Discretization of variational regularization in Banach spaces
aa r X i v : . [ m a t h . NA ] A p r Discretization of variational regularization inBanach spaces
Christiane P¨oschl ∗ , Elena Resmerita † , and Otmar Scherzer ‡ November 10, 2018
Abstract
Consider a nonlinear ill-posed operator equation F ( u ) = y where F isdefined on a Banach space X . In general, for solving this equation numer-ically, a finite dimensional approximation of X and an approximation of F are required. Moreover, in general the given data y δ of y are noisy. Inthis paper we analyze finite dimensional variational regularization, whichtakes into account operator approximations and noisy data: We show(semi-)convergence of the regularized solution of the finite dimensionalproblems and establish convergence rates in terms of Bregman distancesunder appropriate sourcewise representation of a solution of the equation.The more involved case of regularization in nonseparable Banach spacesis discussed in detail. In particular we consider the space of finite totalvariation functions, the space of functions of finite bounded deformation,and the L ∞ –space. Key words:
Ill-posed problem, Regularization, Bregman Distance,Strict Convergence
Let F : X → Y be a nonlinear operator with domain D ( F ), where X is aBanach space and Y is a Hilbert space. We would like to approximate solutionsof the ill-posed equation F ( u ) = y (1)via variational regularization. ∗ Dept. of Information & Communication Technologies Universitat Pompeu Fabra C/T´anger 122-140, 08018 Barcelona, Spain † Department of Industrial Mathematics, Johannes Kepler University, Altenbergerstraße 69,A-4040 Linz, Austria, [email protected] ‡ Computational Science Center, University of Vienna, Nordbergstraße 15, A-1090 Vi-enna, Austria and Radon Institute of Computational and Applied Mathematics, Altenberg-erstraße 69, A-4040 Linz, Austria [email protected] R : X → [0 , + ∞ ] be a penalty functional with nonempty domain D ( R ).An element ¯ u ∈ D ( F ) ∩ D ( R ) is called an R -minimizing solution of (1) if itsolves the constraint optimization problemmin R ( u ) subject to F ( u ) = y. (2)We assume that noisy data y δ are given such that (cid:13)(cid:13) y δ − y (cid:13)(cid:13) ≤ δ. (3)In order to solve equation (1) numerically, the space X has to be approxi-mated by a sequence of finite dimensional subspaces X n . The situation whenthe spaces X and Y are Hilbert and the regularization is quadratic has beenanalyzed in [24], [26] (linear problems) and [25],[27] (nonlinear problems). How-ever, recent advances in regularization theory deal with general Banach spaces.Although convergence of regularization methods in the general setting has beenestablished (see, e.g., [29]. [18], a unifying discretization approach is still not athand. In comparison with the Hilbert space theory a significant complicationis due to the fact that a non-separable Banach space cannot be approximatedby a nested sequences of finite dimensional subspaces, with respect to the normtopology. We have in mind the case of the space of bounded variation functionsBV or of bounded deformation functions BD, which are not separable. However,as mentioned in [3, page 121] in the context of BV, “the norm topology is toostrong for many applications”, and in particular also when considering finitedimensional approximations.In this work, we show how ill-posed nonlinear equations can be approximatedby solving associated finite dimensional convex regularization problems. Thecase of nonseparable Banach spaces is particularly emphasized. We proposeto approximate such spaces X by subspaces with respect to a topology whichis weaker than the norm topology. Instead of the norm topology on X (as inthe separable Hilbert space or in the separable Banach space setting) we usea metric on X , which requires to have available an adequate superspace of X .Our investigation covers a large class of not necessarily separable Banach spaceswhich are frequently used for regularizing imaging and other inverse problems.The paper is organized as follows. Section 2 specifies the assumptions, showswell-posedness and convergence of the discretized regularization method in thecase that the Banach space X is nonseparable and the regularization term is anot necessarily convex function. Also, convergence rates with respect to Breg-man distances are obtained in the convex regularization setting, under a stan-dard source condition. The finite dimensional approximation of solutions ofthe equation in separable Banach spaces X is briefly discussed in Section 3,following the analysis done in Section 2. Section 4 studies in some detail thediscretization of several relevant nonseparable Banach spaces such as the spaceof bounded variation functions, the space of bounded deformation functions andthe space of essentially bounded functions. The inverse ground water filtrationproblem is analyzed in the natural setting of L ∞ (Ω), in Section 5.2 The case of nonseparable Banach spaces
Let X be a not necessarily separable Banach space which can be embedded intoa separable Banach space Z . Let τ be a topology on X which is weaker than thenorm topology on X . Therefore, we refer to τ as the weak topology . In additionlet R : X → [0 , + ∞ ] be a proper functional.We define a metric on the space X by the norm of Z induced on X and apseudometric generated by the function R : d ( u, v ) = k u − v k Z + |R ( u ) − R ( v ) | . (4)We shall denote by τ d the topology generated by this metric. Relating to theabove discussion, this is an intermediary topology between the norm topologyon X and the weak topology τ .The following elementary result gives us a motivation for approximating anonseparable Banach space in a topology that is weaker than the norm topology. Proposition 2.1.
A Banach space X is separable if and only if there exists anested (increasing) sequence of finite dimensional subspaces { X n } such that ∪ n ∈ N X n = X, where the closure is considered with respect to the norm topology of X . Consider a sequence of nested, finite dimensional subspaces { X n } whichsatisfies ∪ n ∈ N X nd = X ; (5)That is, X is the closure of the reunion of the subspaces X n with respect to thetopology of the metric d . This property holds for many nonseparable Banachspaces - see several examples in Section 4.We are given approximation operators F m of F , which have the same do-main D ( F ) as F . We assume that the operators F, F m satisfy the followingapproximation property: k F ( u ) − F m ( u ) k ≤ ρ m for all u ∈ D ( F ) ∩ D ( R ) . (6)Here, the constant ρ m should only depend on m and satisfylim m →∞ ρ m = 0 . Denote D n := X n ∩ D ( F ) ∩ D ( R ) , n ∈ N , and assume that the sets D n are nonempty. We are interested in approximating R -minimizing solutions of equation (1) by solutions u α,δm,n ∈ D n of the problemmin n(cid:13)(cid:13) F m ( u ) − y δ (cid:13)(cid:13) + α R ( u ) o subject to u ∈ D n . (7)3n order to pursue the analysis, we make several (standard) assumptions onthe spaces X , Y , the operator F , the functional R (see also ([29, 18, 30]), andon the approximations X n , F m as well: Assumption 2.2.
1. The Banach space X is provided with a topology τ suchthat • The topology τ d is finer than the topology τ . • The norm topology is finer than the topology τ d .2. The domain D ( F ) is τ -closed.3. The operator F : D ( F ) ⊆ X → Y is sequentially τ -weakly closed. Thatis, { u k } ⊂ D ( F ) , u k τ → u and F ( u k ) w → v imply u ∈ D ( F ) and v = F ( u ) .Moreover, the operator F is continuous from D ( F ) ⊂ Z with the normtopology to ( Y, k · k )
4. For every m ∈ N , the operator F m is sequentially τ -weakly closed.5. The function R is sequentially τ - lower semi-continuous.6. For every M > , α > and every m, n ∈ N , the sets { u ∈ X n : k F ( u ) k + α R ( u ) ≤ M } (8) are τ -sequentially relatively compact.7. For every u ∈ X , there exists some u n ∈ X n such that d ( u n , u ) → as n → ∞ . We emphasize again one of the main ideas in this work: When discretizing anonseparable Banach space, one could work with topologies which are weakerthan the original norm topology and which might be more natural than the lat-ter. A well-posedness result for problem (7) can be stated now in this theoreticalsetting.
Proposition 2.3.
Let m, n ∈ N and α, δ > be fixed. Moreover, let assump-tions 2.2 and (6) , (3) be satisfied.Then, for every y δ ∈ Y , there exists at least one minimizer u of (7) .Moreover, the minimizers of (7) are stable with respect to the data y δ inthe following sense: if { y k } k ∈ N converges strongly to y δ , then every sequence { u k } k ∈ N of minimizers of (7) where y δ is replaced by y k has a subsequence { u l } l ∈ N which converges with respect to the topology τ to a minimizer ˜ u of (7) and such that {R ( u l ) } l ∈ N converges to R (˜ u ) , as l → ∞ . Theorem 2.4.
Let assumptions 2.2 be satisfied. Moreover, assume that:( i ) Equation (1) has an R -minimizing solution ¯ u in the interior of D ( R ) ∩D ( F ) , considered in the norm topology;( ii ) v n ∈ D ( F ) for n sufficiently large, where v n ∈ X n and d ( v n , ¯ u ) → as n → ∞ ;( iii ) The parameter α = α ( m, n, δ ) is such that α → , δ α → , ρ m α → and k F ( v n ) − y k√ α → as δ → and m, n → ∞ .If (6), (3) also hold, then every sequence { u k } , with u k := u α k ,δ k m k ,n k and α k := α ( m k , n k , δ k ) where δ k → , n k → ∞ , m k → ∞ as k → ∞ and u k is a solutionof (7), has a subsequence { u l } which converges with respect to the topology τ toan R -minimizing solution ˜ u of equation (1) and such that {R ( u l ) } l ∈ N convergesto R (˜ u ) , as l → ∞ . Moreover, if ¯ u is the unique solution of (1), then the entiresequence { u k } converges to ¯ u in the sense of τ and R as above.Proof. From (4) and Assumption 2.2, Item 7 it follows that R ( v n ) → R (¯ u ) for n → ∞ . From the definition of u α,δm,n , the estimate (3) and (6), it follows that k F m ( u α,δm,n ) − y δ k + α R ( u α,δm,n ) ≤ k F m ( v n ) − y δ k + α R ( v n ) ≤ ( k F m ( v n ) − F ( v n ) k + k F ( v n ) − F (¯ u ) k + k F (¯ u ) − y δ k ) + α R ( v n ) ≤ ( ρ m + k F ( v n ) − y k + δ ) + α R ( v n ) . (10)Therefore, R ( u α,δm,n ) ≤ ( ρ m + k F ( v n ) − y k + δ ) α + R ( v n ) . Assumption ( iii ) now guarantees thatlim sup R ( u α,δm,n ) ≤ R (¯ u ) . (11)Observe that lim n →∞ k F ( v n ) − F (¯ u ) k = 0 since F is continuous at ¯ u (compareAssumption 2.2 Item 3) and (i), (ii)). By (iii), the quantity ( ρ m + k F ( v n ) − F (¯ u ) k + δ ) in (10) converges to zero as δ → m, n → ∞ , and thenlim k F m ( u α,δm,n ) − y δ k = 0 . Therefore, by applying again (6) and (3), we also have that F ( u α,δm,n ) → y, (12)5ith respect to the norm of Y , as δ, α → m, n → ∞ . Consider α k := α ( m k , n k , δ k ) and u k := u α k ,δ k m k ,n k . Note that {k F ( u k ) k + α k R ( u k ) } is bounded.Hence, by (6) and by the compactness hypothesis in Assumption 2.2, there existsa subsequence { u j } j ∈ N which is τ -convergent to some ˜ u ∈ D n . Due to the lowersemicontinuity of R and (11), we get R (˜ u ) ≤ lim inf j →∞ R ( u j ) ≤ lim sup j →∞ R ( u j ) ≤ R (¯ u ) . We also have that F (˜ u ) = y , as j → ∞ due to (12) and Item 4 in Assumption2.2. Therefore, ˜ u is an R minimizing solution of equation (1) and R (˜ u ) = lim j →∞ R ( u j ) . If the solution ¯ u is unique, the only limit point of { u k } with respect to τ and R is ¯ u . Remark 2.5.
In some situations, convergence of a sequence { u k } to u withrespect to the topology τ and such that R ( u k ) → R ( u ) implies d ( u k , u ) → as k → ∞ . This happens for BV and BD which are embedded into Z = L , and τ is chosen as the weak ∗ topology, R is the total variation and total deformationseminorm, respectively - see Section 4. In this section, we establish error estimates for the approximation method weanalyze, with respect to Bregman distances. To this end, assume throughoutthis section that the regularization functional R is convex.Recall that the Bregman distance with respect to a possibly non-smoothconvex functional R is defined by D R ( v, u ) = { D ξ R ( v, u ) : ξ ∈ ∂ R ( u ) = ∅} , u, v ∈ D ( R ) , where D ξ R ( v, u ) = R ( v ) − R ( u ) − h ξ, v − u i . More information about Bregman distances and their role in optimization andinverse problems can be found in [28]. Error estimates for variational or iterativeregularization of (1) by means of a non-quadratic penalty have been shown in[6, 28, 29, 18, 7, 16]. The Bregman distance D R associated with R was naturallychosen as the measure of discrepancy between the error estimates.We assume Frechet differentiability of the operator F around ¯ u which isconsidered to be in the interior of D ( F ) ∩ D ( R ); moreover, assume that itsextension to the space Z is also Frechet differentiable around ¯ u . In fact, ourstudy is based on the following source-wise representation: There exists ω ∈ Y such that ξ = F ′ (¯ u ) ∗ ω ∈ ∂ R (¯ u ) , (13)6nd on the following nonlinearity condition (see also [29]): There exist ε > and c > such that k F ( u ) − F (¯ u ) − F ′ (¯ u )( u − ¯ u ) k ≤ cD R ( u, ¯ u ) , (14) is satisfied for all u ∈ D ( F ) ∩ U ε (¯ u ) with respect to the above subgradient ξ andsuch that c k ω k < . (15)Here D R ( u, ¯ u ) = R ( u ) − R (¯ u ) − h ξ, u − ¯ u i , where ξ ∈ ∂ R (¯ u ) satisfies (13). Let us denote by γ n := k F ′ (¯ u )( v n − ¯ u ) k , (16) λ n := D R ( v n , ¯ u ) . (17)Here { v n } is a sequence as in Theorem 2.4.In the following we derive a relation between the Bregman distance and themetric d at ¯ u . Observe that D R ( v n , ¯ u ) = R ( v n ) − R (¯ u ) − h ω, F ′ (¯ u )( v n − ¯ u ) i . One has R ( v n ) − R (¯ u ) → d ( v n , ¯ u ) →
0, as n → ∞ (see (4)). Moreover, F ′ (¯ u )( v n − ¯ u ) →
0. As a consequence, lim n →∞ λ n = 0. Thus, convergence withrespect to the metric d is stronger than convergence with respect to the relatedBregman distance. Theorem 2.6.
Suppose that Assumption 2.2, the assumptions in the previousresult, inequalities (3), (13) and (14) hold. Moreover, assume that ρ m = O ( δ + λ n + γ n ) , with λ n , γ n given by (17), (16). If α ∼ max { δ, λ n , γ n } , then D R ( u α,δm,n , ¯ u ) = O ( δ + λ n + γ n ) . (18) Proof.
We have k F m ( u α,δm,n ) − y δ k + α R ( u α,δm,n ) ≤ k F m ( v n ) − y δ k + α R ( v n ) ≤ ( k F m ( v n ) − F ( v n ) k + k F ( v n ) − F (¯ u ) k + k F (¯ u ) − y δ k ) + α R ( v n ) ≤ ( ρ m + k F ( v n ) − F (¯ u ) − F ′ (¯ u )( v n − ¯ u ) k + k F ′ (¯ u )( v n − ¯ u ) k + δ ) + α R ( v n ) ≤ ( ρ m + cλ n + γ n + δ ) + α R ( v n ) . Denote β n := ( ρ m + cλ n + γ n + δ ) . (19)7e use (14) and get k F m ( u α,δm,n ) − y δ k + αD R ( u α,δm,n , ¯ u ) ≤ β n + α R ( v n ) − α R (¯ u ) − α h ξ, u α,δm,n − ¯ u i = β n + αD R ( v n , ¯ u ) − α h ξ, u α,δm,n − v n i = β n + αλ n − α h ω, F ′ (¯ u )( u α,δm,n − v n ) i = β n + αλ n − α h ω, F ′ (¯ u )( u α,δm,n − ¯ u ) i + α h ω, F ′ (¯ u )( v n − ¯ u ) i≤ β n + αλ n + αc k ω k D R ( u α,δm,n , ¯ u )+ α k ω kk F ( u α,δm,n ) − F (¯ u ) k + α k ω k γ n . Therefore, k F m ( u α,δm,n ) − y δ k + α (1 − c k ω k ) D R ( u α,δm,n , ¯ u ) ≤ β n + αλ n + α k ω k ( ζ n + γ n ) , (20)where ζ n = k F m ( u α,δm,n ) − y k . Due to (15), the term α (1 − c k ω k ) D R ( u α,δm,n , ¯ u ) isnon-negative. Therefore, k F m ( u α,δm,n ) − y δ k ≤ β n + αλ n + α k ω k ( ζ n + γ n ) . (21)Using (3) we have ζ n ≤ (cid:0) k F m ( u α,δm,n ) − y δ k + k y δ − y k (cid:1) ≤ k F m ( u α,δm,n ) − y δ k + 2 δ . This together with inequality (21) implies ζ n ≤ β n + 2 αλ n + 2 α k ω k ζ n + 2 α k ω k γ n + 2 δ , which yields ζ n ≤ α k ω k + (cid:0) α k ω k + 2 δ + 2 β n + 2 αλ n + 2 α k ω k γ n (cid:1) / . (22)From (20), it follows that α (1 − c k ω k ) D R ( u α,δm,n , ¯ u ) ≤ β n + αλ n + α k ω k ζ n + α k ω k γ n , with ζ n estimated above. Using (19) and taking α ∼ max { δ, λ n , γ n } yield theabove convergence rate. Let X be a separable Banach space. Consider a nested sequence of finite di-mensional subspaces X n , n ∈ N , such that ∪ n ∈ N X n = X, where the closure is considered with respect to the norm topology of X . Byletting Z = X , a weak topology τ on X , a regularization functional R : X → [0 , + ∞ ] and using the assumptions employed for the results in Section 2, oneobtains stability and convergence results similar to Proposition 2.3 and Theorem2.4. Moreover, if R is convex, then convergence rates can also be established.8 emark 3.1. In some situations, convergence of a sequence { u k } to u withrespect to the topology τ and such that R ( u k ) → R ( u ) implies k u k − u k → ,as k → ∞ . This is the case of locally uniformly convex reflexive Banach spaceswhen τ is chosen as the weak topology on X and R = k · k p , with p ∈ (1 , + ∞ ) such as Hilbert spaces, L p spaces, W m,p spaces, but also in L when τ is theweak topology and R is the Shannon entropy (see [5]). Sparsity regularization
Let { φ i } be an orthonormal basis of L (Ω). Denote by X the Banach space ℓ which is identified with the functions with bounded ℓ Fourier coefficients.Let X n be the linear span of the first n Fourier modes.For sparsity regularization one usually takes R ( u ) = P i | u i | , where u = P i u i φ i . Thus, we consider the regularization method of minimizing the func-tional u → k F ( u ) − y δ k + α X i | u i | , In this case, the topology τ on X is the weak topology of ℓ . The regular-ization results apply also in this setting.Another example of regularization term which promotes sparsity is R ( u ) = X i | u i | p , p ∈ (0 , . (23)This setting with X = ℓ and τ taken as the weak topology of ℓ is also coveredby the regularization theory analyzed in this work. Moreover, convergence of asequence { u k } to u with respect to the topology τ and such that R ( u k ) → R ( u )implies convergence of { u k } to u relative to the quasinorm (23), as k → ∞ , cf.[17]. Recall that, for a bounded Lipschitz domain Ω ⊂ R N and for a given N ∈ N ,the space BV (Ω) of L (Ω)-functions of bounded variation mapping Ω into R can be defined as the set of functions w ∈ L (Ω) such that the total variationof w is finite, that is, Z Ω | Dw | p = sup (cid:26)Z w ( x ) ψ ( x ) dx : ψ ∈ C c (Ω) , | ψ ( x ) | p ′ ≤ x ∈ Ω (cid:27) < ∞ . Here, |·| p ′ denotes the l p ′ vector norm where p ′ = p/ ( p −
1) is the conjugateexponent to p . In particular we are interested in the cases p = 1 ,
2, where l p ′ = l ∞ , l . The case p = 2 corresponds to isotropic total variation.Let us recall several properties of the space BV (Ω):9 It is the dual of a separable Banach space (see [3, Remark 3.12]) whenprovided with the norm k u k BV = k u k L + Z Ω | Du | p . • The space BV (Ω) is continuously embedded in L r (Ω), where 1 ≤ r ≤ NN − .We consider the setting X = BV (Ω), with τ being the weak ∗ topology on BV (Ω), and Z = L (Ω).The functional R is the total variation seminorm. Consider d ( u, v ) = k u − v k L (Ω) + (cid:12)(cid:12)(cid:12)(cid:12)Z Ω | Du | p − Z Ω | Dv | p (cid:12)(cid:12)(cid:12)(cid:12) . The metric d is the metric used also in [8], which gives the so-called strictconvergence , according to [3]. A similar idea is developed in [21, 22]. Moreover,the strict convergence of a sequence { u k } to u is equivalent to convergencewith respect to the topology τ together with R Ω | Du k | → R Ω | Du | as k → ∞ ,since weak ∗ convergence of a sequence { u k } to u in BV (Ω) is equivalent toboundedness of {k u k k BV ) } together with convergence of { u k } to u in L (Ω) -see, e.g., Proposition 3.13 in [3].The choice of the vector norm in the definition of the bounded variationseminorm is of special importance for approximating the BV space by subspacesconsisting of piecewise constant functions. Assume that { Ω j } is a decompositionof Ω, and consider the following finite dimensional spaces: X n = u n = n X j =1 u j χ Ω j : u j ∈ R , ≤ j ≤ n . When considering a partition of Ω into uniform parallelepipeds we can onlyguarantee the density assumption (5) when considering the l -norm in the defi-nition of BV - see [9]. If one wants an isotropic behavior of the regularizationterm one has to consider the l norm in the definition of the BV -seminorm.The problem is that in the case of uniform parallelepipeds (for instance pixelsin imaging), there is no convergence with respect to this isotropic BV -seminorm.One has to consider a more general partition of the domain Ω, that allows toapproximate level lines with any direction. One idea is to use an irregular trian-gulation. These observations have been made by [8] and [9], here we only statetheir main result, concerning anisotropic and isotropic total variation: Theorem 4.1 ([8, 9]) . Let Ω ⊂ R n be a polygonal domain and let h > . • Given u ∈ BV (Ω) , then there exists a family { t h : 0 < h ≤ h } of trian-gulations of Ω such that the mesh-size of t h is at most h , and functions u h ∈ A h , where A h denotes the space of piecewise constant functions cor-responding to the triangulation t h , such that k u − u h k L + (cid:12)(cid:12)(cid:12)(cid:12)Z Ω | Du | − Z Ω | Du h | (cid:12)(cid:12)(cid:12)(cid:12) → as h → . Given u ∈ BV (Ω) . Let { Ω j } be a decomposition of Ω , into parallelepipeds ,such that the maximal length of a parallelepiped is smaller then h . Thenthere exist functions u h ∈ V h , where V h denotes the space of piecewiseconstant functions corresponding to the partition { Ω i } , such that k u − u h k L + (cid:12)(cid:12)(cid:12)(cid:12)Z Ω | Du | − Z Ω | Du h | (cid:12)(cid:12)(cid:12)(cid:12) → as h → . Another type of approximation of X is the ’ J µ -approximation property’ em-ployed in [15] which, adapted to our notation reads as follows: X n is a Φ α -approximation of X if, for each u ∈ X , there exists a sequence { u n } ⊂ X n such that k u − u n k Z → α ( u n ) → Φ α ( u ) as n → ∞ , for any α ≥
0, where Φ α is given byΦ α ( u ) = (cid:13)(cid:13) F ( u ) − y δ (cid:13)(cid:13) + α R ( u ) . In [15] one aims at approximating minimizers of Φ α (which depends on α ) in X by minimizers of Φ α in X n for a fixed α > α in X n when theregularization parameter α depends on the dimension n .We summarize the results of this section in an example Example 4.2.
Let X = BV (Ω) with the weak* topology τ . Moreover, let Z = L (Ω) and let d be the metric d ( u, v ) = k u − v k L (Ω) + (cid:12)(cid:12)(cid:12)(cid:12)Z Ω | Du | − Z Ω | Dv | (cid:12)(cid:12)(cid:12)(cid:12) . (24) Let F m and F : D ( F ) ⊆ BV (Ω) → L ( ˆΩ) satisfy the related conditions inAssumption 2.2.Then according to Theorem 4.1, for every u ∈ BV (Ω) there exists an approx-imating sequence of piecewise constant functions. Consequently, minimizationof the discretized regularized problem is well–posed, stable, and convergent (cf.Proposition 2.3). The piecewise constant regularizers { umn } approximate the R -minimizing solution ¯ u in the sense of the metric (24). In the following let Ω = (0 , N the open unit cube. We choose the simplegeometry not to be forced to take into account approximations of Ω, or irregularmeshes, for the finite element method considered below.The space BD (Ω) [33] of functions with bounded deformation in an openset Ω ⊂ R N is defined as the set of functions u = ( u , . . . , u N ) ∈ L (Ω; R N )such that the symmetric distributional derivative E ij u := 12 ( D i u j + D j u i )11s a (matrix-valued) measure with finite total variation in Ω: BD (Ω) := n u ∈ L (Ω; R N ) , E ij ( u ) ∈ M (Ω) , i, j = 1 , . . . , N o , where M (Ω) denotes the space of bounded measures. BD (Ω) is a nonseparableBanach space provided with the norm k u k BD = k u k L (Ω; R N ) + N X i,j Z Ω | E ij ( u ) | | {z } =: R Ω | E u | . This space is strictly larger than the space of bounded variation functions BV (Ω; R N ). It was introduced in [31] and [23] and has been widely consideredin the literature (see [32, 33]) in connection with the mathematical theory ofplasticity. Several interesting properties of the BD space are as follows: It isthe dual of a separable Banach space (see [33]); the space BD (Ω) is contin-uously embedded in L p (Ω; R N ), where 1 ≤ p ≤ NN − . In addition the space BD (Ω; R N ) is not separable – if it would be, the space BV (Ω; R N ), whichis the subspace of BD (Ω; R N ) where all components u j , j = 2 , . . . , N vanish,would be as well. However, this is not true as stated already in the previousexample.Let us return to the finite dimensional regularization framework we investi-gate in this work.Consider the setting X = BD (Ω), τ the weak ∗ topology on BD (Ω) and Z = L (Ω; R N ). We associate with n ∈ N the discretization size h n := n , usemulti-indices α := ( α , . . . , α N ), and set A := { , . . . , n } N .We consider the finite-dimensional product space of piecewise linear splines X n := ( u n = X α ∈ A N X k =1 u k α ∆ (cid:18) x − ξ α h n (cid:19) : u k α ∈ R ) where ξ α ∈ h n { , , . . . , n } N , and ∆ is the following function∆( x ) := N Y i =1 max(0 , − (cid:12)(cid:12) x i (cid:12)(cid:12) ) . This finite element discretization has already been used in [10] for numericalminimization of variational energies. For x ∈ (0 , h n ) N the derivative of ∆(( x − ξ α ) /h n ) in direction e j is given by D j ∆ (cid:18) x − ξ α h n (cid:19) = sign ( ξ j α − x j ) 1 h n N Y i = j max , − (cid:12)(cid:12) x i − ( ξ α ) i (cid:12)(cid:12) h n !! . For every α with α i < n and for every k, ≤ k ≤ N define A k α := { β ∈ A α , β k = α k } .Additionally define Ω α as the N − dimensional cube, spanned by the vectors { ξ α + e k h n } k =1 ...N , and A α = S Nk =1 A k α (see Figure 1).12 α ξ α +(1 , ξ α +(0 , ξ α +(1 , Ω α A α = { α , α + (0 , , α + (1 , , α + (1 , } A α = { α , α + (0 , } A α = { α , α + (1 , } Figure 1: an example for the sets A α Moreover, for β ∈ A α we have Z Ω α D j ∆ (cid:18) x − ξ β h n (cid:19) d x = ( − (cid:0) h n (cid:1) N − if β j = α j + (cid:0) h n (cid:1) N − if β j = α j + 1 (25)In the following we prove the main result on a pseudometric. Theorem 4.3.
We assume that h n → when n → ∞ . Then for every u ∈ BD (Ω , R N ) ∩ L r (Ω , R N ) , ≤ r < ∞ , we can find a sequence { u n } , with u n ∈ X n , such that lim Z Ω | u − u n | r dx = 0 and lim n →∞ Z Ω | E u n | = Z Ω | E u | . Setting d ( u , u ) = k u − u k L (Ω; R N ) + (cid:12)(cid:12)(cid:12)(cid:12)Z Ω | E u | − Z Ω | E u | (cid:12)(cid:12)(cid:12)(cid:12) , (26) we obtain lim n →∞ d ( u n , u ) = 0 . In order to prove this theorem, we need some additional facts on BD , givenby the following Lemmas. Lemma 4.4.
For every n ∈ N the inclusion X n ⊂ BD (Ω , R N ) holds and foreach u = ( u , . . . , u N ) ∈ X n , Z Ω | E u | = X α N X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X β ∈ A kα (cid:0) u k β + e k − u k β (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18) h n (cid:19) N − + X α N X k = l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X β ∈ A lα (cid:0) u k β + e l − u k β (cid:1) + X β ∈ A kα (cid:0) u l β + e k − u l β (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18) h n (cid:19) N − Proof.
From the definition of u ∈ X m we obtain D l u k ( x ) = X α u k α D l ∆ (cid:18) x − ξ α h n (cid:19) . S α : α i 1. If { u n } ⊂ BD (Ω) and u n → u in ( L (Ω)) N , then Z Ω | E ij u | ≤ lim inf n →∞ Z Ω | E ij u n | . 2. For every u ∈ BD (Ω) ∩ L r (Ω) , ≤ r < ∞ , there exists a sequence { u n } ⊂C ∞ (Ω) such that lim n →∞ Z | u − u n | r dx = 0 lim n →∞ Z Ω | E ij u n | = Z Ω | E ij u | . Proof. 1. Follows from standard properties of convex measures.2. See [32, Theoreme 3.2, Chapitre II].Now we are ready for the proof of Theorem 4.3. Theorem 4.3. The L p -convergence of the picewise polinomial functions u n canbe found in the book of Ciarlet [12]. Due to Lemma 4.5, we can assume that u ∈ C ∞ (Ω). Set the coefficient of u n as ( u n ) k α := u k ( ξ α ), then we have u n = N X k =1 X α u k ( ξ α )∆ (cid:18) x − ξ α h n (cid:19) . From Lemma 4.4 it follows that Z Ω α | E kl u n | = Z Ω α (cid:12)(cid:12) D l u kn + D k u ln (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X β ∈ A lα (cid:0) u k ( ξ β + e l ) − u k ( ξ β ) (cid:1) + X β ∈ A kα (cid:0) u l ( ξ β + e k ) − u l ( ξ β ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18) h n (cid:19) N − (27)14ext we use the mean value theorem: For m, n ∈ { l, k } , m = n we can findpoints η β ,m,n between ξ β and ξ β + e m such that u k ( ξ β + e l ) − u k ( ξ β ) = h n D l u k ( η β ,k,l ) ,u l ( ξ β + e k ) − u l ( ξ β ) = h n D k u l ( η β ,l,k ) , hence from (27) it follows that Z Ω α | E kl u n | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X β ∈ A lα D l u k ( η β ,k,l ) + X β ∈ A kα D k u l ( η β ,l,k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h Nn |{z} | Ω α | (cid:18) (cid:19) N − Summing over all α and l, k = 1 , . . . N and taking the limit h n → n →∞ Z Ω | E ij u n | = Z Ω | E ij u | . We summarize the results of this section as follows: Example 4.6. Let X = BD (Ω) and τ the weak* topology. Moreover, let Z = L (Ω , R N ) and let d be the metric d ( u , v ) = k u − v k L (Ω , R N ) + (cid:12)(cid:12)(cid:12)(cid:12) Z Ω | E u | − Z Ω | E v | (cid:12)(cid:12)(cid:12)(cid:12) . (28) Let F m and F : D ( F ) ⊆ BV (Ω) → L ( ˆΩ) satisfy the related conditions inAssumption 2.2.Take h = 1 /n . Then according to Theorem 4.3, for every u ∈ BD (Ω , R N ) ∩ L r (Ω , R N ) , ≤ r < ∞ there exists an approximating sequence of piecewiseconstant functions in the sense of metric d . Consequently, minimization of thediscretized regularized problem is well–posed, stable, and convergent accordingto Proposition 2.3. The piecewise constant regularizers { u α,δm,n } approximate the R -minimizing solution u in the sense lim u α,δm,n = u in the weak star topologyand lim R Ω (cid:12)(cid:12) E u α,δm,n (cid:12)(cid:12) = R Ω | E u | . L ∞ space In this section we analyze the following regularization method u → k F ( u ) − y δ k + α k u k ∞ . In the sequel we show that there exist finite dimensional subspaces { X n } of L ∞ (Ω) satisfying equality (5). More precisely, there exist finite dimensionalsubspaces { X n } of L ∞ (Ω) such that for any u ∈ L ∞ (Ω) one can find u n ∈ X n , n ∈ N with lim n →∞ ( k u n − u k L p + |k u n k ∞ − k u k ∞ | ) = 0 , p ∈ [1 , + ∞ ) . (29)15t is known that L ∞ (Ω) is not separable, while L p (Ω), for every p ∈ [1 , + ∞ ),is separable. However, every function in L ∞ (Ω) can be approximated uniformlyand thus, in the L ∞ (Ω) - norm by a sequence of simple functions. The proof ofthis statement is constructive. However, it provides a nonlinear approximation,in the sense that the piecewise constant functions which approximate the L ∞ (Ω)function do not yield linear subspaces - see, e.g., [13, Section 3.2]. This doesnot fit the theoretical framework we consider here. An alternative is to considerapproximations of L ∞ (Ω) functions by piecewise constant functions in a weakertopology, as shown in the sequel.For the sake of simplicity let Ω = (0 , N . Assume that { Ω j } is a decom-position of Ω in parallelepipeds with maximal diagonal length h n as in [9], andconsider the following finite dimensional subspaces of L ∞ (Ω): X n = u n = n X j =1 u j χ Ω j : u j ∈ R , ≤ j ≤ n . The following results are essential in proving the main statement of thissection:First denote by J ǫ ∗ u the mollification of u , for every ǫ > Theorem 4.7. [p. 36 [2]] Let u be a function which is defined on R N andvanishes identically outside Ω .If u ∈ L p (Ω) , then J ǫ ∗ u ∈ C ∞ ( R N ) , in fact J ǫ ∗ u ∈ C ∞ (Ω) and J ǫ ∗ u ∈ L p (Ω) , for every p ∈ [1 , + ∞ ) . Also, lim ǫ → + k J ǫ ∗ u − u k p = 0 , (30) k J ǫ ∗ u k p ≤ k u k p , for all ǫ > . (31) Lemma 4.8. For every n ∈ N , one has X n ⊂ L ∞ (Ω) and, for each u n ∈ X n , k u n k ∞ = max ≤ j ≤ n | u j | . Proof. Follows immediately from the definition of u n , taking into account that µ (Ω j ) > 0, for 1 ≤ j ≤ n . Lemma 4.9. If { u n } ⊂ L ∞ (Ω) and u ∈ L ∞ (Ω) are such that k u − u n k L p → as n → ∞ , for some p ∈ [1 , + ∞ ) and k u n k ∞ ≤ k u k ∞ for every n ∈ N , then { u n } converges weakly ∗ to u and k u n k ∞ → k u k ∞ as n → ∞ .Proof. Since {k u n k ∞ } is bounded, there exists a subsequence { u k } which con-verges weakly ∗ to some v ∈ L ∞ (Ω), cf. Alaoglu-Bourbaki Theorem, [19, p.70]. By the definition of weak ∗ convergence, one obtains that { u k } convergesalso weakly, with respect to L p (Ω) to some v . Therefore u = v . In fact, everysubsequence of { u n } converges weakly ∗ to u , which yields weak ∗ convergenceof the entire sequence { u n } to u . In addition, the weak ∗ lower semi continuityof the L ∞ -norm implies k u k ∞ ≤ lim inf n →∞ k u n k ∞ . Thus the assertions areproved. 16ote that the previous result can be relaxed by assuming only weak conver-gence of { u n } in L p (Ω). Lemma 4.10. For every u ∈ C ∞ ( ¯Ω) , there exists a sequence { u n } with u n ∈ X n such that lim n →∞ ( k u n − u k L p + |k u n k ∞ − k u k ∞ | ) = 0 , p ∈ [1 , + ∞ ) . Proof. Let ξ j be the gravity centers of the parallelepipeds Ω j . Define u n = n X j =1 u ( ξ j ) χ Ω j . Then k u n − u k L p → k u n k ∞ = max ≤ j ≤ n | u ( ξ j ) | ≤ max x ∈ ¯Ω | u ( x ) | = k u k ∞ , for all n ∈ N , where the last equality holds due to the continuity of u on ¯Ω. Thus, Lemma 4.9applies and yields |k u n k ∞ − k u k ∞ | → n → ∞ . Theorem 4.11. Assume that h n → when n → ∞ . Then for every u ∈ L ∞ (Ω) one can find u n ∈ X n such that (29) holds.Proof. Let u ∈ L ∞ (Ω) and p ∈ [1 , + ∞ ). By Theorem 4.7, there exists { u j } ⊂ C ∞ ( R N ), in fact in C ∞ ( ¯Ω), with { u j } ⊂ L p (Ω) such thatlim j →∞ k u j − u k p = 0 , and k u j k p ≤ k u k p , for all j ∈ N . By letting p → + ∞ in the last inequality and using Theorem 2.8 on p. 25 in[1], one also has k u j k ∞ ≤ k u k ∞ , for all j ∈ N . Lemma 4.9 applies and yields k u j k ∞ → k u k ∞ as j → ∞ .By consequence, every function in L ∞ (Ω) can be approximated in the sensestated at (29) by functions from C ∞ (Ω). Since every function v ∈ C ∞ ( ¯Ω) canbe approximated by u n ∈ X n as in (29) due to Lemma 4.10, the conclusionfollows immediately. Remark 4.12. One can define the subspaces X n in the previous theorem alsoby means of piecewise polynomial functions of degree no bigger than one in eachvariable, which are continuous on ¯Ω . Moreover, one can employ n -simplicesinstead of parallelipipeds and piecewise linear functions, according to Remark3.8 in [9]. We summarize the results of this section as follows:17 xample 4.13. Let X = L ∞ (Ω) and τ the weak* topology. Moreover, let Z = L p (Ω) , p ∈ (1 , + ∞ ) and let d be the metric d ( u, v ) = k u − v k L p (Ω) + |k u k ∞ − k v k ∞ | . (32) Let F m and F : D ( F ) ⊆ BV (Ω) → L ( ˆΩ) satisfy the related conditions inAssumption 2.2.Take h = 1 /n . Then according to Theorem 4.11, for every u ∈ L ∞ (Ω) thereexists an approximating sequence of piecewise constant functions in the sense ofmetric d . Consequently, minimization of the discretized regularized problem iswell–posed, stable, and convergent according to Proposition 2.3. The piecewiseconstant regularizers { u α,δm,n } approximate the R -minimizing solution ¯ u in thesense lim u α,δm,n = ¯ u in the weak star topology and lim k u α,δm,n k ∞ = k ¯ u k ∞ . This convergence described in the previous sentence is weaker than con-vergence with respect to the metric (32). The fact that these two types ofconvergence are not equivalent, by contrast to the BV and BD cases with thetwo corresponding convergence types, is shown by the following counterexample.Consider the Rademacher functions f n : [0 , → {− , } , f n ( t ) = ( − i +1 if x ∈ (cid:20) i − n , i n (cid:19) , ≤ i ≤ n . This sequence converges weakly star to zero in L ∞ ([0 , L ([0 , g n : [0 , → R , g n ( t ) = f n ( t ) , if t ∈ [0 , g n ( t ) = 1 for t ∈ [1 , { g n } converges weakly star to χ [1 , , thecharacteristic function of [1 , 2] in L ∞ ([0 , L ([0 , k g n k ∞ = k χ [1 , k ∞ = 1. Thus, { g n } is a sequence in L ∞ ([0 , χ [1 , and such that lim n →∞ k g n k ∞ = k χ [1 , k ∞ , butlim n →∞ (cid:0)(cid:13)(cid:13) g n − χ [1 , (cid:13)(cid:13) L + |k g n k ∞ − k χ [1 , k ∞ | (cid:1) = 0. Remark 4.14. Given a direct problem formulated in L ∞ . Is it worth to formu-late the inverse problem in L ∞ or is more appropriate to formulate the problemin L , where we can get also L -approximations? L approximations are quiteadvantageous because in the L ∞ case, one might not even get convergence withrespect to the L -norm, as demonstrated by the above counterexample. However,it is sometimes desirable that the regularized solutions are guaranteed to belongalso to the L ∞ space. Moreover, the extremal behavior of the solution can beevaluated by L ∞ regularization, since convergence of the L ∞ -norm of the regu-larized solutions to the L ∞ norm of the true solution is achieved. Also, in someinverse problems high interest is given to estimating a linear functional of thesolution rather than the solution - according to the mollifier idea in [4], whichcorresponds to the weak-star approximation of our L ∞ regularization results. The inverse ground water filtration problem We consider the problem of recovering the diffusion coefficient in − ( au x ) x = f in Ω ,u (0) = 0 = u (1) , with f ∈ L [0 , F is defined as the parameter-to-solution map-ping F : D ( F ) := { a ∈ L ∞ [0 , 1] : a ( x ) ≥ c > , a.e. } → L [0 , ,a F ( a ) := u ( a ) , where u ( a ) is the unique solution of the above equation and c is a constant.More details about this ill posed problem can be found in [20] and [11, Chapter1]. Note that D ( F ) is a subset of the interior of the nonnegative cone of L ∞ [0 , L ∞ is the natural function space when formulating this problem. How-ever, to the best of our knowledge, previous literature dealing with the problemfrom the regularization viewpoint has usually considered D ( F ) in H which isembedded in L ∞ , mainly due to the Hilbert space setting which is enforced byusing H . The operator F is Fr´echet differentiable from L ∞ to L - see [20].Note that D ( F ) is closed and convex with respect to L , so it is weakly closedin L . This implies that D ( F ) is weakly ∗ closed in L ∞ . Similarly one can arguethat the operator F is sequentially weakly ∗ -weakly closed.In the sequel we are going to employ approximation operators F m as in [25].Let Y m be the space of linear splines on a uniform grid of m + 1 points in [0 , u m ( a ) ∈ Y m be the unique solution of( a ( u m ) x , v x ) L = ( f, v ) L for all v ∈ Y m . The operators F m are defined as F m : D ( F ) := { a ∈ L ∞ [0 , 1] : a ( x ) ≥ c > , a.e. } → L [0 , ,a F m ( a ) := u m ( a ) , Then (6) holds for ρ m = m − cf. [12, Theorems 3.2.2, 3.2.5], k F m ( a ) − F ( a ) k L = k u m ( a ) − u ( a ) k L = O ( k a k L ∞ · m − ) . Then by choosing the discretization of L ∞ [0 , 1] as in Section 4, one obtainsthe convergence results of the discretized regularization method as in Example4.13. 19 cknowledgment The authors thank Prof. J.B. Cooper (Kepler University, Linz) for provid-ing them the counterexample in Section 4. They are grateful to S. Pereverzev(Radon Institute, Linz) for helpful discussions, and to A. Rieder (Karlsruhe Uni-versity) and V. Vasin (Institute of Mathematics and Mechanics, Ekaterinburg)for interesting references. Also, they acknowledge the support by the AustrianScience Fund (FWF) within the national research networks Industrial Geometry,project 9203-N12, and Photoacoustic Imaging in Biology and Medicine, projectS10505-N20 (C. P. and O. S.) and by an Elise Richter scholarship, project V82-N18 (E. R.). References [1] R. Adams. Sobolev Spaces . Academic Press, New York, 1975.[2] R. Adams and J. Fournier. Sobolev Spaces . Academic Press, New York,2003.[3] L. Ambrosio, N. Fusco, and D. Pallara. Functions of Bounded Variationand Free Discontinuity Problems . Oxford Mathematical Monographs. TheClarendon Press Oxford University Press, New York, 2000.[4] R.S. Anderssen. The linear functional strategy for improperly posed prob-lems. Inverse problems, Oberwolfach , pages 11–30, 1986.[5] J. Borwein and A. Lewis. Convergence of best entropy estimates. SIAM J.Optim. , 1:191–205, 1991.[6] M. Burger and S. Osher. Convergence rates of convex variational regular-ization. 20(5):1411–1421, 2004.[7] M. Burger, E. Resmerita, and L. He. Error estimation for Bregman itera-tions and inverse scale space methods in image restoration. 81(2–3):109–135, 2007. Special Issue on Industrial Geometry.[8] P. Bˇel´ık and M. Luskin. Approximation by piecewise constant functionsin a BV metric. Math. Models Methods Appl. Sci. , 13(3):373–393, 2003.Dedicated to Jim Douglas, Jr. on the occasion of his 75th birthday.[9] E. Casas, K. Kunisch, and C. Pola. Regularization by functions of boundedvariation and applications to image enhancement. Appl. Math. Optim. ,40(2):229–257, 1999.[10] A. Chambolle. An approximation result for special functions with boundeddeformation. Journal de Math´ematiques Pures et Appliqu´es , 83(7):929–954,2004. 2011] G. Chavent. Non Linear Least Squares For Inverse Problems. ScientificComputation . Springer, 2009.[12] P. Ciarlet. The finite element method for elliptic problems . Number 40 inSIAM Classics in Applied Mathematics. SIAM, Philadelphia, 2002.[13] R.A. DeVore. Nonlinear approximation. Acta Numerica , 7:51–150, 1998.[14] H. Kunisch K. Engl, A. Neubauer, and K. Kunisch. Convergence rates fortikhonov regularisation of non-linear ill-posed problems. Inverse Problems ,5(4):523, 1989.[15] B.G. Fitzpatrick and S.L. Keeling. On approximation in total variationpenalization for image reconstruction and inverse problems. 1997.[16] K. Frick and O. Scherzer. Regularization of ill-posed linear equations bythe non-stationary augmented lagrangian method. J. Integral EquationsAppl. , to appear, 2009.[17] M. Grasmair, M. Haltmeier, and O. Scherzer. Sparse regularization with l q penalty term. Inverse Problems , 24(5):055020, 13, 2008.[18] B. Hofmann, B. Kaltenbacher, C. P¨oschl, and O. Scherzer. A convergencerates result in Banach spaces with non-smooth operators. Inverse Problems ,23(3):987–1010, 2007.[19] R.B. Holmes. Geometric Functional Analysis and Its Applications .Springer, 1975.[20] K. Ito and K. Kunisch. On the injectivity and linearization of thecoefficient-to-solution mapping for elliptic boundary value problems. J.Math. Anal. Appl. , 188:1040–66, 1994.[21] A.S. Leonov. Regularization of ill-posed problems in sobolev space w . J.Inverse Ill-Posed Probl. , 13:595—-619, 2005.[22] A.S. Leonov. Total variation convergence of regularizing algorithms forsolving ill-posed problems. Zh. Vychisl. Mat. Mat. Fiz. , 47:732–747, 2007.[23] H. Matthies, G. Strang, and E. Christiansen. The saddle point of a differen-tial program. In Energy methods in finite element analysis , pages 309–318.Wiley, Chichester, 1979.[24] A. Neubauer. Tikhonov regularization for non-linear ill-posed problems:optimal convergence rates and finite-dimensional approximation. InverseProblems , 5(4):541–557, 1989.[25] A. Neubauer and O. Scherzer. Finite-dimensional approximation oftikhonov regularized solutions of nonlinear ill-posed problems. Numer.Funct. Anal. Optim. , (11):85–99, 1990.2126] R. Plato and G. Vainikko. On the regularization of projection methods forsolving ill-posed problems. Numer. Math. , 28:63–79, 1990.[27] J. Qinian. Applications of the modified discrepancy principle to tikhonovregularization of nonlinear ill-posed problems. SIAM J. Numer. Anal. ,36:475–490, 1999.[28] E. Resmerita. Regularization of ill-posed problems in Banach spaces: con-vergence rates. Inverse Problems , 21(4):1303–1314, 2005.[29] E. Resmerita and O. Scherzer. Error estimates for non-quadratic regular-ization and the relation to enhancement. Inverse Problems , 22(3):801–814,2006.[30] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen. Variational Methods in Imaging . Applied Mathematical Sciences. Springer,2008.[31] P.-M. Suquet. Existence et r´egularit´e des solutions des ´equations de laplasticit´e. C. R. Acad. Sci. Paris S´er. A-B , 286(24):A1201–A1204, 1978.[32] R. Temam. Probl`emes Math´ematiques en Plasticit´e , volume 12 of M´ethodesMath´ematiques de l’Informatique [Mathematical Methods of InformationScience] . Gauthier-Villars, Montrouge, 1983.[33] R. Temam and G. Strang. Functions of bounded deformation.