ERROR CONTROL AND ADAPTIVITY FOR A VARIATIONAL MODEL PROBLEM DEFINED ON FUNCTIONS OF BOUNDED VARIATION

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1 ERROR CONTROL AND ADAPTIVITY FOR A VARIATIONAL MODEL PROBLEM DEFINED ON FUNCTIONS OF BOUNDED VARIATION SÖREN BARTELS Abstract. We derive a fully computable, optimal a posteriori error estimate for te finite element approximation of a total variation regularized model problem and devise an adaptive refinement strategy. Numerical experiments reveal a significant improvement over related approximations on uniformly refined triangulations. 1. Introduction A simple model problem in te calculus of variations defined on te space of functions of bounded variation seeks a function u : R tat minimizes te functional Eu = Du + α u g L wit a given function g L, a parameter α > 0, and a bounded Lipscitz domain R d, d =, 3. Te first term is te total variation of te distributional derivative Du. It coincides wit te semi-norm in W 1,1 if u belongs to tis space and is finite if Du is a Radon measure wit bounded total variation, i.e., u BV. Te minimization of te functional E as been proposed in [ROF9] as a simple model in image processing, in wic g is a given noisy image and te minimizer u serves as a smooter reconstruction. Te function u may ave discontinuities, e.g., if u is piecewise constant ten Du coincides wit te perimeter of te discontinuity set. Closely related energy functionals occur in te modeling of perfect plasticity [Suq78, BMR1] and te description of material damage [To11]. Te metods discussed in tis paper transfer to minimization problems tat ave te structure of te sum of te total variation norm plus a uniformly convex lower order term. Suc problems often occur in te implicit time-discretization of gradient flows. Te finite element discretization of te minimization problem and te iterative solution of te resulting finite dimensional problems are now well understood: if for every > 0 te set V BV is a finite element space suc tat te spaces V W 1,1 define a dense family of subspaces in W 1,1 ten te restriction of E to V leads to a Γ-convergent approximation. Tis is true if V contains piecewise affine, globally continuous finite element functions but not for te space of piecewise constant functions on a nested sequence of triangulations, cf. [Bar1] for details. Te resulting discrete problems can be solved effectively wit primal-dual metods recently developed and analyzed in [CP11, Bar1]. For a discretization wit piecewise affine functions on a triangulation wit maximal mes-size > 0 it can be sown tat if R is star-saped and g L ten te exact and discrete minimizers u and u are related by u u L c 1/4 so tat a large number of degrees of freedom is required to guarantee a small error wit respect to te norm in L. Tis may be suboptimal and te Date: July 17, Matematics Subject Classification. 65N30, 65N50. Key words and prases. Total variation, functions of bounded variation, finite elements, adaptivity, error estimation. 1

2 bound does not reflect te special structure of te exact solution wic is often discontinuous but piecewise smoot wit a simple jump set. Terefore, te simultaneous accurate resolution of te lower dimensional interface tat seperates regions in wic u is smoot and te approximation of te piecewise smoot functions can benefit from a treatment wit different scales. Wile tis is difficult to realize on te basis of a priori information, a posteriori error estimates tat control te approximation error in terms of computable quantities can realize tis goal automatically. An abstract approac to te a posteriori error control for minimization problems wit functionals J of te form Ju = F u + GΛu for a bounded linear operator Λ : V Q, and proper, convex, lower-semicontinuous functionals F : V R and G : Q R for Banac spaces V and Q and R = R {+ } as been developed in [Rep00]. It employs te dual formulation tat consists in te minimization of te functional J q = F Λ q + G q or more precisely te maximization of q J q wit te Fencel conjugates F and G of F and G. Provided tat F or G as some coercivity properties te result controls te approximation error u u by te primal-dual gap wit an arbitrary admissible function q, e.g., if F is quadratic ten 1.1 γ u u F u + F Λ q + GΛu + G q = Ju + J q. Te estimate can only be efficient if te primal and dual formulation satisfy a strong duality principle, i.e., te solutions u and p of te primal and dual problem satisfy Ju = J p. We will provide a refined, optimal version of te estimate 1.1. We remark tat u and q in 1.1 are arbitrary admissible functions for te primal and dual problem, respectively. In particular, no exact solution of te discretized primal problem is required. Owing to te lack of reflexivity of te Banac space BV te dual of te minimization problem defined by te functional E : BV R is difficult to caracterize. It as owever been sown in [HK04] tat te problem itself is te Fencel-dual of te minimization problem defined by te functional Dp = 1 α div p + αg L α g L + I K 1 0p on te space H N div; = {q L ; R d : div q L, q ν = 0 on }, were ν is te outer unit normal on, and wit te indicator functional I K1 0 tat vanises for vector fields satisfying p 1 in. Tis important observation implies tat strong duality olds and tus tat te error estimate 1.1 is sarp in te sense tat te rigt-and side vanises if u = u and q = p. Notice tat minimizers of D are non-unique in general. Te abstract error estimate 1.1 is closely related to recovery error estimators for elliptic problems. In particular, for te Poisson problem it is known tat a simple averaging of te discrete flux u leads to reliable and efficient error control, i.e., up to iger order terms it is possible to sow tat, u u L c u A u L wit c = 1 in some situations and a converse estimate also applies, cf. [Bra07, CB0]. Te estimate 1.1 actually implies tis upper bound wit c = 1 if div A u = f and tereby justifies a certain locality property of te rigt-and side in 1.1. Te proofs of te estimates for te Poisson problem in [Bra07, CB0] make essential use of te quadratic structure of te Diriclet functional and cannot be transferred to te problem of minimizing te non-smoot energy functional E. Surprsingly, even for simple Helmoltz type problems tat fit into te above framework simple averaging does not lead to efficient error control on unstructured triangulations.

3 To obtain a good bound on te approximation error, i.e., to find a good function q in 1.1, we will consistently discretize te dual formulation and solve it iteratively. Tis poses several difficulties. First, a subspace of H N div; as to be cosen in wic te constraint p 1 can be imposed efficiently. Second, te discretization of te dual problem as to be done in suc a way tat it can be solved reliably wit a computational effort comparable to te solution of te primal problem. We will sow tat lowest order H 1 conforming elements allow to establis bot requirements. Te rest of tis article is organized as follows. Notation and some auxiliary results are specified and stated in Section. In Section 3 we give a refined version of Repin s abstract a posteriori error estimate for convex optimization problems. Te predual problem for te minimization problem defined by E and its discretization and iterative solution will be discussed in Section 4. Numerical experiments tat illustrate te performance of te error estimate and te induced refinement indicators will be presented in Section 5.. Preliminaries We include in tis section some elementary facts about finite element spaces and a result on te approximation of te model problem..1. Function spaces. We use standard notation for Lebesgue and Sobolev spaces and abbreviate te inner product and te norm in L by, and, respectively. Te Banac space BV consists of all functions v L 1 for wic, wit te space of compactly supported, continuously differentiable vector fields Cc 1 ; R d, Dv = sup v div q dx < q Cc 1;Rd, q 1 and is equipped wit te norm v BV = v L 1 + Dv. We let H Ndiv; denote te space of all q L ; R d for wic div q L wit norm q HN div; = q + div q 1/... Discrete time derivatives. Given a time-step size τ > 0 and a sequence of functions or real numbers v n n N in an inner product space X we define d t v n+1 = v n+1 v n /τ and notice tat for every v X we ave d t v n+1 v n+1 v = d t v vn+1 + τ d tv n+1. We also note tat for sequences a n n N and b n n N we ave te summation by parts formula τ N n=0 { dt a n+1 b n+1 + a n d t b n+1 } = a N+1 b N+1 a 0 b Finite element spaces. For a sequence of regular triangulations T >0 of into triangles or tetraedra wit maximal diameters = max T T diamt we define te finite element spaces L 0 T = {q L 1 : q T is constant for eac T T }, S 1 T = {v C : v T is affine for eac T T }. Wit te set N of vertices of triangles or tetraedra te nodal basis of S 1 T is defined by te functions ϕ z : z N S 1 T wic satisfy ϕ z y = 0 for distinct y, z N and ϕ z z = 1 for all z N. An elementwise inverse estimate sows tat tere exists c > 0 suc tat div q c 1 min q wit min = min T T diamt for all q S 1 T d. Te nodal interpolant of a function v C is defined by I v = z N vzϕ z. A discrete inner product tat is equivalent to, on S 1 T d is for q, r S 1 T d defined by q, r = I [q r ] dx = β z q z r z z N 3

4 wit β z = ϕ z dx for all z N. We let q = q, q 1/ denote te corresponding norm. Te L -projection P : L S 1 T is caracterized by P v v, w = 0 for all w S 1 T. We recall tat te lowest order Raviart-Tomas finite element space RT 0 T consists of all q Hdiv; wit q T x = a + bx for a R, b R, and all x T T. We define RTN 0 T = RT 0 T H N div;..4. Duality and optimality. As above we consider lower semicontinuous, proper, convex functionals F : V R and G : Q R for R = R {+ } on Banac spaces V and Q wit duals V and Q and a bounded linear operator Λ : V Q. We assume tat Q is reflexive and recall tat te Fencel conjugates F : V R and G : Q R are defined by F w = sup v, w F v, v V G q = sup r, q Gr. r Q Wit te adjoint operator Λ : Q V tat is defined by Λ q, v = q, Λv for q Q and v V and te identity G = G we verify tat, formally intercanging extrema, we ave inf F v + GΛv = inf sup F v + q, Λv G q v V v V q Q = sup sup Λ q, v F v G q = sup F Λ q G q. q Q v V q Q In general, we only ave tat te left-and side is an upper bound for te rigt-and side. Sufficient conditions for equality can be found in [ET99, Roc97]. Te latter maximization problem defines te dual problem. Te calculations sow tat u and p are optimal for te primal and dual formulation, reseptively, if and only if Λu G p, wit te subdifferentials F u and G p defined by Λ p F u F u = {w V : w, v u F v F u for all v V }, G p = {q Q : q, r p G r G p for all r Q}. We finally remark tat te inclusions are equivalent to, cf., e.g., [Roc97, ET99], p GΛu, u F Λ p..5. Minimization of E. A consistent discretization of te problem of minimizing E on BV seeks a minimizer in S 1 T. It can be sown tat discrete minimizers converge wit respect to intermediate convergence in BV to te minimizer of E as 0. Te representation Ev = sup v q dx + α q L 0 T d v g I K1 0q, for v S 1 T allows to formulate te minimization of E as a discrete saddle-point problem. Tis observation motivates te following algoritm tat approximates te minimizer u of E restricted to S 1 T if τ c min. We refer te reader to [Bar1] for details. Algoritm A. Let u 0, p0 S1 T L 0 T d, set d t u 0 = 0, and solve for n = 0, 1,... wit = u n + τd tu n te equations ũ n+1 d t p n+1 + ũ n+1, q p n+1 0, d t u n+1, v + p n+1, v n+1 = αu n+1 g, v subject to p n+1 1 in for all v, q S 1 T L 0 T d wit q 1 in. 4

5 Remarks.1. i Te algoritm computes a piecewise constant approximation of a in general non-unique solution of te dual problem, i.e., te iterates p n n 0 converge to some p L 0 T d but in general tis vector field does not belong to H N div;. ii Te equation and te variational inequality in te algoritm decouple and can be solved explicitly up to te inversion of a lumped mass matrix. In particular, we ave for te piecewise constant vector field p n+1 tat p n+1 = p n + τ ũn+1 max{1, p n + τ ũn+1 }. 3. A sarp version of Repin s error estimate For Banac spaces V and Q wit duals V and Q, a bounded linear operator Λ : V Q, and convex, lower-semicontinuous, proper functionals F : V R and G : Q R we consider te problem of finding u V wit Ju = inf v V Jv, Jv = F v + GΛv. Te associated dual problem consists in finding p Q wit J p = sup q Q J q, J q = F Λ q + G q We let Φ G and Φ F be non-negative functionals suc tat for all q 1, q Q and v 1, v V we ave G q 1 + q / + Φ G q q 1 Gq 1 + Gq /, F v 1 + v / + Φ F v v 1 F v 1 + F v /. By convexity we ave tat, e.g., Φ G = Φ F 0 satisfy te estimates. Te primal and dual optimization problems are related by a weak complementarity principle wic states tat Ju = inf Jv sup J q = J p. v V q Q We say tat strong duality applies if equality olds. Our final ingredient for te error estimate is a caracterization of te optimality of te solution of te primal problem. For all w Ju we ave wit a non-negative functional Ψ J tat and u is optimal if and only if 0 Ju. w, v u + Ψ J v u Jv Ju Teorem 3.1 [Rep00]. For te solution u V of te primal problem and arbitrary v V and q Q we ave Φ G Λu v + ΦF u v + Ψ J u v/ 1/ [ Jv + J q ]. Proof. Te convexity estimates imply tat Φ G Λu v + ΦF u v 1/ [ F v+gλv+f u+gλu ] F v + u/ G Λv + u/. Te optimality of u sows tat we ave It follows tat F u + GΛu + Ψ J u u + v/ F u + v/ + G Λu + v/. Φ G Λu v + ΦF u v 1/ [ F v + GΛv F u GΛu ] Ψ u v/. Te weak complementarity principle yields te asserted estimate. F u + GΛu = Ju J q = F Λ q G q 5

6 Remarks 3.1. i Notice tat te only relation between te problems defined by te functionals J and J needed in te proof is te weak duality Ju = inf v V Jv sup q Q J q, in particular, J q can be replaced by any quantity tat is a lower bound for Ju. ii If te primal and dual problem are related by a strong complementarity property ten te estimate of te teorem is sarp in te sense tat te rigt-and side vanises if v = u and q solves te dual problem. iii Te estimate differs from te one given in [Rep00] by te term Ψ J wic is necessary to obtain optimal constants, cf. Example 3.1 below. iv Notice tat te coercivity of a convex functional φ is often defined by, cf., e.g., [NSV00], σw, v = φv φw sup q φw q, v w. Te map σ is also known as te Brègman distance defined by φ, cf. [Brè67, MO08]. v Assume tat φ is single-valued. If ten wit Ψ φ w = Φ φ w + Φ φ w we ave Dφu, v u + Φ φ v u φv φu φ v 1 + v / + Ψ φ v v 1 φv 1 + φv /. Example 3.1. For te Poisson problem u = f in, u = 0, we ave V = H0 1, Y = L ; R d, Λ =, GΛv = 1/ v dx, and F v = fv dx. Since F w = I { f} w, G q = 1/ q dx, Λ = div : L ; R d H0 1, we find tat te rigt-and side η v, q of te estimate of Teorem 3.1 is given by [ η v, q = 1/ ] fv dx + I { f} div q + 1/ v dx + 1/ q dx [ = 1/ div qv dx + 1/ v + 1/ q ] = 1/4 v q provided tat div q = f. We also ave tat 1/ q 1 + q / 1/4q 1 + q = 1/8q 1 + q 1 q + q q 1 q = 1/8q 1 q so tat Φ G q = 1/8 q and 1/q 1 1/q q 1 q 1 q = 1/q 1 1/q + q 1 q = 1/q 1 q, i.e., Ψ J v u/ = 1/8 v u. Wit Φ F 0 Teorem 3.1 implies te estimate and equality occurs for q = u. u v inf v q div q=f Example 3.. For te Helmoltz type problem defined by te minimization of Jv = 1 v dx + α v g for v V = H 1 and wit : H 1 H N div; given by v, q = v, div q for v H 1 and q Q = H N div; te dual problem is for q H N div; defined by J q = 1 q dx + 1 α div q + αg α g. 6

7 We may coose Φ F v = α/8 v, Φ G q = 1/8 q, and Ψ J v = 1/ v + α/ v. Teorem 3.1 leads to te error estimate u v + α u v α v g + 1 α div q + αg α g + v dx + q dx = v q v, div q + 1 α div q + αg + α v g α g = v q + 1 α div q αv g. Here equality occurs for q = u since u + αu g = 0. One may also regard te minimization of J q as te primal problem in wic Λ = div : H N div; L and ten identify J as te corresponding dual. 4. Approximation of te Fencel predual Te identification of te dual problem defined by te problem of minimizing Ev among v BV is difficult since BV is not reflexive and its dual is difficult to caracterize. It turns out tat te predual can be described very efficiently, i.e., a minimization problem wose dual consists in te minimization of E on BV. We recall in tis section te result on Fencel duality of [HK04] and discuss te discretization and iterative solution of te predual formulation Fencel predual. For q H N div; let Dq = 1 α div q + αg α g + I K1 0q, were I K1 0q = 0 if q 1 almost everywere in and I K1 0q = oterwise. Te essential link between te functionals E and D is an equivalent caracterization of te total variation norm of v BV, i.e., we ave Dv = sup v div q dx I K1 0q. q H N div; It as been verified in [HK04] tat E is te dual functional related to D and tat Fencel duality teory in te sense of [ET99] applies, i.e., tat strong duality olds. Tis allows us to deduce te following result. Teorem 4.1. Let u BV L be minimal for E. Ten for every u BV L and q H N div; we ave α u u = Du + α u g + 1 α div q + αg α g + I K1 0q. Proof. Owing to te results of [HK04] we ave Eu Dq for every q H N div;. Te teorem is terefore a consequence of Teorem 3.1 upon noting tat we may coose Φ F v = α/8 v, Φ G = 0, and Ψ J = α/ v. Te remainder of tis section is devoted to te computation of a discrete element q H N div; tat leads to a finite and nearly optimal upper error bound. 7

8 4.. Discretization of D. For a regular triangulation T of we set S 1 NT d = {q S 1 T d : q ν = 0 on } and note tat S 1 N T d H N div;. We ave tat a vector field q S 1 N T d satisfies q 1 in if and only if q z 1 for all z N. We ten consider te minimization problem defined by te functional D q = 1 α P div q + αg α P g + I K1 0q, wit te L projection P : L S 1 T. We formally extend D to vector fields in H N div; by setting D q = if q H N div; \ S 1 N T d. Teorem 4.. For a sequence of triangulations T >0 wit maximal mes-size 0 te functionals D converge in te sense of Γ-convergence to te functional D wit respect to strong convergence in H N div;, i.e., i for every sequence q >0 H N div; wit q S 1 N T d for all > 0 and lim 0 q = q in H N div; we ave Dq lim inf 0 D q and conversely, ii for every q H N div; tere exists a sequence q >0 H N div; wit q S 1 N T d for every > 0 and lim 0 q = q suc tat Dq = lim 0 D q. Proof. To sow te first statement let q >0 and q be as in te teorem. By lower semicontinuity of D it suffices to sow tat P div q + αg div q + αg as 0. Letting ψ = div q + αg and ψ = div q + αg and noting tat P is a bounded operator on L wit operator norm P = 1 we ave P ψ ψ 1 P ψ ψ + ψ ψ + ψ ψ 3 ψ ψ + ψ P ψ 0 as 0 by density of S 1 T in L and properties of te projection. To prove te second statement we may assume by density of compactly supported smoot vector fields in H N div; tat q Cc 1 ; R d H ; R d and employ te nodal interpolant q = I q SN 1 T d for every > 0. Te convergence ten follows from standard results on nodal interpolation. Remarks 4.1. i For vector fields in te lowest order Raviart-Tomas finite element space RT N T H N div; te condition q 1 is difficult to formulate in terms of te natural degrees of freedom, i.e., te normal components on edges or faces. ii Te reason for te inconsistent discretization of D, i.e., for incorporating te projection operator P is tat tis allows for a reliable iterative solution of te discrete problems Discrete duality. Te following lemma sows tat te discretization of te predual may be regarded as te discrete pre-dual of a discretization of te functional E. Lemma 4.1. Let,N : S 1 T S 1 N T d be for v S 1 T defined by for all q SN 1 T d. functional,n v, q = v, P div q Ten te Fencel dual of te minimzation of D is defined troug te E u = I,N u dx + α u P g 8

9 Proof. We identify te spaces S 1 T and SN 1 T d wit teir duals via te L inner product, and te discrete L inner product,, respectively, so tat in particular P div =,N = Λ. For F v = α/ v P g we ave F w = 1/α w + αp g α/ P g. Te functional G q = I q dx can be written in te form G r = G z N β z r z = sup β z r z q z = q SN 1 T d : q 1 sup r, q I K1 0q, q SN 1 T d i.e., G = I K 1 0. Since te discrete spaces are reflexive te statement follows from te fact tat = G, F = F, and Λ = Λ. Remark 4.1. It can be sown tat te discrete functionals E are Γ-convergent to te functional E wit respect to intermediate convergence in BV. Numerical experiments reported below sow owever tat minimizers of E develop oscillations at discontinuities. To caracterize solutions of te discrete formulations more precisely, we define te discrete subdifferential I K1 0p at p S 1 N T d wit p 1 in as te set of all elements ξ S 1 N T d wit ξ, q p 0 for all q S 1 N T d wit q 1 in. Lemma 4.. Minimizers u S 1 T and p S 1 N T d of te discrete functionals E and D, respectively, are saddle points of te functional S v, q = α v P g v, P div q I K1 0q, in particular, tey are solutions if and only if αu P g P div p = 0,,N u I K1 0p. Proof. Te first statement follows from noting tat we ave, as in te proof of Lemma 4.1, sup v, div q I K1 0q = sup,n v, q I K1 0q = I,N v dx q SN 1 T d q SN 1 T d and α inf v S 1 T v P g v, P div q = 1 α P div q + αg α P g. Te optimal q and v satisfy te equations stated in te lemma Iterative solution. We approximately solve te saddle-point formulation of Lemma 4. by a simultaneous gradient flow for u and p in descent and ascent directions, respectively, i.e., we consider temporal discretizations of te system of ordinary differential equations and inclusions t u = αu P g + P div q, t p +,N u I K1 0p. Te following algoritm defines a time-stepping sceme and states te equations and inclusions in variational form. Algoritm A. Let τ > 0 and u 0, p0 S1 T SN 1 T d wit p 0 z 1 for all z N, set d t u 0 = 0, and solve for n = 0, 1,... wit ũn+1 = u n + τd tu n te equations d t p n+1 +,N ũ n+1, q p n+1 0, d t u n+1, v div p n+1, v n+1 = αu n+1 P g, v subject to p n+1 z 1 for all z N and for all v, q S 1 T SN 1 T d wit q z 1 for all z N. 9

10 Remark 4.. Notice tat p n+1 is te unique minimizer of and is for every z N given by q 1 τ q p q,,n ũ n+1 + I K1 0q p n+1 z = p n z + τ,n ũ n+1 z max{1, p n z + τ,n ũ n+1 z }. Te iterates of Algoritm A converge to a stationary point, e.g., if τ c min. We denote,n = sup 0 v S 1 T,N v / v and owing to an inverse estimate on SN 1 T we ave,n c 1 min. Proposition 4.1. Let u S 1 T be te unique minimizer for E in S 1 T. If θ = τ,n < 1 ten te iterates of Algoritm A satisfy for every N 1 τ N n=0 1 θ τ dt u n+1 + d t p n+1 + α u u n+1 C. In particular, we ave tat u n u and p n p for a minimizer p S 1 N T d of D as n. Proof. Let p SN 1 T d be as in Lemma 4.. Upon coosing v = u u n+1 and q = p in Algoritm A and q = p n+1 in te variational inclusion of Lemma 4. and using u n+1 P g, u u n+1 + u u n+1 = u P g, u u n+1 we find tat d t u u n+1 + p p n+1 τ + dt u n+1 + d t p n+1 + α u u n+1 = d t u n+1, u u n+1 d t p n+1, p p n+1 + α u u n+1 p n+1,,n u u n+1 + αu n+1 P g, u u n+1 p p n+1,,n ũ n+1 + α u u n+1 = p n+1,,n u u n+1 p p n+1,,n ũ n+1 + αu P g, u u n+1 = p n+1,,n u u n+1 p p n+1,,n ũ n+1 p,,n u u n+1 = p p n+1,,n u n+1 ũ n+1 + p n+1 p,,n u p p n+1,,n u n+1 ũ n+1 = τ p p n+1,,n d t u n+1, were we used u n+1 ũ n+1 = τ d t u n+1. Multiplication by τ, summation over n = 0,..., N, discrete integration by parts, Young s inequality, and d t u 0 = 0 sow tat for te rigt-and side we ave τ 3 N n=0 p p n+1,,n d t u n+1 = τ 3 τ N n=0 N d t p n+1,,n d t u n + τ p p n,,n d t u n n=0 N+1 n=0 τ,n d t u n + d t p n p p N+1 + τ 4,N d t u N+1. A combination of te estimates proves te asserted bound. Te bound implies tat u n u and p n p for some p SN 1 T d as n. Since d t u n+1 0 and d t p n+1 0 we ave tat te pair u, p is a saddle point as in Lemma

11 5. Numerical experiments We discuss in tis section te practical performance of te proposed error estimate. To illustrate some of its features in a well understood setting, we first report our experience in te case of elliptic model problems. Te teory is ten applied to te non-smoot functional E from Section 4 defined on BV Elliptic problems. We consider two elliptic problems defined on H 1 tat reveal some fundamental properties of te error estimate provided by Teorem 3.1. Example 5.1. For = 1, 1 \ [0, 1 1, 0] and f 1 consider Jv = 1 v dx fv dx for v V = H 1 0. For te unique minimizer u H1 0 and a Galerkin approximation u S 1 0 T we ave according to Example 3.1 u u u q for every q Hdiv; wit div q = f in. Te employed initial triangulation was obtained from two uniform refinements of a coarse triangulation of consisting of 6 triangles wit diameters tat are alved squares along te diagonal 1, 1. Our first coice of q Hdiv; is obtained by an averaging of te gradient of te approximate solution u, i.e., we employ q = A u S 1 T defined by q = z N q z ϕ z wit q z = 1 u dx ω z ω z for every z N and ω z = supp ϕ z. In general, te vector field q does not satisfy div q = f. Te second coice results from te solution of a discretization of te dual problem wit te lowest order Raviart-Tomas finite element space, i.e., we compute p, u RT 0 T L 0 T wit p, q + div q, u = 0, div p, v = f, v for all q, v RT 0 T L 0 T. Te corresponding error estimators are given by η A = u A u, η DP = u p. We associate to η DP te elementwise refinement indicator specified by η DP T = u p L T for T T. Notice tat η DP is a reliable error estimator owing to te results of Section 3 and te fact tat f is elementwise constant. Te reliability of η A does not follow from te arguments above but can be proved up to a generic constant wit different arguments, cf. [CB0]. Figure 1 sows te error estimators and te error for sequences of uniformly and adaptively refined triangulations. Te adaptively refined meses were obtained troug a red-green-blue refinement strategy and a set of marked elements given by M = {T T : η DP T 1/ max T T η DP T }. Te error δ = u u = u u 1/ was computed wit an approximation of u obtained by an extrapolation of corresponding values for finite element approximations on a sequence of uniform triangulations. We see tat te estimator η A provides an accurate approximation of te error δ but is not a reliable upper bound. Tis is satisfied for te estimator η DP wic leads to some overestimation but defines a guaranteed upper bound for te error. Te adaptive strategy improves te suboptimal experimental convergence rate δ N 1/3 of uniform mes-refinement to te quasi-optimal rate δ N 1/, were N = #N is te number of nodes tat define te approximation u on a triangulation T. 11

12 δ uniform η A η DP δ adaptive η A 1/ η DP 1 1/ Figure 1. Error δ = u u and error estimators η A and η DP versus degrees of freedom N for a Poisson problem on an L-saped domain defined in Example 5.1 for uniformly and adaptively refined triangulations. All quantities decay wit te same rates N 1/3 and N 1/ for uniform and adaptive mes-refinement, respectively δ η A η DP δ η A η DP δ η A η DP uniform perturbed adaptive 1/ Figure. Error δ and error estimators η A and η DP versus degrees of freedom N for a Helmoltz type problem on a square defined in Example 5. for uniformly, perturbed uniformly, and adaptively refined triangulations. Te reliable estimator η DP decays at te same optimal rate as te error on all sequences of triangulations wile te reliable estimator η A is efficient only on uniformly refined, igly symmetric triangulations. 1

13 Example 5.. For = 1, 1 and gx 1, x = 1 + π /α cosπx 1 consider Jv = 1 v dx + α v g dx for v V = H 1. For te unique minimizer ux 1, x = cosπx 1 and a Galerkin approximation u S 1 T we ave according to Example 3. u u + α u u u q + 1/α div q αu g for every q H N div;. We start wit a triangulation tat is obtained from two uniform refinements of a coarse triangulation of consisting of triangles wit diameters tat are alved squares along te diagonal 1, 1. Again, we consider estimators obtained by simple averaging and by a numerical solution of te dual problem. To satisfy te condition q ν = 0 on, we modify te coefficients in te definition of A u above by projecting q z onto te ortogonal complement of te space spanned by te normals at a boundary node z, i.e., q z = Π N z q z, were N z = span{ν E : E E, z E} is te span of all normals of boundary edges E of te triangulation tat ave te node z as an endpoint. We ten set à u = z N q z ϕ z S 1 NT H N div;. In contrast to Example 5.1 te vector field q = à u leads to a finite guranteed upper bound for te error. Our employed numerical solution of te discretized dual problem is te unique pair p, u RT 0 N T L 0 T wit p, q + div q, u = 0, div p, v αu, v = αg, v for all q, v RT 0 N T L 0 T. For te Helmoltz problem defined in Example 5. we ave tat bot estimators η A = u à u + 1/α div à u αu P g, η DP = u p + 1/α div p αu P g are according to Example 3. reliable upper bounds for te error δ = u u + α u u if g = P g. Data oscillation terms tat are related to an approximation error g P g are neglected in te following. Te numerical results sown in Figure confirm tat te estimators η A and η DP serve as guaranteed upper bounds for te error. In case of uniformly refined triangulations tey converge wit te same optimal rate as te error δ. Te error estimator η DP obtained troug te solution of te discrete dual problem is also efficient on adaptively refined and perturbed uniform triangulations were inner nodes z are randomly perturbed by a vector ξ z wit ξ z /10, i.e., it decays at te same rate as te error δ, and is nearly insensitive to mes perturbations. Tis is not te case for te estimator η A wic remains almost constant for adaptively and perturbed uniformly refined triangulations wit more tan 00 nodes. A more careful investigation of te contributions to te estimators sow tat te failure of η A on non-symmetric triangulations is due to te term div à u αu g. 13

14 Remark 5.1. We also tried an estimator defined troug averaging in te Raviart-Tomas space RT 0 T defined troug A RT u = { } u E ν Eψ E, E E were { u } E is te average of u on an edge E, ν E a fixed unit normal for every E E, and ψ E te basis of RT 0 T satisfying ψ E ν E = δ EE for all E, E E. Te corresponding error estimator led to similar results as te estimator η A. In particular, te failure of estimation by averaging cannot be attributed solely to te limited approximation properties of te H 1 conforming space S 1 T d in Hdiv;. 5.. Total variation regularization. Te experiments for elliptic problems sow tat simple averaging may not lead to efficient error control. Terefore, we solve te dual problem of te non-smoot model problem as discussed in Section 4 to define a reliable error estimator. Example 5.3. Set = 1, 1, α = 100, gx = χ B 1/ x, were B1/ = {x 1, x R : max{ x 1, x } 1/}, and consider te minimization problem defined troug te functional Ev = Dv + α v g for v BV L. For te minimal u BV L, arbitrary q H N div; wit q 1 in, and an arbitrary appoximation u W 1,1 L we ave according to Teorem 4.1 α u u u dx + α u g dx + 1 div q + αg dx α g dx. α Te initial triangulation was obtained from two uniform refinements of a coarse triangulation wit triangles tat are alved squares wit diameter. An approximation p SN 1 T d of a solution of te dual problem tat satisfies p z 1 for all nodes z N and ence p 1 in is computed wit Algoritm A. For tis and te approximate solution of te discretized primal problem wit Algoritm A we always used te step-size τ = min /10 wit te minimial mes-size min of a triangulation T. Te approximations u and p define te error estimator ηdp = u dx + α u g dx + 1 div p + αg dx α g dx. α To te reliable estimator η DP we associate te elementwise defined refinement indicators η DP T = u dx + α u g dx + 1 div p + αg dx α g dx 1/ T T α T T tat steer te adaptive algoritm. Figure 3 sows te beaviour of te estimator η DP for a sequence of uniformly and adaptively refined triangulations. On te uniform triangulations we observe tat η DP 1/4 wic coincides wit te teoretically predicted convergence rate. Te adaptive strategy leads to a smaller error estimator and an improved experimental rate of convergence η DP 1/ wic is te optimal rate for te approximation of a function u BV L on a sequence of uniform triangulations. Te refinement strategy refines te grid in a neigbourood of te discontinuity set of te data were a discontinuity of te exact solution is expected. Tis is illustrated in te sequence of adaptively generated triangulations sown in Figure 4. Te refinement indicators η DP T : T T can also be used for de-refinement of a triangulation, i.e., for mes coarsening. Tis is of particular importance wen a fine initial grid is required 14

15 /8 η DP uniform η DP adaptive 1/ Figure 3. Error estimator η DP versus degrees of freedom N for te total variation minimization problem defined in Example 5.3. Te estimator decays at a rate N 1/8 for uniform refinement. Te numbers are reduced on adaptively refined meses wit comparable numbers of degrees of freedom and te rate is improved to η DP N 1/ Figure 4. Adaptively generated grids and numerical solutions u in Example 5.3. Te adaptive strategy steered by te refinement indicators η DP T automatically refines a neigbourood of te discontinuity set of te data function g wic is expected to coincide wit te singuarity set of te exact solution u. 15

16 Figure 5. Adaptively refined and coarsened grids and numerical solutions u in Example 5.4. Te mes is significantly coarsened away from te circular discontinuity set and refined terein. 1 0 Figure 6. Numerical solution obtained troug adaptive refinement and coarsening based on te indicators η DP T : T T in Example 5.4. Te algoritm automatically coarsens te triangulation away from te discontinuity set and refines te mes in a neigbourood of it leading to an efficient approximation sceme. 16

17 to resolve given data g. To illustrate tis we consider te following experiment in wic te discontinuity set of g is circular and not exactly resolved on te employed triangulations. Example 5.4. Let = 1, 1, α = 100, and gx = χ B 1/ x for B 1/ = {x R : x 1 +x 1/ 1/}. Te initial triangulation was obtained from 13 global bisection steps of a coarse triangulation wit triangles and wose refinement edges are cosen as longest edges. Our adaptive mes-refinement and coarsening strategy consisted in marking elements in a triangulation T for coarsening if η DP T 1/ max T T η DP T. Subsequently, te mes was refined according te rule η DP T 1/ max T T η DP T using te elementwise function η DP obtained from a restriction of η DP onto te coarsened mes. Te mes coarsening was based on te algoritm proposed and analyzed in [BS1]. Te sequence of meses sown in Figures 5 and 6 sow tat adaptive mes coarsening can substantially improve te efficiency of te image regularization and it can be regarded as an adaptive image compression tecnique. Since we use an approximation g L 0 T of g on te initial mes and prolongate and restrict tis function to coarser and finer meses we cannot expect convergence to a circular interface. Te role of data oscillation and numerical integration is not investigated in tis article. Figure 7 reveals te limited applicability of Algoritm A to compute solutions of te primal problem. Artificial oscillations occur at te discontinuity set and a discrete maximum principle is violated. 1 Figure 7. Numerical approximation u obtained wit Algoritm A in Example 5.4. In contrast to te numerical solution computed wit Algoritm A oscillations occur at te discontinuity set. Te purpose of Algoritm A is to compute a conforming solution of te dual problem. 0 References [Bar1] Sören Bartels, Total variation minimization wit finite elements: Convergence and iterative solution, SIAM J. Numer. Anal , no. 3, [BMR1] Sören Bartels, Alexander Mielke, and Tomáš Roubiček, Quasi-static small-strain plasticity in te limit of vanising ardening and its numerical approximation, SIAM J. Numer. Anal , no., [Bra07] Dietric Braess, Finite elements, tird ed., Cambridge University Press, Cambridge,

18 [Brè67] Lev M. Brègman, A relaxation metod of finding a common point of convex sets and its application to te solution of problems in convex programming, Z. Vyčisl. Mat. i Mat. Fiz , [BS1] Sören Bartels and Patrick Screier, Local coarsening of simplicial finite element meses generated by bisections, BIT Numerical Matematics 01, 1 11, /s [CB0] Carsten Carstensen and Sören Bartels, Eac averaging tecnique yields reliable a posteriori error control in FEM on unstructured grids. I. Low order conforming, nonconforming, and mixed FEM, Mat. Comp , no. 39, electronic. [CP11] Antonin Cambolle and Tomas Pock, A first-order primal-dual algoritm for convex problems wit applications to imaging, J. Mat. Imaging Vision , no. 1, [ET99] Ivar Ekeland and Roger Témam, Convex analysis and variational problems, englis ed., Classics in Applied Matematics, vol. 8, Society for Industrial and Applied Matematics SIAM, Piladelpia, PA, [HK04] Micael Hintermüller and Karl Kunisc, Total bounded variation regularization as a bilaterally constrained optimization problem, SIAM J. Appl. Mat , no. 4, electronic. [MO08] Antonio Marquina and Stanley J. Oser, Image super-resolution by TV-regularization and Bregman iteration, J. Sci. Comput , no. 3, [NSV00] Ricardo H. Nocetto, Giuseppe Savaré, and Claudio Verdi, A posteriori error estimates for variable timestep discretizations of nonlinear evolution equations, Comm. Pure Appl. Mat , no. 5, [Rep00] Sergey I. Repin, A posteriori error estimation for variational problems wit uniformly convex functionals, Mat. Comp , no. 30, [Roc97] R. Tyrrell Rockafellar, Convex analysis, Princeton Landmarks in Matematics, Princeton University Press, Princeton, NJ, 1997, Reprint of te 1970 original, Princeton Paperbacks. [ROF9] Leonid I. Rudin, Stanley Oser, and Emad Fatemi, Nonlinear total variation based noise removal algoritms, Pysica D: Nonlinear Penomena , no. 1-4, [Suq78] Pierre-Marie Suquet, Existence et régularité des solutions des équations de la plasticité, C. R. Acad. Sci. Paris Sér. A-B , no. 4, A101 A104. [To11] Marita Tomas, Quasistatic damage evolution wit spatial bv-regularization, Weierstraß Institute Berlin, Preprint No. 1638, 011. Abteilung für Angewandte Matematik, Albert-Ludwigs-Universität Freiburg, Hermann-Herder Str. 10, Freiburg i.br., Germany address: bartels@matematik.uni-freiburg.de 18