Adaptive Wavelet Schemes for an Elliptic Control Problem ... - CiteSeerX

Adaptive Wavelet Schemes for an Elliptic Control Problem with Dirichlet Boundary Control Angela Kunoth

Abstract An adaptive algorithm based on wavelets is proposed for the fast numerical solution of control problems governed by elliptic boundary value problems with Dirichlet boundary control. A quadratic cost functional representing Sobolev norms of the state and a regularization in terms the control is to be minimized subject to linear constraints in weak form. In particular, the constraints are formulated as a saddle point problem that allows to handle the varying boundary conditions explicitly. In the framework of (biorthogonal) wavelets, a representer for the functional is derived in terms of `2 –norms of wavelet expansion coefficients and the constraints are written in form of an `2 automorphism. Standard techniques from optimization are then used to deduce the resulting first order necessary conditions as a (still infinite) system in `2 . Applying the machinery developed in [8, 9] which has been extended to control problems in [14], an adaptive method is proposed which can be interpreted as an inexact gradient method for the control. In each iteration step, in turn the primal and the adjoint saddle point system are solved up to a prescribed accuracy by an adaptive iterative Uzawa algorithm for saddle point problems which has been proposed in [10]. Under these premises, it can be shown that the adaptive algorithm containing now three layers of iterations is asymptotically optimal. This means that the convergence rate achieved for computing the solution up to a desired target tolerance is asymptotically the same as the wavelet–best N –term approximation of the solution, and the total computational work is proportional to the number of computational unknowns.

Key words. Optimal control, elliptic boundary value problem, Dirichlet boundary control, saddle point problem, wavelets, infinite `2 –system, preconditioning, adaptive refinement, inexact iteration, convergence, convergence rate, optimal complexity. AMS subject classification. 65K10, 65N99, 93B40.

1

Introduction

Approximating the solution to a stationary partial differential equation numerically by an adaptive method allows to resolve problem–dependent singularities like re–entrant corners in an underlying domain, or non–smooth right hand sides or boundary conditions, without having to resort to an unnecessarily large number of unknowns caused by a uniform fine grid. Depending 1

on a target accuracy prescribed by the user, in adaptive finite element based methods one aims at locally equidistributing the error by introducing additional degrees of freedom where the error is large. A somewhat different route for a class of variational problems has been proposed recently in [8, 9] by viewing the problem from the angle of nonlinear approximation theory. Assuming for a moment that the solution is already known, one can ask for the best possible approximation with a given number of degrees of freedom N . Depending on the regularity of the solution in certain Besov spaces, which are much larger than Sobolev spaces with the same regularity parameter and which allow for singularities as they appear in stationary partial differential equations, it has been assured that the wavelet–best N –approximation can be achieved with a particular rate. Now setting the target accuracy for numerically computing the solution of the variational problem depending on this rate, an adaptive algorithm has been proposed in [8] for elliptic boundary value problems that computes this best N –approximation. Moreover, is has been proved there that the adaptive method produces the best N –approximation in an asymptotically optimal way, that is, with an amount of arithmetic operations that is proportional to N . Corresponding numerical results in [3] confirm the asymptotic estimate. A particular feature of the proposed adaptive strategy is that the solution of a discretized but still infinite–dimensional operator equation is computed only approximately by applying appropriate sections of the infinite–dimensional operator. In fact, by discretizing the operator equation in terms of wavelets which constitute a Riesz basis for the underlying function space, the discretization by itself does not a priori prescribe a finest level of resolution. The adaptive wavelet strategy and convergence rate estimates in [8, 9] for stationary variational problems has been specified to saddle point problems in [10]. In particular, an adaptive Uzawa algorithm is derived there that automatically takes care of the LBB condition which in uniform discretizations usually restricts the choice of the two different discretization spaces, see e.g. [5, 13]. For linear–quadratic control problems involving distributed or Neumann boundary control, an extension of these ideas has been proposed in [14]. The problems treated there are such that the constraints can still be formulated in terms of one single operator equation involving a linear elliptic operator. In this paper, this is carried even a step further to consider also a class of problems with Dirichlet boundary control for which we construct along the above lines adaptive wavelet methods. Consider as an illustrative example the problem to minimize for some given data y∗ the quadratic functional 1 ω J(y, u) = ky − y∗ k2Z + kuk2U , (1.1) 2 2 where the state y and the control u are coupled through the linear elliptic boundary value problem −∇ · (a∇y) + ky = f in Ω, y = u on Γ, (1.2) (a∇y) · n = 0 on ∂Ω \ Γ.

2

Here Ω ⊂ Rn is a domain with Lipschitz continuous boundary ∂Ω and the control boundary Γ ⊂ ∂Ω is a set of positive Lebesgue measure on which the control is exerted. The term n = n(x) denotes the outward normal at x ∈ ∂Ω \ Γ, the coefficient a(x) = (ai,j (x))i,j is assumed to be uniformly positive definite on Ω, and ky := b · ∇y + a0 y with bi , a0 ∈ L∞ (Ω). The norms appearing in the functional in (1.1) are usually chosen as L2 or H 1 norms on the domain or on some part of the boundary. The first norm called the observation term requires given data y∗ to be fitted in a least squares sense while the second term involving some given weight ω > 0 attains the role of a regularization in order to enforce well–posedness. The main goal is then reformulated as follows: given the right hand side f and data y∗ to be matched, determine the boundary control u such that an appropriate weak formulation of (1.2) has a unique solution y and J in (1.1) is minimized. The appearance of the control u as a Dirichlet boundary condition in (1.2) is referred to as a Dirichlet boundary control. As it will be required to allow for repeated updates of the control, this suggests to formulate the constraints (1.2) weakly as a saddle point problem which results from appending the Dirichlet boundary conditions by Lagrange multipliers as in [1]. For such problems, fast solution strategies based on wavelets for uniform discretizations have been introduced in [22, 23]. The verification of the necessary ingredients to apply the paradigm for adaptive wavelet methods requires in addition to discuss the compressibility of the involved operators, in particular, trace operators, as we will see below. The paper is structured as follows. In Section 2 we formulate the class of linear–quadratic control problems which is considered here. This is represented in terms of wavelet coordinates in Section 3 involving a representer of the minimization functional. Furthermore, the necessary conditions for optimality are derived. Section 4 recalls the basis concepts from [8, 9, 14] and formulates the adaptive scheme. In Section 5 we realize the corresponding basic ingredients for the situation at hand, containing three levels of iterations, and summarize in Section 6 the resulting complexity estimates along with a discussion of the compressibility of the involved operators.

2

Problems in Optimal Control

A class of problems in optimal control for which the subsequent analysis will apply will first be described in terms of an abstract framework. We denote by Y the function space of states and by Q the control space, which are assumed to be (closed subspaces of) Hilbert spaces, with Y 0 , Q0 denoting their topological duals Y 0 , Q0 , and associated dual forms h·, ·iY 0 ×Y , h·, ·iQ0 ×Q , or shortly h·, ·i. Usually norms for function spaces will be indexed by corresponding subscripts. Moreover, we will employ the following notational conventions throughout the paper, unless constants have to be identified specifically. > The relation a ∼ b stands for a < ∼ b and a ∼ b, i.e., a can be estimated from above and below by a constant multiple of b independent of all parameters on which a or b may depend. A particular feature of the class considered here is that the constraints appear in form of a linear saddle point problem. 3

2.1

Abstract Saddle Point Problems

Let a(·, ·) : Y × Y → R be a continuous bilinear form, a(v, w) < ∼ kvkY kwkY ,

v, w ∈ Y,

(2.1)

which defines a linear continuous operator A : Y → Y 0 by hAv, wi := a(v, w). Suppose there is a second bilinear form b(·, ·) : Y × Q → R which is also continuous, b(v, q) < ∼ kvkY kqkQ ,

v ∈ Y, q ∈ Q,

(2.2)

and which defines a linear operator B : Y → Q0 and its adjoint B 0 : Q → Y 0 by hBv, qiQ0 ×Q = hv, B 0 qiY ×Y 0 := b(v, q) for all v ∈ Y , q ∈ Q. In view of the cases that will be relevant here, it will suffice to consider the restriction of B being surjective, i.e., range B = Q0 and ker B 0 = {0}. A general linear saddle point problem in operator form can then be formulated as follows. Given (f, u) ∈ Y 0 × Q0 , find (y, p) ∈ Y × Q such that y f A B0 = (2.3) B 0 p u holds. Existence and uniqueness of such a pair (y, p) is assured under the conditions that A is invertible on ker B ⊆ Y and that the range of B is closed in Q0 , in addition to the continuity of A and B, see e.g. [4, 5, 19]. The condition that range B is closed in Q0 is called the inf-sup condition since it means that sup v∈Y

hBv, qiQ0 ×Q kvkY

> kqkQ , ∼

q ∈ Q,

holds. Under these conditions on A and B, the linear saddle point operator A B0 L := : Y × Q → Y 0 × Q0 B 0 is an isomorphism, and one has the norm equivalence

v

v

v

cL ≤ L ≤ CL

q q Y ×Q q Y 0 ×Q0 Y ×Q

(2.4)

(2.5)

(2.6)

for any (v, q) ∈ Y × Q, where the finite positive constants cL , CL can be derived from the continuity and the other constants for the estimates from below for A and B [22]. In the examples considered below, A is elliptic on ker B and B is surjective so that the inf–sup condition (2.4) is always satisfied.

2.2

Abstract Linear–Quadratic Control Problems

Suppose that Z, U are additional Hilbert spaces which will be specified later. We define a cost functional in terms of the state and control variables y and u as J(y, u) :=

1 ω ky − y∗ k2Z + kuk2U , 2 2 4

(2.7)

where y∗ ∈ Z are given observation data to be matched, and the regularization ω > 0 determines the relative weight of the control term. Because of its role in (2.7), the space Z is often called the observation space. With the notation introduced in Section 2.1, the abstract linear–quadratic control problem (the notion referring to a quadratic functional and linear constraints in form of a partial differential equation) is then the following. (ACP): For given observations y∗ ∈ Z, a right hand side f ∈ Y 0 and a weight parameter ω > 0, minimize the quadratic functional (2.7) over (y, u) ∈ Y × Q0 subject to the linear constraints (2.3), y f A B0 = . B 0 p u Remark 2.1 Of course, arbitrary choices of Z and U would render a solution to (ACP) impossible. The essential requirement for the observation space is that Y ,→ Z with continuous embedding [25]. Here for simplicity we will always consider the case where the control space U is the whole space Q0 . More general cases of Sobolev spaces on the boundary of different fractional smoothness could be handled as in [14]. In fact, once a cost functional representing (2.7) is formulated in terms of wavelet coefficients, a variation of the smoothness index corresponds to a diagonal scaling.

2.3 2.3.1

Examples Classical Situation Γ ⊆ ∂Ω

The classical case involving natural Sobolev spaces and norms appears in the weak formulation of the second order elliptic boundary value problem (1.2) by appending the Dirichlet boundary conditions by Lagrange multipliers [1]. The function spaces that come into play here are Y = H 1 (Ω) and Q = (H 1/2 (Γ))0 where Γ ⊆ ∂Ω of positive measure is the part of the boundary where essential boundary conditions are posed, and the Neumann boundary conditions are incorporated naturally, see e.g. [22]. (Recall that if Γ ≡ ∂Ω, the dual space of H 1/2 (Γ) is H −1/2 (Γ), see [20].) In the weak formulation, B acting on H 1 (Ω) is just the ordinary trace operator, that is, Bv := v|Γ , and its trace is known to be in H 1/2 (Γ). Thus, the bilinear form b(v, q) := hBv, qiH 1/2 (Γ)×(H 1/2 (Γ))0

(2.8)

is well–defined on H 1 (Ω) × (H 1/2 (Γ))0 , and the associated trace theorem directly implies the inf–sup condition, see e.g. [20, 23]. Moreover, the bilinear form a(·, ·) : H 1 (Ω) × H 1 (Ω) → R is defined as Z a(v, w) := (a∇v · ∇w + (kv)w) dx (2.9) Ω

H01 (Ω),

which is invertible on ker B = see e.g. [4]. With these choices, the corresponding saddle point problem (2.3) would have for any given (f, u) ∈ (H 1 (Ω)) × H 1/2 (Γ) a unique solution. The variable p introduced in the formulation of the problem is called the Lagrange multiplier. 5

Viewing (2.3) as constraints for the cost functional (2.7), two particular natural choices for the observation space Z in which Y is continuously embedded are Z ∈ {H 1 (Ω), H 1/2 (Γy )}.

(2.10)

Here Γy ⊆ ∂Ω of positive Lebesgue measure is an observation boundary which should not coincide with the control boundary Γ. In order to treat both cases simultaneously, we introduce a linear observation operator T : Y → Z whose specific form depends on the choice of Z. If Z = Y , then T is just the identity, and the state is observed in the H 1 (Ω)–norm. For Z = H 1/2 (Γy ), T is a trace operator T : H 1 (Ω) → H 1/2 (Γy ) which like B also satisfies a trace theorem. 2.3.2

Problems Involving Fictitious Domains

In the context of problems with moving boundaries, so–called fictitious domains offer the advantage that most of the computationally heavy calculations can be made on a simple domain , typically a cube, called the fictitious domain, which satisfies ⊇ Ω. Here one assumes that, given an elliptic boundary value problem (1.2), one can define an extended bilinear form a(·, ·) : H 1 ()×H 1 () → R with the properties required above. For the trace operator B acting as an operator from H 1 () to H 1/2 (Γ), one can show like in the case above continuity and an inf–sup condition [23]. Natural choices for Z would then be Z ∈ {H 1 (), H 1/2 (Γy )},

(2.11)

where as above the observation boundary Γy ⊆ ∂Ω should not coincide with Γ. Like in Section 2.3.1, we treat both cases in conjunction by introducing the observation operator T : Y → Z which is either the identity or the trace operator T : H 1 () → H 1/2 (Γy ).

3

Representation of (ACP) in Wavelet Coordinates

Once a control problem like (ACP) is formulated, one could apply standard techniques from optimization to derive the necessary conditions for optimality in a functional analytic setting, see e.g. [25, 27], leading to a weakly coupled system of stationary partial differential equations. These could then be solved numerically with respect to suitable finite dimensional trial spaces, see e.g. [21]. Here we deviate from this road map in several ways, following the strategy in [8, 9, 22, 23]. In a first step, we will derive a representer for the abstract control problem (ACP) in terms of a still infinite dimensional problem in wavelet coordinates which is formulated entirely in `2 . All norms appearing in the cost functional will be represented by `2 norms. The constraint (2.3) will be rewritten in terms of a well–conditioned `2 –system. Such a representation of (ACP) in wavelet coordinates will be a crucial point of departure for adaptive wavelet methods discussed later for the approximative solution of the infinite systems. This in turn hinges on having suitable wavelet bases specified next. 6

3.1 3.1.1

Wavelet Bases Basic Properties

Suppose that we have a wavelet basis for each of the Hilbert spaces H ∈ {Y, Q, Z}, that is, a collection of functions ΨH := {ψH,λ : λ ∈ IIH } ⊂ H (3.1) with the following properties. By IIH we denote an infinite index set whose elements λ comprise information such as the refinement scale or level of resolution j =: |λ| and a spatial location k = k(λ) ∈ Zn . In the simplest case of wavelets on R, members in ΨH are of the shift–invariant form ψH,λ = 2j/2 ψ(2j ·−k), j, k ∈ Z, where the factor 2j/2 accounts for normalization in L2 . Here one has λ = (j, k). For intervals in R, wavelets are of the same type except for local boundary modifications, see [15] for a detailed construction which is further optimized in [16]. For Hilbert spaces on domains in more than one space dimensions, ψH,λ is constructed by taking tensor products of univariate (generator and wavelet) functions resulting in 2n − 1 different ‘types’ of wavelets which will be encoded in a further index. On boundary manifolds, one can construct composite wavelets as in [7, 17] or by more sophisticated techniques by means of topological isomorphisms as in [18, 24]. Without going into details, the crucial properties of wavelet basis that are relevant in the sequel are the following. Riesz basis property (R): Every v ∈ H has a unique expansion in terms of ΨH , X v= vλ ψH,λ =: vT ΨH , v := (vλ )λ∈IIH ,

(3.2)

λ∈IIH

and its expansion coefficients satisfy a norm equivalence: for any v = {vλ : λ ∈ IIH } one has cH kvk`2 (IIH ) ≤ kvT ΨH kH ≤ CH kvk`2 (IIH ) ,

v ∈ `2 (IIH ),

(3.3)

where cH , CH are some finite positive constants. Thus, wavelet expansions induce isomorphisms between certain function and sequence spaces in terms of wavelet coefficients. We will for convenience write `2 norms without subscripts as k · k := k · k`2 (IIH ) . If the precise format of the constants does not matter, we will also abbreviate (3.3) as kvk ∼ kvT ΨH kH ,

v ∈ `2 (IIH ).

(3.4)

Locality (L): The functions ψH,λ are local with decreasing support for increasing level, i.e., diam (supp ψH,λ ) ∼ 2−|λ| .

(3.5)

Cancellation property (CP): There exists an integer m ˜ =m ˜ H such that −|λ|(d/2+m) ˜ , hv, ψH,λ i < |v|W∞ m ˜ (supp ψ H,λ ) ∼ 2

(3.6)

where n is the dimension of the underlying domain or manifold. Hence, integrating against a wavelet mimics an mth ˜ order difference which annihilates the smooth part of v. The cancellation 7

property entails quasi–sparse representations of a wide class of operators including the differential operators considered here. By duality arguments one can show that (3.3) is equivalent to the existence of a biorthogonal collection which is dual or biorthogonal to ΨH , ˜ H := {ψ˜H,λ : λ ∈ IIH } ⊂ H 0 , Ψ

hψH,λ , ψ˜H,µ i = δλ,µ ,

λ, µ ∈ IIH ,

(3.7)

˜ H ∈ H 0 one has ˜T Ψ which is a Riesz basis for H 0 , that is, for any v˜ = v −1 ˜ H kH 0 ≤ c−1 k˜ CH k˜ vk ≤ k˜ vT Ψ H vk,

(3.8)

˜ H is a dual basis to a primal one for see [11, 12, 22]. Here the tilde always is to express that Ψ ˜ the space identified by the subscript so that ΨH = ΨH 0 . Computations for the present applications in fact only employ the primal wavelets while the duals are mainly needed for analysis purposes. In the cases which have been implemented in [3, 6], the primal wavelets consist of linear combinations of tensor products of cardinal B–Splines and are explicitly known. As above in (3.3), we make systematic use of the following shorthand notation. ΨH will be viewed both as a collection of functions as well as a (possibly infinite) column vector containing all functions always assembled in some fixed unspecified order. For a countable collection of functions Θ and some single function σ, the term hΘ, σi is to be understood as the column vector with entries hθ, σi, θ ∈ Θ, and correspondingly hσ, Θi the row vector. For two collections Θ, Σ, the quantity hΘ, Σi is then a (possibly infinite) matrix with entries (hθ, σi)θ∈Θ, σ∈Σ for which hΘ, Σi = hΣ, ΘiT . This also implies for a (possibly infinite) matrix C that hCΘ, Σi = ChΘ, Σi and hΘ, CΣi = hΘ, ΣiCT . In this notation, the biorthogonality or duality conditions in (3.7) can be abbreviated as ˜ =I hΨ, Ψi (3.9) with the infinite identity matrix I.

3.2

Representation of Operators

A wavelet representation of linear operators in terms of wavelets is now the following. Let H, V ˜ H, Ψ ˜ V , and suppose be Hilbert spaces with wavelet bases ΨH , ΨV and corresponding duals Ψ 0 0 0 0 that C : H → V is a linear operator with dual C : V → H defined by hv, C wi := hCv, wi for all v ∈ H, w ∈ V . Then Cv = w ∈ V 0 can be represented in the equivalent formulation Cv = w ˜ V ), where in terms of the wavelet coefficients v for v (expanded in ΨH ) and w (in terms of Ψ C := hΨV , CΨH i,

(3.10)

see [14]. The infinite matrix C is therefore a standard representation of C with respect to the underlying wavelet bases ΨV , ΨH . For such matrices, we write CT for the adjoint just like for finite matrices. 8

3.3

Control Problems in `2

After these preliminaries, we are in the position to formulate a representer of the abstract control problem (ACP) in terms of wavelet coordinates, having asserted that wavelet bases with the above properties exist for H ∈ {Y, Z, Q} which will be indexed by a corresponding letter. From now on, we will only be working in `2 coordinates. Starting with the constraints (2.3) we follow the recipe from Section 3.2, and expand y = yT ΨY , p = pT ΨQ and test with the elements of ΨY , ΨQ . Then (2.3) attains the form y y f A BT L := = , (3.11) B 0 p p u where A := hΨY , AΨY i

f

:= hΨY , f i,

B := hΨQ , BΨY i,

u := hΨQ , ui.

(3.12)

In view of the above assertions, the operator L is an `2 –automorphism, i.e., for every (v, q) ∈ `2 (II) = `2 (IIY × IIQ ) one has

v

v

v

(3.13) ≤ C L ≤ cL L

q

q q with constants cL , CL only depending on cL , CL from (2.6) and the constants in the norm equivalences (3.3) and (3.8). Correspondingly, depending on the choice of Z, let the representation of T : Y → Z be either ˜ Z , T ΨY i. In view of the T = I or the trace operator defined according to (3.10) by T := hΨ norm equivalence(3.3), the representer of the minimization functional (2.7) is now defined as e u) := 1 kTy − y∗ k2 + ω kuk2 , J(y, 2 2

(3.14)

˜ Z , y∗ i are the wavelet coefficients for y∗ expanded in ΨZ . It should be pointed out where y∗ := hΨ that the specific form of the second term, emerging just as kuk (without further shift operators as in [6, 14]) stems from measuring u in (2.7) in its natural H 1/2 (Γ) and then applying the norm equivalence (3.3). A discrete infinite control problem in wavelet coordinates is then formulated as follows. (DCP): For given observations y∗ , right hand side f and a weight parameter ω > 0, minimize the quadratic functional (3.14) over (y, u) subject to the linear constraints (3.11). Remark 3.1 We use the notion of a representer for the control problem (ACP) for the following reason. Although (3.14) is the exact representation of problem (2.3), the functional (3.14) is only equivalent to the cost functional (2.7) in the sense of equivalent norms (3.3). Thus, employing (3.14) still captures the main features of the linear–quadratic control but does not yield exactly the same solution. This issue together with several numerical experiments on the quality of solutions, taking also norms with different Sobolev smoothness indices into account, is discussed at length in [6]. 9

Once (DCP) is posed, we can now apply the standard formalism for control problems and derive the necessary conditions for optimality as e.g. in [27]. Because of the linear–quadratic nature of the problem, these conditions are also sufficient for a unique solution. The following has been proved in [23]. Theorem 3.2 The unique solution (y, u) of (DCP) is given as part of solving the coupled system y f L = p u −ωTT (Ty − y∗ ) T z (EE) L = (3.15) µ 0 u = µ

(3.16)

for (y, p, z, µ, u), where z, µ are like p additional Lagrange multipliers. The first system appearing in the energy equations (EE) are just the constraints (3.11) which we call primal system while (3.15) will be referred to as dual or adjoint system. The specific form of the dual system involving the adjoint of L emerges from the particular formulation of the minimization functional (3.14). Moreover, the trivial equation (3.16) stems from measuring u just in `2 , representing measuring the control in its natural trace norm. Instead of replacing µ by u in (3.15) and trying to solve the resulting equations, (3.16) will be essential to devise an e u) inexact gradient scheme. In fact, since L in (3.11) is an isomorphism, one can rewrite J(y, e by formally inverting (3.11) as a functional of u, that is, J(u) := J(y(u), u), see [23]. Then we make use of the following facts proved in [23]. Proposition 3.3 The first variation of J satisfies δJ(u) = u − µ,

(3.17)

where (u, µ) are part of the solution of (EE). Moreover, J is convex so that a unique minimizer exists. Thus, (3.16) in fact equals δJ(u) = 0. This naturally suggests a simple gradient method for u as uk+1 := uk − α δJ(uk ), k = 0, 1, 2, . . . (3.18) with some initial guess u0 . It has been asserted in [23] that under the above assumptions on L and because of the structure of the control problem (DCP), there exists a fixed parameter α, depending on bounds for the second variation of J, such that (3.18) converges and reduces the error in each step by at least a fixed factor ρ < 1, i.e., ku − uk+1 k ≤ ρku − uk k,

k = 0, 1, 2, . . . ,

(3.19)

where u is part of the exact solution of (EE) and ρ is determined by ρ := kI − αQk < 1. 10

(3.20)

Since the gradient method (3.18) is still formulated in terms of infinite systems, we need to discuss the actual realization of the ideal iteration (3.18). To this end, it is very convenient that by formally inverting (3.11) and (3.15), one can rewrite (EE) in a straightforward fashion in compact form as a linear equation for u alone, namely Qu = g.

(3.21)

Here we have employed the abbreviations Q := ZT Z + ωI, and −1

Z := T L

I ,

g := ZT (y∗ − T L−1 I f ) 0 I := , I

T := (T 0).

(3.22)

(3.23)

Proposition 3.4 The vector u as part of the solution vector (y, p, z, µ, u) of (EE) coincides with the unique solution u of the condensed equations (3.21). One can then in turn identify δJ(u) = Qu − g so that (3.18) can be rewritten as uk+1 := uk + α (g − Quk ),

k = 0, 1, 2, . . . .

(3.24)

Since on account of (3.22), Q is symmetric positive definite, it can also be seen here that (3.24) converges for any initial guess u0 . Moreover, since Z is bounded on `2 , so is Q, i.e., we have that Q has uniformly bounded condition numbers. This means that there exist finite positive constants cQ , CQ such that cQ kvk ≤ kQvk ≤ CQ kvk,

v ∈ `2 ,

(3.25)

where we can actually take cQ = ω. Even if we were already in the situation of finite systems, in order to make the above iterative schemes practically feasible, of course, the explicit inversion of L in the definition of Q must be avoided and will by carried out by an iterative solver for (3.11). Thus, the idea is to use (3.24) as an outer iteration in which for the computation of the residual g − Quk each of the two systems (3.11) and (3.15) are solved approximately by an iterative method. Since these systems are of saddle point type, this in turn requires to use an iterative solver. Our ultimate goal will be to carry out the ideal iteration (3.24) approximately with dynamically updated tolerances. Remark 3.5 It should be pointed out that the simple gradient iteration can, of course, be replaced by a conjugate gradient iteration as in [6] where then the direction vectors conjugate with respect to the hQ·, ·i inner product are computed approximately, resulting in an inexact conjugate gradient method. This would not require any restrictions on a parameter α as in (3.18). However, since using the conjugate gradient method in this context would introduce an additional level of technicalities, we dispense with this generalization here. 11

Later we will also need a realization of the following basic ideal iteration for the infinite dimensional saddle point problem (3.11) (for given right hand side uf ) called the Uzawa algorithm. Given any initial guess p0 one computes for k = 1, 2, 3, . . . Ayk = f − BT pk−1 pk = pk−1 + β(Byk − u).

(3.26)

Here β is some positive constant depending on cL , CL which can be determined such that the contraction factor kI − βBA−1 BT k is less than one, enforcing convergence of the method [10].

4

Basic Concepts

In this section the basic concepts and results which turn (3.24) into a computable routine and provides a complexity analysis are assembled briefly from [8, 9, 14] without further reference.

4.1

Perturbed Iterations

Of course, the convergent iteration (3.24), uk+1 = uk + α(g − Quk ),

k = 0, 1, 2, . . . ,

cannot be carried out exactly because Q is an infinite matrix and the data g could be an infinite array. However, we will devise perturbed iterations with dynamically suitably updated accuracy tolerances which will still converge. To this end, a routine with the following property will be designed. Res [η, Q, g, v] → rη determines for a given tolerance η > 0 a finitely supported sequence rη satisfying kg − Qv − rη k ≤ η. (4.1) Moreover, the following ingredient will eventually play a crucial role in controlling the complexity of the scheme. Coarse [η, w] → wη determines for any finitely supported input vector w a vector wη with smallest possible support such that kw − wη k ≤ η.

(4.2)

The realization of Coarse essentially consists of performing a quasi–sorting of the entries of w by size into bins of the same refinement level and subtracting squares of their moduli starting from the smallest one until the sum reaches η 2 . Thus, the complexity of Coarse is linear in # supp w [2] in contrast to an element–wise sorting which would introduce an additional logarithmic factor.

12

Suppose now that the routine Res is already at our disposal. Given (an estimate of) the reduction rate ρ and the step size parameter α from (3.20), define the control parameter K := min{` ∈ N : ρ`−1 (α` + ρ) ≤

1 10 }.

(4.3)

Given any target accuracy ε > 0, a perturbed iteration reads as follows. Solve [ε, Q, g, u0 , ε0 ] → uε (i) Fix a target accuracy ε > 0. Given an initial guess u0 along with an error bound ku − u0 k ≤ ε0 , set j = 0. (ii) If εj ≤ ε, stop and set uε := uj . Otherwise set u0 := uj . (ii.1) For k = 0, . . . , K − 1 compute Res [ρk εj , Q, g, vk ] → rk and vk+1 := vk + αrk .

(4.4)

(ii.2) Apply Coarse [ 25 εj , vK ] → uj+1 ; set εj+1 := 12 εj , j + 1 → j and go to (ii). If no specific initial guess is known, step (i) is replaced by (i)’ Fix a target accuracy ε > 0. Set j = 0 and ε0 := c−1 Q kgk.

u0 = 0,

(4.5)

In this case we shortly write Solve [ε, Q, g] → uε . The choice of the interior tolerance ρk εj in step (ii.1) yields for the final iterate vK resulting from step (ii.1) the estimate εj kvK − uk ≤ . (4.6) 10 Theorem 4.1 For every target accuracy ε > 0 the scheme Solve [ε, Q, g, u0 , ε0 ] → uε produces after finitely many steps a finitely supported approximation uε of the exact solution u of (3.21) such that ku − uε k ≤ ε. (4.7) Moreover, the iterates uj generated by Solve [ε, Q, g] satisfy ku − uj k ≤ εj , where εj = 2−j ε0 ≤ ε.

13

j = 1, 2, . . .

(4.8)

4.2

Complexity Analysis

A suitable framework for the complexity analysis of an actual realization of the routine Res and its complexity may be the following. Optimal work/accuracy rate: We say that the scheme Solve has an optimal work/accuracy rate s if the following holds: if the error of best N –term approximation σN (u) := ku − uN k :=

min

# supp v≤N

ku − vk

(4.9)

decays like O(N −s ), then the solution uε is produced by Solve at an expense that also stays proportional to ε−1/s and,thus, matches the best N –term rate. Of course, this implies that # supp uε also stays proportional to ε−1/s . Thus, ‘best N –term approximation’ is the benchmark. If u is known, this best N –term approximation uN of u is given by taking the N largest terms in modulus from u. This is, however, an information that is not available when u is the unknown solution of (3.21). Nevertheless, appropriate complexity criteria will be based on a characterization of those sequences v for which the best N –term approximation error decays at a particular rate. To this end, we consider sequences that are sparse in the following sense. Set for some 0 < τ < 2 −τ `w , for all 0 < η ≤ 1} τ := {v ∈ `2 : #{λ ∈ II : |vλ | > η} ≤ Cv η

(4.10)

which determines a strict subspace of `2 only for τ < 2, the sequence being the sparser the smaller τ is. Denote for a given v ∈ `w τ by Cv the smallest constant for which (4.10) holds. 1/τ 1/τ ∗ Then one has |v|`wτ := supn∈N n vn = Cv , where v∗ = (vn∗ )n∈N denotes a non–decreasing rearrangement of v. Moreover, kvk`wτ := kvk + |v|`wτ is a quasi–norm for `w τ , and the continuous w w embeddings `τ ⊂ `τ ⊂ `τ +ε ⊂ `2 for τ < τ + ε < 2 mean that `τ is very close to `τ , therefore calling `w τ weak `τ . The following relation between sequences in such Lorentz spaces `w τ and best N –term approximation will be crucial. Theorem 4.2 Let positive real numbers s and τ be related by 1 1 =s+ . τ 2

(4.11)

Then a sequence v belongs to `w τ if and only if −s kv − vN k < ∼ N

and

−s σN (v) < ∼ N kvk`wτ ,

(4.12)

where as before vN denotes a best N –term approximation of v. Depending on the space H which is characterized by the wavelet basis ΨH , the fact that T a vector of wavelet coefficients v is in `w τ is equivalent to the expansion v ΨH belonging to a certain Besov space which describes a much weaker regularity measure than a Sobolev space of 14

corresponding order. Theorem 4.2 therefore says how much loss of regularity can be compensated by judiciously placing the degrees of freedom in a nonlinear way for retaining a certain optimal order of error decay. As will be seen in Theorem 4.3 below, a criterion for a scheme Solve to have an optimal work/accuracy rate can be formulated through the following property. τ ∗ –Sparsity: The routine Res is called τ ∗ –sparse for some 0 < τ ∗ < 2 if the following holds: ∗ Whenever the solution u of (3.21) is in `w τ for some τ < τ < 2, then for any v with finite support the output rη of Res [η, Q, g, v] satisfies (i) krη k`wτ # supp rη

< ∼ < ∼

max{kvk`wτ , kuk`wτ },

(4.13)

1/s

1/s

τ

τ

η −1/s max{kvk`w , kuk`w },

where s and τ are related as before by (4.11); (ii) the number of floating point operations needed to compute rη stays proportional to # supp rη . Furthermore, the constants in (i), (ii) depend only on τ as τ → τ ∗ . In this context we always assume that given data can be considered to be accessible completely. Practically, this could mean that depending on the final target accuracy sufficiently many of the corresponding coefficients of explicitly given data are determined in a preprocessing step and then ordered by size so that Coarse can be applied to an array of finite support. Theorem 4.3 If Res is τ ∗ –sparse and if the exact solution u of (3.21) is in `w τ for some τ > τ ∗ , then for every ε > 0 Solve [ε, Q, g] produces after finitely many steps an output uε which, according to Theorem 4.1, always satisfies ku − uε k < ε with the following properties: For s and τ related by (4.11), one has 1/s −1/s # supp uε < kuk`w , ∼ ε τ

kuε k`wτ < ∼ kuk`wτ ,

(4.14)

and the number of floating point operations needed to compute uε remains proportional to # supp uε . Hence, τ ∗ -sparsity of the routine Res implies asymptotically optimal work/accuracy rates of the scheme Solve for a certain range of decay rates given by τ ∗ . It should be emphasized that the algorithm itself does not require any a–priori knowledge about the solution such as its actual best N –term approximation rate. Theorem 4.3 also reveals that controlling the `w τ –norm of all quantities generated in the course of the computation is crucial, explaining the role of Coarse in step (ii.2) of Solve. In fact, one has the following result.

15

Lemma 4.4 (Coarsening Lemma) Let v ∈ `w τ and let w be any finitely supported approximation such that kv − wk ≤ 15 η. Then the output wη of Coarse [ 54 η, w] satisfies 1/τ −1/s # supp wη < , ∼ kvk`wτ η

kv − wη k < ∼ η,

and

kwη k`wτ < ∼ kvk`wτ .

(4.15)

Knowing an error bound for a given finitely supported approximation w, a specific coarsening using only knowledge about w produces a new approximation to the unknown v which gives rise to a slightly larger error but realizes up to a uniform constant the optimal relation between support and accuracy. In Solve this means that by the coarsening step the `w τ –norms of the iterates vK are controlled. The choice of the constant K in (4.3), which controls the number of iterations in step (ii.1), guarantees that in the (j + 1)st outer iteration of Solve the iterate vK 1 satisfies ku − vK k ≤ 10 εj . The threshold 25 εj in step (ii.2) assures, on account of (4.15), that j the error after coarsening is still bounded by 12 εj . In addition, if u ∈ `w remains τ , then ku k`w τ −1/τ

bounded and # supp uj increases at most like εj . Recall from (4.12) that this is the best w possible N –term rate for sequences in `τ . Thus, to ensure an overall optimal work/accuracy k rate one has to show that the `w τ –norms of the intermediate iterates v in step (ii.1) of Solve ∗ cannot grow too much. This is in turn guaranteed by τ –sparsity of Res. It remains to construct a concrete realization of Solve for (DCP) which will be called SolveDCP such that the corresponding routine ResDCP is τ ∗ -sparse. Essentially we build on the construction given in [14] by introducing an additional level of inner iteration for the approximate solution of each of the saddle point problems (3.11) and (3.15) in terms of an adaptive Uzawa algorithm devised in [10].

5

The Scheme SolveDCP

Since Q from (3.21) involves two inversions of matrices it is not so clear how to realize a residual approximation in an economical way. Here the auxiliary formulation (EE) which avoids the explicit inversion of L comes into play.

5.1

Realization of the Routine ResDCP

The realization of the routine ResDCP for the problem (3.21) is based on the residual representation (3.21) and requires in turn solving the two auxiliary systems in (EE). Since the residual only has to be approximated, these systems will have to be solved only approximately. These approximate solutions, in turn, will be provided again by calls of the scheme Solve but now with respect to suitable residual schemes tailored to the systems in (EE). Although from a conceptual point of view use the fact that a gradient iteration for the reduced problem (3.21) reduces the error for u in each step by a fixed amount, employing (EE) for the evaluation of the residuals will generate as byproducts approximate solutions to the full quintuple (y, p, z, µ, u) of (EE). The saddle point matrix L appearing in (EE) is by definition indefinite with (3.13). Hence, one could in principle multiply the saddle point systems in (EE) by LT , respectively L, yielding a 16

least squares formulation of (3.11) and (3.15) with LT L, respectively LLT , as system matrices. Each of them still satisfies (3.13) but is now a symmetric positive definite system to which all of the results in Section 4.1 apply. Here we avoid working with the normal equations and instead employ an iterative algorithm directly for saddle point problems. Next we formulate the ingredients for suitable versions of Solve for the saddle point systems in (EE). The main task there is to apply the involved linear operators, that is, one has to assure that for each M ∈ {A, AT , B, BT , T, TT } we have a scheme at our disposal with the following property. Apply[η, M, v] → wη determines for any finitely supported input vector v and any tolerance η > 0 a finitely supported output wη which satisfies kMv − wη k ≤ η.

(5.1)

A scheme SolveSPP for the saddle point system (3.11) or (3.15) in (EE) with the following property has been devised and analyzed in [10]. SolveSPP [η, L, f , v, y, p] → (yη , pη ) determines for any positive tolerance η, given finitely supported f , v and any finitely supported input (y, p) finitely supported output (yη , pη ) satisfying

f yη

(5.2)

v − L pη ≤ η. Without recalling the details here, one essentially combines Apply schemes for A and B with an approximate solution of the first equation in (3.26) by a scheme Solve, followed by an appropriate coarsening, similar of the type exhibited in ResDCP below. Recall that the exact solution u of (3.21) is the last component of the solution quintuple (y, p, z, µ, u) of the energy equations (EE). This vector of wavelet coefficients shall always denote the exact solutions. Now we can define the residual scheme for SolveDCP applied to (3.21), employing the constants from (3.13) and a uniform constant cT := kTk. ResDCP [η, Q, g, py˜˜ , δy , µz˜˜ , δz , v, δv ] → (rη , py˜˜ , δy , µz˜˜ , δz ) determines for any approxi˜, z ˜, µ, ˜ v) of the system (EE) satisfying mate solution quintuple (˜ y, p

z

y ˜ ˜ z y

≤ δz , kw − vk ≤ δv ,

(5.3) − ≤ δy ,

p − p

˜ ˜ µ µ an approximate residual rη such that kg − Qv − rη k ≤ η.

(5.4)

˜, p ˜, z ˜, µ ˜ are overwritten by new approxiMoreover, the initial approximations y ˜, p ˜, z ˜, µ ˜ satisfying (5.3) with new bounds δy and δz defined in (5.6) below, mations y as follows: 17

cL η (i) SolveSPP [ 3C 2 , L, T

(ii) SolveSPP [ η3 , LT ,

f v

˜, p ˜ , δy ] → (yη , pη ); ,y

−ωTT (Tyη −y∗ ) 0

˜, µ, ˜ δz ] → (zη , µη ); ,z

(iii) set rη := µη − ωv; (iv) set ξy :=

1 cL δv + 2 η, cL 3CT

ξz :=

2 CT 2 δv + η, 2 3 cL

(5.5)

˜, p ˜ , δy and z ˜, µ, ˜ δz by and replace y ˜ y ˜ p ˜ z ˜ µ

:= Coarse[4ξy , yη ], := Coarse[4ξy , pη ], := Coarse[4ξp , zη ], := Coarse[4ξy , µη ].

δy := 5 ξy , δz := 5 ξz .

(5.6)

The relation (6.2) below between η and δv in the context of SolveDCP emerges from the complexity analysis given in [14], where also the confirmation of the claimed estimates (5.4) and (5.6) is elaborated and an initialization for SolveDCP for some initial tolerance ε0 in terms of the data kf k and ky∗ k and corresponding initial thresholds δy,0 , δz,0 in terms of ε0 is provided. Finally, the scheme SolveDCP attains the following form with the error reduction factor ρ = ρ(Q) from (3.19) and K given by (4.3) with α from (3.24). SolveDCP [ε, Q, g] → (yε , pε , zε , µε , uε ) (i) Let u0 := 0 and let ε0 , δy := δy,0 , δz := δz,0 be initialized as mentioned above. ˜ := 0, p ˜ := 0, z ˜ := 0, µ ˜ := 0, and set j = 0. Moreover, let y ˜ , pε = p ˜ , zε = z ˜, µε = µ. ˜ (ii) If εj ≤ ε, stop and set uε := uj , yε = y 0 j Otherwise set v := u . (ii.1) For k = 0, . . . , K − 1, compute ResDCP [ρk εj , Q, g, py˜˜ , δy , µz˜˜ , δz , vk , δk ] → (rk ,

˜ ˜ y z ˜ , δy , µ ˜ , δz ), p

where δ0 := εj and δk := ρk−1 (αk + ρ)εj ; set

vk+1 := vk + αrk .

(5.7)

(ii.2) Apply Coarse [ 25 εj , vK ] → uj+1 ; set εj+1 := 12 εj , j + 1 → j and go to (ii). ˜, p ˜, z ˜, µ ˜ at the last stage prior to the termination of SolveDCP , one has δk ≤ ε When overwriting y and η ≤ ε so that the following fact is an immediate consequence of (5.6).

18

Proposition 5.1 The outputs yε , pε , zε , µε which are produced by SolveDCP in addition to uε are approximations to the exact solutions y, p, z, µ of (EE) satisfying 1 cL + ky − yε k, kp − pε k ≤ 5ε , (5.8) 2 cL 3CT 2 CT 2 kz − zε k, kµ − µε k ≤ 5ε . (5.9) + 3 c2L

6

The Complexity of SolveDCP

The scheme SolveDCP ultimately hinges on the availability of suitable schemes Apply for the operators M ∈ {A, AT , B, BT , T, TT }. Conditions on the Apply schemes that ensure τ ∗ – sparsity of ResDCP as formulated in Section 4.2. is that the approximate application of each of these operators has a work/accuracy rate that is within some range comparable to best N –term approximation. Precisely, Apply[·, M, ·] is called τ ∗ –efficient for some 0 < τ ∗ < 2 if for any ∗ finitely supported v ∈ `w τ , for 0 < τ < τ < 2, the output wη of Apply[η, M, v] satisfies kwη k`wτ < ∼ kvk`wτ ,

1/s −1/s #supp wη < kvk`w ∼ η τ

η → 0,

(6.1)

where the constants depend only on τ as τ → τ ∗ and where s is related to τ by (4.11). Moreover, the number of floating point operations needed to compute wη is to remain proportional to #supp wη . The existence of a τ ∗ –efficient scheme for an operator M has the following important implication that follows immediately from Theorem 4.2. Proposition 6.1 If one can find a τ ∗ –efficient scheme for M then M is bounded on `w τ for ∗ every τ > τ . The following has been shown in [14]. Proposition 6.2 If the Apply schemes in SolveSPP for (3.11) and (3.15) are τ ∗ –efficient for some τ ∗ < 2, then ResDCP is τ ∗ –sparse whenever there exists a constant C such that Cη ≥ δv

(6.2) ˜ `wτ , kvk`wτ } ≤ C kyk`wτ + kpk`wτ + kzk`wτ + kµk`wτ + kuk`wτ , max {k˜ yk`wτ , k˜ pk`wτ , k˜ zk`wτ , kµk (6.3) ˜ ˜ ˜ ˜ where v is the current finitely supported input and where y, p, z, µ are the initial guesses for the exact solution components (y, p, z, µ). Finally, we also have the following.

19

Theorem 6.3 Suppose that the Apply schemes appearing in SolveSPP for (3.11) and (3.15) are τ ∗ -efficient for some τ ∗ < 2 and that the solution (y, p, z, µ, u) of (EE) are all in the ∗ respective space `w τ for some τ > τ . Then the approximate solutions yε , pε , zε , µε , uε produced by SolveDCP for any target accuracy ε satisfy kyε k`wτ + kpε k`wτ + kzε k`wτ + kµε k`wτ + kuε k`wτ < ∼ kyk`wτ + kpk`wτ + kzk`wτ + kµk`wτ + kuk`wτ (6.4) and (#supp yε ) + (#supp pε ) + (#supp zε ) +(#supp µε ) + (#supp uε ) 1/s 1/s 1/s 1/s < ε−1/s kyk1/s +kpk + kzk + kµk + kuk , w w w w w ` ` ` ` ` ∼ τ

τ

τ

τ

τ

(6.5) (6.6)

where the constants depend only on τ when τ approaches τ ∗ . Moreover, the number of floating point operations required during the execution of SolveDCP remains proportional to the right hand side of (6.5). Hence, the practical realization of SolveDCP providing optimal work/accuracy rates for a possibly large range of decay rates of the error of best N –term approximation hinges on the availability of τ ∗ –efficient Apply schemes with possibly small τ ∗ for the involved operators. For the examples considered in Section 2.3, T is either the identity mapping or a trace operator to Γy . Since the identity mapping is τ ∗ –efficient for any τ ∗ < 2, we therefore have to discuss the τ ∗ –efficiency of A defined by (3.12) and of the trace operator B (and possibly T). As for A being a second order elliptic differential operator, this is in fact a consequence of the cancellation properties (3.6) of wavelets together with the norm equivalences (3.3) for the relevant function spaces. A detailed description and analysis of a resulting routine Apply for A satisfying (5.1) has been given in [8], and it has been shown that this Apply is τ ∗ -efficient for τ ∗ = (s∗ + 1/2)−1 whenever A is s∗ –compressible. One knows that s∗ is the larger the higher the regularity and the order of cancellation properties of the wavelets are for all second order differential operators considered in Section 2.3. Bounds for s∗ in terms of these quantities for biorthogonal spline wavelets can be found in [3]. In summary, the sparsity of A depends only on the cancellation properties and the regularity of ΨY and not on the particular differential operator. This still applies to some extent for the trace operator B (and for T in case of observations on the boundary) in the classical situation Γ ⊆ ∂Ω (Γy ⊆ ∂Ω) for tensor product domains when using tensor product wavelets. As the entries of B (and T) are inner products of wavelets on Ω with wavelets on the control boundary Γ (or on the observation boundary Γy ), the s∗ – compressibility of large parts of B (and T) in this case hinges on the fact that traces of wavelets are again wavelets on the boundary (except for the generator parts). Thus, at least for large parts of B, a similar conclusion as for A can be drawn in this case. As for the more general situation considered in Section 2.3.2 involving fictitious domains, this is different for the trace operators. For general Γ (Γy ) not aligned with ∂, it has been shown in [26] that B is τ ∗ -efficient only for τ ∗ ≥ 1 which means that B is not bounded on `w τ 20

for τ < 1, i.e., that B is at most s∗ -compressible for s∗ = 1/2 (and the same holds for T.) This is a consequence of the fact that in this case traces of wavelets are in no longer wavelets on Γ in general. Thus, this factor does not have any cancellation properties which help to keep the entries of B small.

Acknowledgments This work has been supported partly by the Deutsche Forschungsgemeinschaft (SFB 611) at the Universität Bonn and the European Community’s Human Potential Programme under contract HPRN–CT–2002–00286 ‘Breaking Complexity’.

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[13] W. Dahmen, A. Kunoth, Appending boundary conditions by Lagrange multipliers: Analysis of the LBB condition, Numer. Math., 88 (2001), 9–42. [14] W. Dahmen, A. Kunoth, Adaptive wavelet methods for linear–quadratic elliptic control problems, Preprint # 46, SFB 611, Universität Bonn, December 2002, submitted for publication, revised November 2003. [15] W. Dahmen, A. Kunoth, K. Urban, Biorthogonal spline wavelets on the interval – Stability and moment conditions, Appl. Comput. Harm. Anal., 6 (1999), 132–196. [16] W. Dahmen, A. Kunoth, K. Urban, Wavelets in numerical analysis and their quantitative properties, in: Surface Fitting and Multiresolution Methods, A. Le Méhauté, C. Rabut and L.L. Schumaker (eds.), Vanderbilt University Press, Nashville, TN (1997), 93–130. [17] W. Dahmen, R. Schneider, Composite wavelet bases for operator equations, Math. Comp., 68 (1999), 1533–1567. [18] W. Dahmen, R. Schneider, Wavelets on manifolds I: construction and domain decomposition, SIAM J. Math. Anal., 31 (1999), 184–230. [19] V. Girault, P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer, 1986. [20] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, 1985. [21] M.D. Gunzburger, H.C. Lee, Analysis, approximation, and computation of a coupled solid/fluid temperature control problem, Comp. Meth. Appl. Mech. Engrg., 118 (1994), 133–152. [22] A. Kunoth, Wavelet Methods — Elliptic Boundary Value Problems and Control Problems, Advances in Numerical Mathematics, Teubner, 2001. [23] A. Kunoth, Fast iterative solution of saddle point problems in optimal control based on wavelets, Comput. Optim. Appl., 22 (2002), 225–259. [24] A. Kunoth, J. Sahner, On the optimized construction of wavelets on manifolds, Manuscript, October 2003. [25] J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971. [26] M. Mommer, Fictitious domain – Lagrange multiplier approach: Smoothness analysis, IGPM Preprint, June 2003. [27] E. Zeidler, Nonlinear Functional Analysis and its Applications; III: Variational Methods and Optimization, Springer, 1985. Angela Kunoth, Institut f¨ ur Angewandte Mathematik, Wegelerstr. 6, Universität Bonn, 53115 Bonn, Germany, E–Mail: [email protected]

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