Adaptive and multilevel methods for parameter identification in partial differential equations

Transcription

1 Adaptive and multilevel methods for parameter identification in partial differential equations Barbara Kaltenbacher, University of Stuttgart joint work with Hend Ben Ameur, Université de Tunis Anke Griesbaum, University of Heidelberg Boris Vexler, Technical University of Munich AIP09, Vienna, July

2 Overview motivation: parameter identification in PDEs ideas on adaptivity for inverse problems 1st approach: refinement/coarsening based on predicted misfit reduction 2nd approach: goal oriented error estimators outlook AIP09, Vienna, July

3 Motivation: Parameter Identification in PDEs instability: sufficiently high precision (amplification of numerical errors) computational effort: large scale problem: each regularized inversion involves several PDE solves repeated solution of regularized problem to determine regularization parameter AIP09, Vienna, July

4 Motivation: Parameter Identification in PDEs instability: sufficiently high precision (amplification of numerical errors) computational effort: large scale problem: each regularized inversion involves several PDE solves repeated solution of regularized problem to determine regularization parameter Example u = q: refine grid for u and q: at jumps or large gradients or at locations with large error contribution AIP09, Vienna, July

5 Motivation: Parameter Identification in PDEs instability: sufficiently high precision (amplification of numerical errors) computational effort: large scale problem: each regularized inversion involves several PDE solves repeated solution of regularized problem to determine regularization parameter Example u = q: refine grid for u and q: at jumps or large gradients or at locations with large error contribution location of large gradients / large errors a priori unknown AIP09, Vienna, July

6 Motivation: Parameter Identification in PDEs instability: sufficiently high precision (amplification of numerical errors) computational effort: large scale problem: each regularized inversion involves several PDE solves repeated solution of regularized problem to determine regularization parameter Example u = q: refine grid for u and q: at jumps or large gradients or at locations with large error contribution location of large gradients / large errors a priori unknown general strategy for mesh generation possibly separately for q and u (example q(u) u) = f) AIP09, Vienna, July

7 Motivation: Parameter Identification in PDEs instability: sufficiently high precision (amplification of numerical errors) computational effort: large scale problem: each regularized inversion involves several PDE solves repeated solution of regularized problem to determine regularization parameter Example u = q: refine grid for u and q: at jumps or large gradients or at locations with large error contribution location of large gradients / large errors a priori unknown general strategy for mesh generation possibly separately for q and u (example q(u) u) = f) instability regularization necessary! AIP09, Vienna, July

8 Regularization of Inverse Problems unstable operator equation: F (q) = g with F : q u or C(u) AIP09, Vienna, July

9 Regularization of Inverse Problems unstable operator equation: F (q) = g with F : q u or C(u) solution q = F 1 (g) does not depend continuously on g i.e., ( (g n ), g n g q n := F 1 (g n ) F 1 (g) ) AIP09, Vienna, July

10 Regularization of Inverse Problems unstable operator equation: F (q) = g with F : q u or C(u) solution q = F 1 (g) does not depend continuously on g i.e., ( (g n ), g n g q n := F 1 (g n ) F 1 (g) ) only noisy data g δ g available (measurements): g δ g δ AIP09, Vienna, July

11 Regularization of Inverse Problems unstable operator equation: F (q) = g with F : q u or C(u) solution q = F 1 (g) does not depend continuously on g i.e., ( (g n ), g n g q n := F 1 (g n ) F 1 (g) ) only noisy data g δ g available (measurements): g δ g δ making F (q) g δ small good result for q! AIP09, Vienna, July

12 An Example (Identification of a source term in a 1-d differential equation) exact and noisy data g, g δ (δ = 1%) exact solution q and q with F ( q) g δ = 1.e 14 exact and noisy data g, g δ (δ = 1%) exact solution q and q with F ( q) g δ = 2δ AIP09, Vienna, July

13 Regularization of Inverse Problems unstable operator equation: F (q) = g with F : q u or C(u) solution q = F 1 (g) does not depend continuously on g i.e., ( (g n ), g n g q n := F 1 (g n ) F 1 (g) ) only noisy data g δ g available (measurements): g δ g δ making F (q) g δ small good result for q! regularization means approaching solution along stable path: given (g n ), g n g construct q n := R αn (g n ) such that q n = R αn (g n ) F 1 (g) AIP09, Vienna, July

14 Regularization of Inverse Problems unstable operator equation: F (q) = g with F : q u or C(u) solution q = F 1 (g) does not depend continuously on g i.e., ( (g n ), g n g q n := F 1 (g n ) F 1 (g) ) only noisy data g δ g available (measurements): g δ g δ making F (q) g δ small good result for q! regularization means approaching solution along stable path: given (g n ), g n g construct q n := R αn (g n ) such that q n = R αn (g n ) F 1 (g) regularization method: family (R α ) α>0 with parameter choice α = α(g δ, δ) such that worst case convergence as δ 0: sup R α(g δ,δ) (gδ ) F 1 (g) 0 as δ 0 g δ g δ AIP09, Vienna, July

15 Motivation: Parameter Identification in PDEs instability: sufficiently high precision (amplification of numerical errors) computational effort: large scale problem: each regularized inversion involves several PDE solves repeated solution of regularized problem to determine regularization parameter Example u = q: refine grid for u and q: at jumps or large gradients or at locations with large error contribution location of large gradients / large errors a priori unknown general strategy for mesh generation possibly separately for q and u (example q(u) u) = f) AIP09, Vienna, July

16 Motivation: Parameter Identification in PDEs instability: sufficiently high precision (amplification of numerical errors) computational effort: large scale problem: each regularized inversion involves several PDE solves repeated solution of regularized problem to determine regularization parameter Example u = q: refine grid for u and q: at jumps or large gradients or at locations with large error contribution location of large gradients / large errors a priori unknown general strategy for mesh generation possibly separately for q and u (example q(u) u) = f) computational effort efficient numerical strategies necessary! AIP09, Vienna, July

17 Efficient Methods for PDEs l = 4 l = 3 l = 2 l = 1 multilevel iteration: start with coarse discretization refine successively adaptive discretization: coarse discretization where possible fine grid only where necessary AIP09, Vienna, July

18 Efficient Methods for PDEs combined multilevel adaptive strategy: courtesy to [R.Becker&M.Braack&B.Vexler, App.Num.Math., 2005] start on coarse grid sucessive adaptive refinement AIP09, Vienna, July

19 Some Ideas on Adaptivity for Inverse Problems Haber&Heldmann&Ascher 07: Tikhonov with BV type regularization: Refine for u to compute residual term sufficiently precisely; Refine for q to compute regularization term sufficiently precisely Neubauer 03, 06, 07: moving mesh regularization, adaptive grid regularization: Tikhonov with BV type regularization: Refine where q has jumps or large gradients Borcea&Druskin 02: optimal finite difference grids (a priori refinement): Refine close to measurements Chavent&Bissell 98, Ben Ameur&Chavent&Jaffré 02, BK&Ben Ameur 02: refinement and coarsening indicators Becker&Vexler 04, Griesbaum&BK&Vexler 07, Bangerth 08, BK&Vexler 09: goal oriented error estimators... AIP09, Vienna, July

20 Some Ideas on Adaptivity for Inverse Problems Haber&Heldmann&Ascher 07: Tikhonov with BV type regularization: Refine for u to compute residual term sufficiently precisely; Refine for q to compute regularization term sufficiently precisely Neubauer 03, 06, 07: moving mesh regularization, adaptive grid regularization: Tikhonov with BV type regularization: Refine where q has jumps or large gradients Borcea&Druskin 02: optimal finite difference grids (a priori refinement): Refine close to measurements Chavent&Bissell 98, Ben Ameur&Chavent&Jaffré 02, BK&Ben Ameur 02: refinement and coarsening indicators Becker&Vexler 04, Griesbaum&BK&Vexler 07, Bangerth 08, BK&Vexler 09: goal oriented error estimators... AIP09, Vienna, July

21 1st approach: refinement/coarsening based on predicted misfit reduction AIP09, Vienna, July

22 Identification of a Distributed Parameter: Groundwater modelling s u div (q grad u) = f in Ω IR2 t with initial and boundary conditions u... hydraulic potential (ground water level), s(x, y)... storage coefficients, q(x, y)... hydraulic transmissivity, f(x, y, t)... source term, space and time discretization (time step t, mesh size h). AIP09, Vienna, July

23 Parameter Identification s u div (q grad u) = f in Ω t Reconstruction of the transmissivity q (pcw. const.) from measurements of u. Find zonation and values of q such that J(q) := u(q) u obs 2 = min! [Ben Ameur&Chavent&Jaffré 02], [Chavent&Bissell 98], [BK&Ben Ameur 02] AIP09, Vienna, July

24 Refinement Indicators 1 zone 2 zones q := min of J(q) solves min J(( q 1 )) q 2 s.t. d T ( q 1 q 2 ) = q 1 q 2 = B =: 0 ( q 1 q 2 ):= min of J(( q 1 q 2 )) solves min J(( q 1 q 2 )) s.t. d T ( q 1 q 2 ) = q 1 q 2 = B =: q 1 q 2 B J(( qb 1 q B 2 )) = λ B J(( q 1 q )) J(q ) + λ 0 (q1 q2) 2 λ 0 large large possible reduction of data misfit J B opt λ 0 = (1/d T d)d T J(q ) (negligible computational effort) AIP09, Vienna, July

25 Compute all refinement indicators for zonations generated systematically by families of vertical, horizontal, checkerboard and oblique cuts. Mark those cuts that yield largest refinement indicators λ 0 AIP09, Vienna, July

26 Z 11 Z 12 Z 13 Coarsening Indicators Z 2 ( q 1 ):=solution of min J(( q 1 )) q 2 J( q B 11 q B 12 q B 13 q B 2 ) Bi =0,B j =B,j i }{{} optimum if q 1i is aggregated with q 2 J(( q 1 q 2 q 2 solves min J( q 11 q 12 q 13 q 2 q 11 q 2 = B 1 q 12 q 2 = B 2 q 13 q 2 = B 3 with B i := B := q1 q2 )) λ } B i{{} B coarsening indicator ) s.t. AIP09, Vienna, July

27 Multilevel Refinement and Coarsening Algorithm [H.Ben Ameur, G.Chavent, J.Jaffré, 2002] Minimize J on starting zonation Do until refinement indicators = 0 Refinement: compute refinement indicators λ chooose cuts with largest λ Coarsening: if chosen cuts yield several sub-zones: evaluate coarsening indicators and aggregate zones where possible Minimize J for each of the retained zonations and keep those with largest reduction in J AIP09, Vienna, July

28 6 6 mesh, 20 per cent noise: Numerical Tests exact transmissivity starting value third iterate fourth iterate mesh, 10 per cent noise: exact transmissivity seventh iterate AIP09, Vienna, July

29 Abstract Setting for Refinement and Coarsening nonlinear ill-posed operator equation F (q) = g F : X Y, X, Y Hilbert spaces exact solution q exists, F (q ) = g noisy data g δ with g g δ δ misfit minimization min X F (q) g δ 2 discretization: X N = span{φ 1,..., φ N } s.t. X = N IN X N AIP09, Vienna, July

30 Abstract Setting for Refinement and Coarsening discretization: X N = span{φ 1,..., φ N } misfit minimization s.t. X = N IN X N min F (q) g δ 2 = min F ( N q X N a IR N i=1 a iφ i ) g δ }{{} 2 =J (a) AIP09, Vienna, July

31 Abstract Setting for Refinement and Coarsening discretization: X N = span{φ 1,..., φ N } misfit minimization s.t. X = N IN X N min F (q) g δ 2 = min F ( N q X N a IR N i=1 a iφ i ) g δ }{{} 2 =J (a) consider misfit minimization on some index set I {1, 2,..., N}: min F ( a IR I i I a iφ i ) g δ 2 (P I ) solution a I, q I with a i := 0 for i I AIP09, Vienna, July

32 Abstract Setting for Refinement and Coarsening discretization: X N = span{φ 1,..., φ N } misfit minimization s.t. X = N IN X N min F (q) g δ 2 = min F ( N q X N a IR N i=1 a iφ i ) g δ }{{} 2 =J (a) consider misfit minimization on some index set I {1, 2,..., N}: min F ( a IR I i I a iφ i ) g δ 2 (P I ) solution a I, q I with a i := 0 for i I sparsity AIP09, Vienna, July

33 Abstract Setting for Refinement and Coarsening discretization: X N = span{φ 1,..., φ N } misfit minimization s.t. X = N IN X N min F (q) g δ 2 = min F ( N q X N a IR N i=1 a iφ i ) g δ }{{} 2 =J (a) consider misfit minimization on some index set I {1, 2,..., N}: min F ( a IR I i I a iφ i ) g δ 2 (P I ) solution a I, q I with a i := 0 for i I sparsity Find index set I and coefficients a I such that F ( i I a I i φ i ) g δ 2 = min a IR I F ( i I a I i φ i ) g δ 2 = min F (q) g δ 2 q X N AIP09, Vienna, July

34 Refinement Indicators current index set I k with computed solution a Ik of (P Ik ); AIP09, Vienna, July

35 Refinement Indicators current index set I k with computed solution a Ik of (P Ik ); for some index {i } I k consider constrained minimization problem min F ( a IR Ik i I k {i } a iφ i ) g δ 2 +1 }{{} =J (a) s.t. a i = β (P Ik, i β ) solution a β with a i := 0 for i I k {i }; AIP09, Vienna, July

36 Refinement Indicators current index set I k with computed solution a Ik of (P Ik ); for some index {i } I k consider constrained minimization problem min F ( a IR Ik i I k {i } a iφ i ) g δ 2 +1 }{{} =J (a) s.t. a i = β (P Ik, i β ) solution a β with a i := 0 for i I k {i }; note that a β=0 = a Ik solves (P Ik ) AIP09, Vienna, July

37 Refinement Indicators current index set I k with computed solution a Ik of (P Ik ); for some index {i } I k consider constrained minimization problem min F ( a IR Ik i I k {i } a iφ i ) g δ 2 +1 }{{} =J (a) s.t. a i = β (P Ik, i β ) solution a β with a i := 0 for i I k {i }; note that a β=0 = a Ik solves (P Ik ) Lagrange function L(a, λ) = J (a) + λ(β a i ) AIP09, Vienna, July

38 Refinement Indicators current index set I k with computed solution a Ik of (P Ik ); for some index {i } I k consider constrained minimization problem min F ( a IR Ik i I k {i } a iφ i ) g δ 2 +1 }{{} =J (a) s.t. a i = β (P Ik, i β ) solution a β with a i := 0 for i I k {i }; note that a β=0 = a Ik solves (P Ik ) Lagrange function L(a, λ) = J (a) + λ(β a i ) necessary optimality conditions: 0 = L a i (a β, λ β ) = J a i (a β) λ β ( ) AIP09, Vienna, July

39 Refinement Indicators current index set I k with computed solution a Ik of (P Ik ); for some index {i } I k consider constrained minimization problem min F ( a IR Ik i I k {i } a iφ i ) g δ 2 +1 }{{} =J (a) s.t. a i = β (P Ik, i β ) solution a β with a i := 0 for i I k {i }; note that a β=0 = a Ik solves (P Ik ) Lagrange function L(a, λ) = J (a) + λ(β a i ) necessary optimality conditions: 0 = L a i (a β, λ β ) = J a i (a β) λ β Lagrange multipliers = sensitivities: d dβ J (a β) = d dβ L(a β, λ β ) = λ β ( ) AIP09, Vienna, July

40 Refinement Indicators current index set I k with computed solution a Ik of (P Ik ); for some index {i } I k consider constrained minimization problem min F ( a IR Ik i I k {i } a iφ i ) g δ 2 +1 }{{} =J (a) s.t. a i = β (P Ik, i β ) solution a β with a i := 0 for i I k {i }; note that a β=0 = a Ik solves (P Ik ) Lagrange function L(a, λ) = J (a) + λ(β a i ) necessary optimality conditions: 0 = L a i (a β, λ β ) = J a i (a β) λ β Lagrange multipliers = sensitivities: d dβ J (a β) = d dβ L(a β, λ β ) = λ β Taylor expansion J (a β ) J (a 0 ) + d dβ J (a 0) β = J (a Ik ) + λ β=0 β ( ) AIP09, Vienna, July

41 Refinement Indicators current index set I k with computed solution a Ik of (P Ik ); for some index {i } I k consider constrained minimization problem min F ( a IR Ik i I k {i } a iφ i ) g δ 2 +1 }{{} =J (a) s.t. a i = β (P Ik, i β ) solution a β with a i := 0 for i I k {i }; note that a β=0 = a Ik solves (P Ik ) Lagrange function L(a, λ) = J (a) + λ(β a i ) necessary optimality conditions: 0 = L a i (a β, λ β ) = J a i (a β) λ β Lagrange multipliers = sensitivities: d dβ J (a β) = d dβ L(a β, λ β ) = λ β Taylor expansion J (a β ) J (a 0 ) + d dβ J (a 0) β = J (a Ik ) + λ β=0 β r i := λ β=0 ( ) = J a i (aik )... refinement indicator ( ) AIP09, Vienna, July

42 Coarsening Indicators current index set Ĩk with computed solution aĩk of (P Ĩk ); for some index {l } Ĩk consider constrained minimization problem min a IR Ĩk F ( i Ĩk a i φ i ) g δ 2 }{{} =J (a) s.t. a l = γ ( P Ĩk, l γ ) solution a γ note that a γ = aĩk with γ := aĩk l solves (P Ĩk ) Lagrange function L(a, µ) = J (a) + µ(γ a l ) necessary optimality conditions: 0 = L a (a γ, µ γ ) = J l a l (a γ ) µ γ ( ) Lagrange multipliers = sensitivities: d dγ J (a γ) = d dγ L(a γ, µ γ ) = µ γ Taylor expansion J (a γ=0 ) J (a γ ) d dγ J (a γ ) γ = J (aĩk ) µ γ γ c l := µ γ γ ( ) = J a l (aĩk )γ... coarsening indicator AIP09, Vienna, July

43 Multilevel Refinement and Coarsening Algorithm k = 0: Minimize J on starting index set I 0 minimal value J 0 Do until refinement indicators = 0 Refinement: compute refinement indicators r i, i I k chooose index sets I k {i } with largest r i Minimize J on each of these index sets and keep Ĩ := Ik {i } with largest reduction in J Coarsening (only if J < J k ): evaluate coarsening indicators c l J chooose index sets Ĩk \ {l } with largest c l Minimize J on each of these index sets and keep I := Ĩk \ {l } with largest reduction in J J If J J + ρ(j k J ) (coarsening does not deteriorate optimal value too much) set I k+1 := I, J k+1 := J (refinement and coarsening) Else set I k+1 := Ĩ, J k+1 := J (refinement only) AIP09, Vienna, July

44 Convergence Proof For fixed N <, Algorithm stops after finitely many steps k = K; q K := i I K a K i φ i a K solves (P IK ) 0 = J (a K ) 0 = F (q K ) g δ, F (q K )φ i i I K refinement indicators vanish 0 = r i = F (q K ) g δ, F (q K )φ i i I K Proj XN F (q K ) (F (q K ) g δ ) = 0 Stability and convergence follow from (existing) results on regularization by discretization AIP09, Vienna, July

45 Convergence Results Stability for fixed discretization N < (regularization by discretization) convergence with exact data as N noisy data with noise level δ g δ F (q ) : a priori or a posteriori choice N(δ) convergence as δ 0 AIP09, Vienna, July

46 Numerical Tests example 1 (linear system): with (Aq) i x i 0 Aq = g q(t) dt numerical differentiation Identify vector (q i ) i=1,...,n from right hand side g AIP09, Vienna, July

47 Example 1: Linear System... a... a 0... I +... I 0... r 0 AIP09, Vienna, July

48 Example 1: Linear System... a... a 1... I +... I 1... r 1 AIP09, Vienna, July

49 Example 1: Linear System... a... a 2... I +... I 2... r 2 AIP09, Vienna, July

50 Example 1: Linear System... a... a 3... I +... I 3... r 3 AIP09, Vienna, July

51 Example 1: Linear System... a... a 3... I +... I 3... r 3 70 instead of 8000 flops! AIP09, Vienna, July

52 Numerical Tests example 2 (coefficients in system of ODEs): u = Qu, u(0) = u 0 Identify matrix Q = (q ij ) i,j=1,...,n from measurements of u(t l ), l = 1,..., T. AIP09, Vienna, July

53 Example 2: Coefficients in System of ODEs a a I... I 0 r 0 (r max = 5.0) AIP09, Vienna, July

54 Example 2: Coefficients in System of ODEs a a I... I 1 r 1 (r max = 2.5) AIP09, Vienna, July

55 Example 2: Coefficients in System of ODEs a a I... I 2 r 2 (r max = 1.2) AIP09, Vienna, July

56 Example 2: Coefficients in System of ODEs a a I... I 3 r 3 (r max = 1.5) AIP09, Vienna, July

57 Example 2: Coefficients in System of ODEs a a I... I 4 r 4 (r max = 1.0) AIP09, Vienna, July

58 Example 2: Coefficients in System of ODEs a a I... I 5 r 5 (r max = 0.6) AIP09, Vienna, July

59 Example 2: Coefficients in System of ODEs a a I... I 6 r 6 (r max = 0.23) AIP09, Vienna, July

60 Example 2: Coefficients in System of ODEs a a I... I 7 r 7 (r max = 0.21) AIP09, Vienna, July

61 Example 2: Coefficients in System of ODEs a a I... I 8 r 8 (r max = 0.2) AIP09, Vienna, July

62 Example 2: Coefficients in System of ODEs a a I... I 9 r 9 (r max = 0.22) AIP09, Vienna, July

63 Example 2: Coefficients in System of ODEs a a I... I 10 r 10 (r max = 3.e 14) AIP09, Vienna, July

64 Example 2: Coefficients in System of ODEs a a instead of flops! +... I... I 10 r 10 (r max = 3.e 14) AIP09, Vienna, July

65 Remarks for 1st Approach more systematic coarsening based on problem specific properties (related dofs due to local closeness in groundwater example) Lagrange multipliers = gradient components (but we do not carry out gradient steps!): possible improvement by taking into account Hessian information (Newton type) Greedy type approach (Burger&Hofinger 04, Denis&Lorenz&Trede 09) relation active set strategy semismooth Newton (Hintermüller&Ito&Kunisch 03) AIP09, Vienna, July

66 2nd approach: goal oriented error estimators AIP09, Vienna, July

67 Tikhonov Regularization and the Discrepancy Principle Parameter identification as a nonlinear operator equation F (q) = g g δ g... given data; noise level δ g δ g F... forward operator: F (q) = (C S)(q) = C(u) where u = S(q) solves A(q, u)(v) = (f, v) v V... PDE in weak form Hilbert spaces Q, V, G: q Q S u V C g G AIP09, Vienna, July

68 Tikhonov Regularization and the Discrepancy Principle Parameter identification as a nonlinear operator equation F (q) = g g δ g... given data; noise level δ g δ g F... forward operator: F (q) = (C S)(q) = C(u) where u = S(q) solves A(q, u)(v) = (f, v) v V... PDE in weak form Hilbert spaces Q, V, G: q Q S u V C g G Tikhonov regularization: Minimize j α (q) = F (q) g δ 2 + α q 2 over q Q, AIP09, Vienna, July

69 Tikhonov Regularization and the Discrepancy Principle Parameter identification as a nonlinear operator equation F (q) = g g δ g... given data; noise level δ g δ g F... forward operator: F (q) = (C S)(q) = C(u) where u = S(q) solves A(q, u)(v) = (f, v) v V... PDE in weak form Hilbert spaces Q, V, G: q Q S u V C g G Tikhonov regularization: Minimize j α (q) = F (q) g δ 2 + α q 2 over q Q, Minimize J α (q, u) = C(u) g δ 2 + α q 2 over q Q, u V, under the constraint A(q, u)(v) = (f, v) v V AIP09, Vienna, July

70 Tikhonov Regularization and the Discrepancy Principle Parameter identification as a nonlinear operator equation F (q) = g g δ g... given data; noise level δ g δ g F... forward operator: F (q) = (C S)(q) = C(u) where u = S(q) solves Tikhonov regularization: A(q, u)(v) = (f, v) v V... PDE in weak form Minimize j α (q) = F (q) g δ 2 + α q 2 over q Q, AIP09, Vienna, July

71 Tikhonov Regularization and the Discrepancy Principle Parameter identification as a nonlinear operator equation F (q) = g g δ g... given data; noise level δ g δ g F... forward operator: F (q) = (C S)(q) = C(u) where u = S(q) solves Tikhonov regularization: A(q, u)(v) = (f, v) v V... PDE in weak form Minimize j α (q) = F (q) g δ 2 + α q 2 over q Q, Choice of α: discrepancy principle (fixed constant τ 1) F (q δ α ) g δ = τδ AIP09, Vienna, July

72 Tikhonov Regularization and the Discrepancy Principle Parameter identification as a nonlinear operator equation F (q) = g g δ g... given data; noise level δ g δ g F... forward operator: F (q) = (C S)(q) = C(u) where u = S(q) solves Tikhonov regularization: A(q, u)(v) = (f, v) v V... PDE in weak form Minimize j α (q) = F (q) g δ 2 + α q 2 over q Q, Choice of α: discrepancy principle (fixed constant τ 1) F (q δ α ) g δ = τδ Convergence analysis: [Engl& Hanke& Neubauer 1996] and the references therein AIP09, Vienna, July

73 Goal Oriented Error Estimators in PDE Constrained Optimization (I) [Becker&Kapp&Rannacher 00], [Becker&Rannacher 01], [Becker&Vexler 04, 05] Lagrange functional: Minimize J(q, u) over q Q, u V under the constraints A(q, u)(v) = f(v) v V, L(q, u, z) = J(q, u) + f(z) A(q, u)(z). First order optimality conditions: L (q, u, z)[(p, v, y)] = 0 (p, v, y) Q V V (1) Discretization Q h Q, V h V discretized version of (1). Estimate the error due to discretization in some quantity of interest I: I(q, u) I(q h, u h ) η AIP09, Vienna, July

74 Auxiliary functional: Goal Oriented Error Estimators (II) M(q, u, z, p, v, y) = I(q, u)+l (q, u, z)[(p, v, y)] (q, u, z, p, v, y) (Q V V ) 2 Consider additional equations: M (x h )(dx h ) = 0 dx h X h = (Q h V h V h ) 2 Proposition (Becker&Vexler, J. Comp. Phys., 2005): I(q, u) I(q h, u h ) = 1 2 M (x h )(x x h ) +O( x x h 3 ) x h X h. }{{} =:η Error estimator η is a sum of local contributions due to either q, u, z, p, v, or y: η = N q i=1 N u η q i + i=1 η u i + N z i=1 η z i + N p i=1 N v η p i + i=1 η v i + local refinement separately for q Q h, u V h, z V h,... N y i=1 η y i AIP09, Vienna, July

75 Choice of Quantity of Interest? aim: recover infinite dimensional convergence results for Tikhonov + discr. princ. in the adaptively discretized setting challenge: carrying over infinite dimensional results is... straightforward if we can guarantee smallness of operator norm F h F huge number of quantities of interest!... not too hard if we can guarantee smallness of F h (q ) F (q ) large number of quantities of interest!... but we only want to guarantee precision of one or two quantities of interest AIP09, Vienna, July

76 Convergence Analysis Choice of Quantity of Interest Proposition [Griesbaum&BK& Vexler 07], [BK& Vexler 09]: α = α (δ, g δ ) and Q h V h V h such that for I(q, u) := C(u) g δ 2 G = F (q) gδ 2 G τ 2 δ 2 I(qh,α δ, u δ h,α ) τδ 2 (i) If additionally I(qh,α δ, u δ h,α ) I(qα δ, u δ α ) ci(qh,α δ, u δ h,α ) for some sufficiently small constant c > 0 then qα δ q as δ 0. Optimal rates under source conditions (logarithic/hölder). AIP09, Vienna, July

77 Convergence Analysis Choice of Quantity of Interest Proposition [Griesbaum&BK& Vexler 07], [BK& Vexler 09]: α = α (δ, g δ ) and Q h V h V h such that for I(q, u) := C(u) g δ 2 G = F (q) gδ 2 G τ 2 δ 2 I(qh,α δ, u δ h,α ) τδ 2 (ii) If additionally for I 2 (q, u) := J α (q, u) I 2 (qh,α δ, u δ h,α ) I 2 (qα δ, u δ α ) σδ 2 for some constant C > 0 with τ σ, then qh,α δ q as δ 0. see also [Neubauer&Scherzer 1990] AIP09, Vienna, July

78 Convergence Analysis Choice of Quantity of Interest Proposition [Griesbaum&BK& Vexler 07], [BK& Vexler 09]: α = α (δ, g δ ) and Q h V h V h such that for I(q, u) := C(u) g δ 2 G = F (q) gδ 2 G τ 2 δ 2 I(qh,α δ, u δ h,α ) τδ 2 (ii) If additionally for I 2 (q, u) := J α (q, u) I 2 (qh,α δ, u δ h,α ) I 2 (qα δ, u δ α ) σδ 2 for some constant C > 0 with τ σ, then qh,α δ q as δ 0. see also [Neubauer&Scherzer 1990] J as quantity of interest [Becker&Kapp&Rannacher 00], [Becker&Rannacher 01], [Bangerth 08] L as quantity of interest [Beilina&Klibanov 09] AIP09, Vienna, July

79 Idea of Proof error bound J α (q δ h,α, u δ h,α ) J α (q δ α, u δ α ) σδ 2 and optimality of q δ α, u δ α imply J α (q δ h,α, u δ h,α ) J α (q δ α, u δ α ) + σδ 2 J α (q, u ) + σδ 2 AIP09, Vienna, July

80 Idea of Proof error bound J α (q δ h,α, u δ h,α ) J α (q δ α, u δ α ) σδ 2 and optimality of q δ α, u δ α imply J α (q δ h,α, u δ h,α ) J α (q δ α, u δ α ) + σδ 2 J α (q, u ) + σδ 2 on the other hand, by the discrepancy principle τ 2 δ 2 F (q δ h,α ) g δ 2 τδ 2 and the definition of the cost functional J α (q, u) = F (q) g δ 2 + α q 2 J α (q δ h,α, u δ h,α ) τ 2 δ 2 + α q δ h,α 2 J α (q, u ) δ 2 + α q 2 AIP09, Vienna, July

81 Idea of Proof error bound J α (q δ h,α, u δ h,α ) J α (q δ α, u δ α ) σδ 2 and optimality of q δ α, u δ α imply J α (q δ h,α, u δ h,α ) J α (q δ α, u δ α ) + σδ 2 J α (q, u ) + σδ 2 on the other hand, by the discrepancy principle τ 2 δ 2 F (q δ h,α ) g δ 2 τδ 2 and the definition of the cost functional J α (q, u) = F (q) g δ 2 + α q 2 J α (q δ h,α, u δ h,α ) τ 2 δ 2 + α q δ h,α 2 J α (q, u ) δ 2 + α q 2 Combining these estimates and the choice τ 2 > 1 + σ we get q δ h,α 2 q α (1 + σ τ 2 )δ 2 q 2. The rest of the proof is standard. AIP09, Vienna, July

82 Idea of Proof error bound J α (q δ h,α, u δ h,α ) J α (q δ α, u δ α ) σδ 2 and optimality of q δ α, u δ α imply J α (q δ h,α, u δ h,α ) J α (q δ α, u δ α ) + σδ 2 J α (q, u ) + σδ 2 on the other hand, by the discrepancy principle τ 2 δ 2 F (q δ h,α ) g δ 2 τδ 2 and the definition of the cost functional J α (q, u) = F (q) g δ 2 + α q 2 J α (q δ h,α, u δ h,α ) τ 2 δ 2 + α q δ h,α 2 J α (q, u ) δ 2 + α q 2 Combining these estimates and the choice τ 2 > 1 + σ we get q δ h,α 2 q α (1 + σ τ 2 )δ 2 q 2. The rest of the proof is standard. (Also works for stationary points q δ h,α instead of global minimizers.) AIP09, Vienna, July

83 Efficient Computation of α Choice of Quantity of Interest Choice of α: discrepancy principle (fixed constant τ 1) 1-d nonlinear equation solve by Newton s method F (q δ α ) g δ = τδ AIP09, Vienna, July

84 Efficient Computation of α Choice of Quantity of Interest Proposition [Griesbaum&BK& Vexler 07], [BK& Vexler 09]: Define i( 1 α ) := I(q, u) := C(u) gδ 2 G = F (q) gδ 2 G I 2 (q, u) := i ( 1 α ) β solution to i(β ) = τ 2 δ 2 (discr.princ.) β k+1 = β k ik h τ 2 δ 2 for k k 1 with k = min{k IN i k h τ 2 δ 2 0} with i k h, i k h satisfying i(β k ) i k h εk, i (β k ) i k h ε k, i k h (Newton) ε k, ε k sufficiently small (see above). Then β k satisfies an asymptotically optimal quadratic convergence estimate and (τ 2 τ 2 )δ 2 i(β k ) (τ 2 + τ 2 )δ 2 AIP09, Vienna, July

85 Remarks computation of error estimators for i(β): just one more SQP type step; evaluation of i (β): can be extracted from quantities computed for error estimators for i(β) error estimators for i (β): stationary point of another auxiliary functional AIP09, Vienna, July

86 Numerical Tests u + q = 0 on Ω = (0, 1) 2 + homogeneous Dirichlet BC Identify q from distributed measurements of u at points in Ω (a), (b) q (x, y) = 1 2πσ 2 exp ( (x 5 11 )2 +(y 5 ) 11 )2 2σ 2, σ = { 0.05 (a) 0.01 (b) (c) q (x, y) = { 1 x x > 1 2 Computations with Gascoigne and RoDoBo. AIP09, Vienna, July

87 Numerical Tests exact par. q: (a), (b), (c) exact state u: (a), (b), (c) AIP09, Vienna, July

88 Numerical Results (I) Computations with 1% random noise: CPU times: (a) (b) (c) adaptive s s 7.976s global s s s adaptive grids: AIP09, Vienna, July

89 Numerical Results (II) exact par. q: (a), (b), (c) computed par. q: (a), (b), (c) AIP09, Vienna, July

90 Numerical Results (III) Convergence as δ 0 for examples (a) and (b): q δ α q δ q 1/α 8% % % % % q δ α q δ q 1/α 8% % % % % AIP09, Vienna, July

91 Regularization by Projection and the Discrepancy Principle P l (F (q) g δ ) = 0, q q 0 F (q ) G l P l = Proj Gl, dim(g l ) <, G 1 G 2 G 3... G F... forward operator: F (q) = (C S)(q) = C(u) where u = S(q) solves A(q, u)(v) = (f, v) v V... PDE in weak form AIP09, Vienna, July

92 Regularization by Projection and the Discrepancy Principle P l (F (q) g δ ) = 0, q q 0 F (q ) G l P l = Proj Gl, dim(g l ) <, G 1 G 2 G 3... G F... forward operator: F (q) = (C S)(q) = C(u) where u = S(q) solves A(q, u)(v) = (f, v) v V... PDE in weak form Φ(q, u, z, λ)(dq, du, dz, dλ) = C(u) g δ, dλ G +f(dz) A(q, u)(dz) + q q 0, dq Q (A q(q, u )dq)(z) +(A u(q, u )du)(z) + λ, C (u )du G = 0 (dq, du, dz, dλ) (Q V V G l ) AIP09, Vienna, July

93 Regularization by Projection and the Discrepancy Principle P l (F (q) g δ ) = 0, q q 0 F (q ) G l P l = Proj Gl, dim(g l ) <, G 1 G 2 G 3... G F... forward operator: F (q) = (C S)(q) = C(u) where u = S(q) solves A(q, u)(v) = (f, v) v V... PDE in weak form Φ(q, u, z, λ)(dq, du, dz, dλ) = C(u) g δ, dλ G +f(dz) A(q, u)(dz) + q q 0, dq Q (A q(q, u )dq)(z) +(A u(q, u )du)(z) + λ, C (u )du G = 0 (dq, du, dz, dλ) (Q V V G l ) Choice of l: discrepancy principle (fixed constant τ 1) l = min{l IN : F (ql δ ) g δ τδ} AIP09, Vienna, July

94 Multilevel Iteration l = 4 l = 3 l = 2 l = 1 P l (F (q) g δ ) = 0 l = min{l IN : F (ql δ ) g δ τδ}... stopping rule [BK 00, 07] AIP09, Vienna, July

95 Convergence Analysis Proposition [BK& Vexler 09]: l = l (δ, g δ ) and Q h V h V h such that for I(q, u) := C(u) g δ 2 G = F (q) gδ 2 G τ 2 δ 2 I(qh,l δ, u δ h,l ) τδ 2 (i) If additionally I(qh,l δ, u δ h,l ) I(ql δ, u δ l ) ci(qh,l δ, u δ h,l ) for some sufficiently small constant c > 0 then ql δ q as δ 0. Optimal rates under source conditions (logarithic/hölder). AIP09, Vienna, July

96 Summary and Outlook for 2nd Approach goal oriented error estimators allow to control the error in some quantity of interest sufficiently small error in residual norm i( 1 α ) and its derivative i ( 1 α ) fast convergence of Newton s method for choosing α (discr. princ.) coarse grids at the beginning of Newton s method save computational effort sufficiently small error in residual norm and Tikhonov functional convergence of Tikhonov regularization preserved other regularization methods: regularization by discretization multilevel method [BK&Vexler 09] other regularization parameter choice strategies: e.g., balancing principle AIP09, Vienna, July

97 Thank you for your attention! AIP09, Vienna, July