∇s b + , ∂n ∂n

(3)

since ∇s · n = 0.

(4)

The following also holds: ∗

n= −∇s vn . Then using the definition (1), and relations (2)-(4), it follows that: · ¸ ¡ ∗ ¢ ¢ ¢ ¡ ¢ ∂¡ D¡ = ∇f = ∇f ∇f + vn ∇ ∇f n Dξ ∂ξ ξ=0 µ ¶ ¡ ¢ ∂f = ∇ + vn ∇ ∇f n ∂ξ ¡ ¢ = ∇f˙ + vn ∇ ∇f n. 172

(5)

(6)

Also ∗ ¢ ¡ f = f˙ + vn ∇f · n ,

(7)

∗ ¡ ¢ f˙ =f −vn ∇f · n .

(8)

and

From (5) and (7), there results: · ¸ ∗ ¡ ∗ ¢ ¡ ¢ ¡ ¢ ∇f = ∇ f −vn ∇f · n + vn ∇ ∇f n · ¸ ∗ ¡ ¢ ¡ ¢ = ∇ f −∇ vn ∇f · n + vn ∇ ∇f n ∗ ¢ ¡ ¢ ¡ ¢ ¡ = ∇ f −∇vn ∇f · n − vn ∇ ∇f · n + vn ∇ ∇f n ¸ · µ ¶ ∗ ¡ ¢ ∂f ∂f − ∇ ∇f n . = ∇ f − ∇vn − vn ∇ ∂n ∂n

It is also true that: µ ¶ · ¸ ¡ ¢ ¡ ¢T ¡ ¢ T ∂f ∇ = ∇ ∇f · n = ∇n ∇f + ∇ ∇f n ∂n · ¸ ¡ ¢T ¡ ¢ = ∇n ∇f + ∇ ∇f n, or,

µ ¶ ¡ ¢ ¡ ¢T ∂f ∇ − ∇ ∇f n = ∇n ∇f, ∂n

(9)

(10)

(11)

and thus, there results: ∗ ¡ ∗ ¢ ¡ ¢T ∂f ∇f = ∇ f − ∇vn − vn ∇n ∇f, ∂n ¸ · ∗ ∗ ¡ ¢T ¡ ¢ ∂f ∇vn · n − vn ∇n ∇f · n ∇f ·n = ∇ f ·n − ∂n

There also holds:

h¡

∇n

¢T

i ¡ ¢ ∇f · n = ∇f · ∇n n, 173

(12) (13)

(14)

but ¡

¢ ∇n n = 0,

(15)

and therefore: ¡

∗ ¢ ∗ ∂f ∂vn ∇f ·n = ∇ f ·n − . ∂n ∂n

174

(16)

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Vita

Seong-Won Na was born in Seoul, Korea on February 16, 1971, the son of UngSang Na and Young-Hee Lee. After graduating from Dankook High School in February 1989, he entered Yonsei University in Seoul, Korea that same spring. In February 1993, he received a Bachelor of Science degree in Civil Engineering. Soon after graduation, he continued his studies there and received the Master of Science degree with the Best Thesis Award in February 1995. From March 1995 he started working in Research Institute of Hyundai Heavy Industries Co., Ltd. as a research engineer and he studied earthquake engineering and computer-aided engineering until May 2000. In August of 2000, he joined in the doctoral program in Civil Engineering Department at the University of Texas at Austin, where he has been working as a research assistant.

Permanent address: 101-1004 Life Apt. Mapyung-Dong, Yongin-Shi, Kyunggi-Do Republic of Korea

This dissertation was typeset with LATEX† by the author. † A LT

EX is a document preparation system developed by Leslie Lamport as a special version of Donald Knuth’s TEX Program.

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