A Review of Parametric Model Order Reduction Techniques Mohammad Abid Bazaz, Mashuq-un-Nabi and S. Janardhanan
Abstract— Parametric Model Order Reduction has been recognized as one of the outstanding issues in reduced order modeling paradigms. Recent years have seen a spurt in research activity related to this area and a number of problems have opened up. This paper presents a critical review of the strategies for parameter dependency preservation in reduced order models, which have been proposed over the years, and attempts to contextualize the problem in proper physical and numerical perspective. The open problems in the area are also identified, along with a discussion on some potential application areas.
I. INTRODUCTION
T
HE importance of numerical simulations has steadily increased across all scientific and engineering disciplines. In many engineering domains, real time numerical capabilities are desired to predict the behavior of complex systems. Modeling of complex systems typically leads to a high order differential equation. Prediction and optimization of large scale systems thus requires the numerical solution of these high order differential equations, which is a computationally costly and time consuming procedure. In order to perform rapid design prediction and optimization, it becomes imperative to build accurate and easy to use reduced order dynamical models for these systems. A sufficiently accurate reduced order model is a lower dimensional computational model which can faithfully reproduce the essential features of a higher dimensional model at a fraction of the computational cost. The art and science of deriving cost effective reduced order model for large scale systems is known as Model Order Reduction (MOR). MOR methods are usually projection based and seek to replace a large scale system by a considerably smaller one while preserving the input-output dynamics. A reduced order model is often constructed by projecting the higher dimensional counterpart onto a lower dimensional subspace in which the dynamics of the large scale system predominantly evolves. This is possible in many large scale systems on account of poor controllability and observability [1]. However, while reducing large scale systems, it has to be made sure that the procedure used for such reduction be computationally stable and efficient. Mohammad Abid Bazaz is with the Electrical Engineering Department, NIT Srinagar, India presently on deputation to IIT Delhi for his Ph.D. (Email:
[email protected]). Mashuq-un-Nabi is with the Electrical Engineering Department, IIT Delhi, India. (E-mail:
[email protected]). S. Janardhanan is with the Electrical Engineering Department, IIT Delhi, India. (E-mail:
[email protected]).
Furthermore, it is desirable that important system theoretic properties like stability, passivity, reciprocity etc. be preserved while performing these reductions. The existence of an apriori error bound is a desirable feature which can be used to design robust controllers for large scale systems, using their reduced order counterparts while treating the modeling error as an uncertainty. Some of the application areas in which MOR strategies have been used include VLSI simulations [2], weather predictions [3], molecular dynamics simulations [4], Micro electro-mechanical systems (MEMS) [5], structural mechanics [6], power systems [7], transient electromagnetic simulations [8], fluid dynamics [9], optimal control of large scale systems [10] etc. Engineering systems are, however, always parameterized to allow variations in shape, material, loading, boundary and initial conditions during their design and analysis. Standard MOR techniques are not robust with respect to parametric variations and a new reduced model needs to be generated each time a parameter is varied in the system under study. This severely limits their applicability for the design flow and system level simulation. As such, it becomes imperative to develop the strategies for reduction of parametric systems which can somehow reflect the parameter dependant behavior in the reduced order systems. Parametric Model Order Reduction (PMOR) methods precisely strive to achieve this goal and have assumed much significance of late. These methods seek to generate a reduced order dynamical system which retains a functional dependence on important design parameters. Expeditious optimization and design space exploration cycles can thus be achieved by carrying out the parametric simulations in the reduced space. PMOR is at an early stage of development. It has been recognized as one of the outstanding issues in model order reduction [11]. In recent years, there has been a spurt in research activity related to the area of PMOR. This paper intends to present a holistic view of the various PMOR strategies that have been proposed from time to time. In this work, we present a critical review of the literature related to PMOR and attempt to contextualize the various aspects of the problem in proper physical and numerical perspective. The next section gives a mathematical formulation of the problem. In section III, the various strategies that have been used for PMOR are discussed and critically reviewed for their pluses and minuses. Section IV discusses two motivating potential application areas for PMOR, so as to
get a better grasp of the problem. In Section V, some open problems in existing PMOR strategies are identified which need immediate attention of the researchers for improved viability. The concluding section summarizes the paper. II. MATHEMATICAL FORMULATION OF THE PROBLEM Consider a Multi-Input Multi-Output (MIMO) state space system in descriptor form: (1) (2) , , . where , , , are the state, input and output vectors and p is a vector of parameters. The system matrices are dependent upon the parameter vector as is evident in the state space formulation. It needs to be emphasized here that the parameter dependency in the system matrices may be explicit or implicit. The idea behind PMOR is to have a reduced model of the form: (3) (4) , where , , , . Parametric MOR implies that it should be possible to extract the parametric dependency of the original system from the reduced order parametric model. Thus the main goal of parametric model order reduction is to find a reduced model that preserves the parameter dependency, allowing the variation of any of the parameters without the need to repeat the reduction step. Thereby, the reduction method should ideally be able to cope with any number of parameters and with systems where no analytical expression of the parameter dependency in the matrices is available. In addition, it should be numerically efficient to be suitable for the reduction of large scale systems, and at the same time, its computational cost should be low enough to keep the reduction step numerically justified. III. EXISTING PMOR STRATEGIES: A CRITICAL REVIEW Having mathematically stated the problem with its idealistic requirements, this section gives a critical review of various PMOR strategies that have been proposed over the years: 1. Multi-dimensional Moment Matching: Moments of the transfer function are the coefficients of its Taylor series expansion. Some of the computationally efficient methods for reduction of large scale systems are based on deriving a reduced order transfer function which has the same moments as that of the original system transfer function. Owing to their simplicity and efficiency, these algorithms have gained a well deserved reputation in nominal MOR. However, nominal moment matching algorithms all preserve the system moments only with respect to frequency. Multi-dimensional moment matching algorithms
are an extension to these methods to parametric systems. They are usually based on the explicit or implicit matching of the coefficients of the Taylor series expansion of the parametric transfer function. These coefficients depend not only on the frequency but also on the set of parameters affecting the system, and thus are denoted as multidimensional moments. The extension of moment matching algorithms to parametric linear systems was first reported in [12][13], where the moment matching algorithm was generalized to a parametric system with the system matrix linearly depending on one parameter. This work was generalized in [14] to the multi-parameter case. It was shown as to how suitable Krylov subspaces can be generated to guarantee matching of the moments with respect to frequency and all the independent design parameters. However, as the number of parameters increase, the order of the reduced system increases rapidly, making the scheme rather infeasible for systems with large number of parameters. In addition, it turns out that it is often difficult in practice to generate parametric models with an analytically expressed parameter dependency, which is a pre-requisite for the application of these methods.. Multi-parameter moment matching has also been used in [15][16] [17][18][19][20][21]. The basic premise on which these methods are based is similar, with the differences being in the way the moments are computed (implicitly or explicitly) and whether mixed moments have been matched or not. Approaches based on explicitly computed moments suffer from numerical instabilities as the moment vectors become linearly dependant only after a few iterations. However, implicit approaches appear to provide a robust realization of these difficulties, at least for low dimensional parameter spaces. The salient features of multi-dimensional moment matching algorithms can thus be stated as: a) Moments are matched with respect to frequency as well as the parameters. b) These algorithms can be used only if the parametric dependency can be analytically expressed in the original large scale model. c) As the number of parameters increase, the order of the reduced system blows up exponentially, as such this method is not suitable for systems with large number of parameters. d) Implicit moment matching approaches are numerically more robust as compared to explicit approaches. 2. Aggregate Projection Matrix Approach: This is another well known technique for the parametric reduction of linear systems [22][23]. This method consists of calculating local projection matrices from several local models in the parametric space, merging these matrices together and then applying a common order reducing
projection to the original parametric model. The common projection matrix is obtained by applying some rank revealing algorithm like SVD or QR decomposition to the aggregate matrix. The dominant directions in the aggregate matrix are identified through these rank revealing algorithms and the less contributing vectors are thrown away. The main advantage of this method is simple and direct way of calculating the projection matrix. However, in order to result in a parametric reduced order model, the parametric dependency needs to be affine. Moreover, the order of the reduced system tends to become very large once many local models are considered and no moment matching can be guaranteed for the reduced models. Furthermore, the discretization scheme to be used for sampling the parametric space is an open problem that needs to be addressed. The salient features of aggregate projection matrix approach can thus be stated as: a) Projection matrices can be calculated in a simple and direct manner. b) Parametric space needs to be explicitly discretized to calculate the projection matrix. c) No moment matching can be guaranteed for the reduced models. d) This method may give large sized models for fine discretizations of the parametric space. e) Parametric dependency in the original system must be at least affine in order to obtain parametric reduced order models. f) Due care must be exercised in the selection of sampling schemes for discretizing parametric space to better capture the parametric dependency. 3. Matrix Interpolatory Framework: This method has been recently suggested for the parametric reduction of linear dynamical systems [24]. In this method, the original model is generated and independently reduced at k different parameter values with individual projection matrices. The reduced parametric model is then obtained by a weighted interpolation between the matrices of these k reduced models. Prior to that, suitable transformations have to be applied to make the interpolation physically meaningful. No affine or analytical parametric dependency is needed for the implementation of this method. This significantly simplifies the modeling process in many practical cases where it is often impossible to assume or obtain an affine or analytical dependency. The order of the reduced model is comparable to that obtained in nonparametric reductions and does not blow up with the increase in the number of local models. This allows increasing the number of local models in order to better capture the generally unknown parameter dependency without increasing the complexity of resulting reduced
model. However, as far as this approach is concerned, there are still a number of open questions like the choice of local large scale systems in the parametric space, the choice of the weighting functions, error estimates to judge the quality of the reduced model, and the stability of the generated parametric reduced order model. In [25], the issue of stability has been addressed to some extent. Based on the matrix measure approach [26], sufficient conditions on the original system matrices are derived. Once they are fulfilled, the stability of each of the reduced models is guaranteed as well as that of the parametric model resulting from interpolation. In addition, it is shown that these sufficient conditions are met by some classes of systems like Port-Hamiltonian systems and by a set of second order systems obtained by the Finite Element Method. However, for general systems, no framework has been suggested to obtain the so called contractive representations. In [27][28], the extension of interpolatory projection methods to parameterized systems has been attempted, but these methods require a structured dependence on the parameters. The salient features of matrix interpolatory framework can thus be stated as: a) Parametric dependency need not be expressed in an analytical or affine fashion. b) This method does not suffer from the curse of dimensionality and the order of the parametric reduced order model obtained is comparable to that obtained in non-parametric reductions. c) Local models may be reduced through any feasible method suiting the physics of the problem and does not affect the interpolatory procedure as such. d) A number of open problems (already discussed) need to be addressed so as to increase the viability of this method. IV. POTENTIAL APPLICATION AREAS In the previous section, the various PMOR strategies proposed in the literature were discussed and critically reviewed. This section provides a fairly detailed account of two potential and motivating application areas in which PMOR strategies can really prove to be beneficial in reducing time delays associated with optimization and design cycles. 1. Parametric Model Order Reduction of MEMS devices: Micro Electro-mechanical Systems (MEMS) are becoming ubiquitous in nearly every domain of our life, with a constantly growing market share in consumer electronics. However, the efficient design of MEMS devices remains a challenging task. The relevant physical phenomena are governed by Partial Differential Equations (PDE’s). Analytical methods for solving
PDE’s are available only for the simplest of geometries. As such, numerical methods like Finite Difference Method (FDM), Finite Element Method (FEM), and Boundary Element Method (BEM) etc. become inevitable for fitting solutions over complicated domains. These methods approximate the geometry with a computational mesh and discretize the underlying partial differential equation based on this mesh. The result of this discretization is a system of ordinary differential equations (ODE’s) of usually several thousands to several hundred thousand or even millions of degrees of freedom for a single device component. This high complexity makes computations clumsy even with present day resources. MOR techniques provide a way out of this impasse. However standard MOR techniques cannot be used if the equation system includes parameters that are to be preserved in the reduced model. For efficient design and optimization of design parameters in MEMS, parametric MOR strategies must be developed so as to capture the parametric dependencies in the reduced model. This way, repeated simulation can be carried out in the reduced space without the need to redo the reduction step. However, while developing the reduction strategies, tackling the coupled multi-physics environments can prove to be a challenging task. 2. Parametric Model Order Reduction for fast Transient Electromagnetic Computations: Simulations of Electromagnetic systems require heavy time stepping computations often involving several thousand time varying variables in the finite element models. Transient Electromagnetic Problems constitute an area of significant investigative effort. The principal computational issue in these problems is the solution of the large system of Differential Algebraic Equations (DAE’s) obtained after FE discretization. Model Order Reduction techniques provide a mechanism to generate reduced order models from the detailed description of the original FE network. MOR strategies are well developed or give better results with ODE models. One way of dealing with DAE models is to first convert them into ODE’s and then apply MOR strategies to the resulting ODE models. While converting the DAE’s into ODE’s, for parametric model order reduction, preservation of parametric dependency in the resulting ODE model has to be ensured. The reduction of resulting ODE model is usually achieved by using moment matching techniques, where the reduced order model matches the moments of the original system to approximate the original system with a low order transfer function. However, these numerical techniques all preserve the original system moments only with respect to frequency. While this provides a suitable cost advantage when performing a single frequency sweep, a
new reduced model is required each time a parameter is varied in the structure under study. This necessitates the use of parametric MOR strategies so as to expedite optimization and design space exploration cycles. Extension of existing PMOR strategies to suit the physics of this problem will be a challenging area to investigate. Recently, some work has been reported in the area of parameterized reduction of electromagnetic systems [29][30]. V. PROBLEMS IN EXISTING PMOR STRATEGIES The existing PMOR strategies, as discussed in Section III, throw open some challenges for researchers which need immediate attention so as to improve the viability of these methods for parametric reduction. Some of the problems identified by the authors are presented below: 1. Optimal Sampling of the Parametric Space: The sensitivity of the system to parametric variation needs to be numerically quantified where it is impossible to assume or obtain an affine or analytical dependency. The possible numerical measures that may be taken include the Eigen values, Hankel Singular Values (HSV’s) and the moments of the reduced order models. This can provide a framework for the choice of sampling method to discretize the parametric space. Analytical measures to quantify the sensitivity can be worked upon in cases where analytical dependency can be easily established. 2. Error Estimates for the Parametric Reduced Model: In model reduction strategies like Truncated Balanced Realization, suitable error estimates exist to judge the quality of the reduced model [1]. How these error estimates can be extended to parametric models obtained through matrix interpolations, will be an interesting problem to investigate. In general, obtaining the error estimates for the parametric reduced models can be a challenging task to be tackled. 3. Stability of the Parametric Reduced Model: Preserving the stability becomes more complex while dealing with parametric systems, as stability of the reduced system for all possible parameter values has to be guaranteed. This is one of the major difficulties behind the generalization of the existing stability preserving methods to parametric systems. Hence, all the reduction methods based on moment matching and those using a common projector cannot ensure stability of the resulting reduced system. Moreover, the reduced parametric model obtained by interpolation of locally reduced ones is not necessarily stable even if each of the involved models is stable. Hence, studies need to be undertaken for the extension of stability preserving standard MOR techniques to parametric reduction. 4. Scope of improvement in Matrix Interpolatory Framework:
As discussed in Section III, this framework can handle parametric reduction cases wherein no analytical dependency can be established in the models. However, following aspects need to be worked upon for better acceptability in MOR community: a) Characterization of the parametric dependency by studying the properties of the local projection matrices used in generating reduced order models at discrete points in the parametric space. b) Using these numerical properties to quantify the sensitivity of the model to parametric variations and use it as criteria to choose the local original models in the parametric space and if possible, exploit it to decide the choice of interpolating functions for matrix interpolation. c) In this framework, it is being assumed apriori that the parametric variations are such that the dynamics at any parametric value is restricted to the same dimensional subspace, which may not be true for all parametric variations as well as for all physical parameters. So, one can relax the condition that the local reduced models be of same size by using a stopping criteria for the local reduction and then devise a proper framework for interpolating between matrices of different sizes. The profile of the reduced models so obtained in the parametric space can directly reveal some useful information about parametric dependence of the system.
[3] [4]
[5] [6]
[7] [8]
[9]
[10] [11]
[12]
[13]
VI. CONCLUSION Optimization and design space exploration are usually performed during a typical design process that consequently requires multiple frequency domain simulations for different design parameter values. Traditional MOR techniques perform model reduction only with respect to frequency and a new reduced model has to be generated each time a design parameter is modified, thereby reducing the CPU efficiency. Parameterized model order reduction methods are therefore needed to efficiently perform these design activities. This paper presents a critical review of strategies for parameter dependency preservation in reduced order models, which have been proposed over the years, and attempts to contextualize the problem in proper physical and numerical perspective. Two potential motivating application areas are discussed to get a better grasp of the problem. The shortcomings in existing PMOR strategies are identified and directions of improvement are suggested. REFERENCES [1] [2]
A. C. Antoulas: Approximation of Large-Scale Dynamical Systems, SIAM, Philadelphia, PA, 2005. L.T.Pillage and R.A.Rohrer, “Asymptotic waveform evaluation for timing analysis,” IEEE Transactions Computer Aided Design,vol. 9, pp. 352-366, 1990.
[14]
[15] [16] [17] [18]
[19] [20]
[21]
[22]
M.Verlaan, “Efficient Kalman filtering algorithms for hydrodynamic models,” Ph.D. Thesis, Technische Universiteit, Delft, The Netherlands, 1998. T.D.Romo, J.B.Clarage, D.C.Sorensen, and G.N. Phillips, Jr.,“ Automatic Identification of discrete substates in proteins: Singular value decomposition analysis of time-averaged crystallographic refinements,” Proteins Structure Function Genetics,22, pp. 311-322, 1995. D. Teegarden, G.Lorenz, and R.Neuf, “How to model and simulate micro-gyroscope systems”, IEEE Spectrum, 35, pp. 66-75, 1998. Zhaojun Bai, Yangfeng Su, “Dimension Reduction of Large Scale Second-Order dynamical systems via a second order Arnoldi Method,” SIAM J. SCI. COMPUT., vol. 26, No. 5, Aug. 1987, pp. 1692-1709, 2005. Chaniotis, D.; Pai, M.A., “Model reduction in power systems using Krylov subspace methods," Power Systems, IEEE Transactions on, vol.20, no.2, pp. 888- 894, May 2005. J.E.Bracken, D.K.Sun, Z.Cendes, "Characterization of electromagnetic devices via reduced order models,” Computational Methods in Appl. Mech. Eng., 169, pp. 311-330, 1999. K. Carlberg, J. Cortial, D. Amsallem, M. Zahr, and C. Farhat,“The GNAT non-linear model reduction method and its application to fluid dynamics problems,” 6th AIAA Theoretical Fluid Mechanics Conference, Honolulu, Hawaii, June 27-30,2011. P. Benner,, “Solving large-scale control problems”, Control Syst. Magazine, vol.24, pp. 44-59, 2004. Joao M.S.Silva, Jorge Fernandez Villena, Paulo Flores and L. Miguel Silveria, “Outstanding Issues in Model Order Reduction” Chapter in book Scientific Computing in Electrical Engineering, vol 11 of Mathematics in Industry, pp. 139-152, Springer Verlag, May 2007. D.S.Wielle, E.Michelssen, E.Grimme and K.Gallivan,“ A method for generating rational interpolant reduced order models of two parameter linear systems,” Appl. Math. Letters, vol. 12, pp. 93-102, 1999. E.J.Grimme, “Krylov Projection Methods for Model Reduction,” Ph.D. Thesis, Department of Electrical Engineering, University of Illnois at Urbana-Champagne, 1997. L.Daniel, O.Siong, K.Lee, and J.White,“ A multi-parameter moment matching model reduction approach for generating geometrically parameterized interconnect performance models,” IEEE Transactions on Computer Aided Design of Integrated Circuits and Systems, vol.5, pp. 678-693, 2004. L.Feng and P.Benner, “A robust algorithm for parametric model order reduction based on implicit moment matching”,Proc. Appl. Math. Mech.,vol.7, pp.501-502, 2008. B.Bond and L.Daniel, “Parameterized Model Order Reduction of non-linear dynamical systems”, IEEE/ACM International Conference on Computer Aided Design, ICCAD-2005, pp. 487-494. R.Eid, B.Salimbahrami, B.Lohmann, E.B.Rudnyi and J.G.Korvink, “Parametric order reduction of proportionally damped second order systems,” Sensors and Materials, Tokyo, vol.19, pp.149-164, 2007. O.Farle, V.Hill, P.Ingelstrom and Dyczij-Edlinger, “Multi-parameter polynomial order reduction of linear finite element models,” Mathematical and Computer Modeling of Dynamical Systems, vol 14, pp. 421-434, 2008.. L.Feng, “Parameter independent Model Order Reduction,” Mathematics in Computers and Simulation, vol. 68, pp. 221-234, 2008. L.Feng, E.B. Rudnyi, J.G. Korvink, “Preserving the film coefficient as a parameter in the compact thermal model for fast electro-thermal simulations,” IEEE Transactions on Computer Aided Design of Integrated Circuits and Systems, vol. 24, pp. 1838-1847, 2005. P.K. Gunupudi, R.Khazaka, M.S. Nakhla, T. Smy, D. Celo, “Passive Parameterized time domain macro-models for high speed transmission line networks,” IEEE Transactions Microwave Theory and Networks, vol. 51, pp. 2347-2354, 2003. A.T.Leung, R.Khazaka, “Parametric Model Order Reduction for Design Optimization,” Proc. Intl. Symp. Circuits and Systems, pp. 1290-1293, 2005.
[23] L.Peng, F.Liu, L.T.Pillegi, S.R. Nassif, “Modeling interconnect variability using efficient parametric model order reduction,” Proc. Of the Design, Automation and Test in Europe Conference and Exhibition, Munich, Germany, pp. 958-963, 2005. [24] H.Panzer, J.Mohring, R.Eid, B.Lohmann, “Parametric Model Order Reduction by Matrix Interpolation,” Automatisierungtechnik, vol 58, pp. 475-484, 2010. [25] R.Eid, R.Costane-Selga, H.Panzer, T.Wolf, B.Lohmann, “Stability Preserving parametric model order reduction by matrix interpolation,” Mathematical and Computer Modeling of Dynamic Systems, vol. 17, pp. 319-335, Aug. 2011. [26] R. Castane-Selga and R. Eid, “Preserving Stability in Krylov-based Model Reduction,” Workshop GAMM-FA Dynamik und Regelungstheorie, München, 27. - 28.03.2009. [27] U.Baur, P.Benner, “Modelereduktion filr paramerisierte systeme durch balanciertes Abschneiden und Interpolation,” Automatisierungtechnik, vol. 57(8), 2009. [28] U.Baur, C.Beattie, P.Benner, S.Gugercin “Interpolatory Projection Methods for Parameterized Model Reduction” Chenmit Scientific Computing Preprints, TU-Chenmitz, Germany, November, 2009 Automatisierungtechnik, vol. 57(8), 2009. [29] M.Ahmadloo, A.Dounavis, “Parameterized model order reduction of electromagnetic systems using multi-order Arnoldi,” IEEE Transactions on Advanced Packaging, Vol. 33, No. 4, pp. 10121020, 2010. [30] M. Abid Bazaz, M. Nabi, S. Janardhanan, “Parameterized Model Order Reduction for fast transient electromagnetic computations,” presented at IEEE-EPSCICON 2012,Thrissur,Kerela.
