Iterative regularization of nonlinear ill-posed problems in Banach space

Transcription

1 Iterative regularization of nonlinear ill-posed problems in Banach space Barbara Kaltenbacher, University of Klagenfurt joint work with Bernd Hofmann, Technical University of Chemnitz, Frank Schöpfer and Thomas Schuster, University of Oldenburg 25th IFIP TC 7, Berlin, September 2011

2 Parameter identification in PDEs (I) e.g., electrical impedance tomography (EIT) (σ φ) = 0 in Ω. Identify conductivity σ from measurements of the Dirichlet-to-Neumann map Λ σ, i.e., all possible pairs (φ, σ n φ) on Ω. e.g., magnetic resonance electrical impedance tomography (MREIT): ( µ 1 E ) + σ t E + ε 2 t 2 E = J in Ω. Identify σ from measurements of the deposited energy σ E 2 in Ω.

3 Parameter identification in PDEs (I) e.g., electrical impedance tomography (EIT) (σ φ) = 0 in Ω. Identify conductivity σ from measurements of the Dirichlet-to-Neumann map Λ σ, i.e., all possible pairs (φ, σ n φ) on Ω. e.g., magnetic resonance electrical impedance tomography (MREIT): ( µ 1 E ) + σ t E + ε 2 t 2 E = J in Ω. Identify σ from measurements of the deposited energy σ E 2 in Ω.

4 Parameter identification in PDEs (II) e.g. a-example (transmissivity in groundwater modelling) (a u) = 0 in Ω. Identify a from measurements of u in Ω. e.g. c-example (potential in stat. Schrödinger equation) u + c u = 0 in Ω. Identify c from measurements of u in Ω.

5 Parameter identification in PDEs (II) e.g. a-example (transmissivity in groundwater modelling) (a u) = 0 in Ω. Identify a from measurements of u in Ω. e.g. c-example (potential in stat. Schrödinger equation) u + c u = 0 in Ω. Identify c from measurements of u in Ω.

6 Phase retrieval Ff = r Reconstruct the real-valued function f : R R from measurements of the intensity r : R R + of its Fourier transform.

7 Forward operator F EIT: F : σ Λ σ where Λ σ : φ σ n φ and (σ φ) = 0 in Ω. MREIT: F : σ σ E 2 where ( µ 1 E ) + σ t E + ε 2 t 2 E = J in Ω + bndy.cond.

8 Forward operator F EIT: F : σ Λ σ where Λ σ : φ σ n φ and (σ φ) = 0 in Ω. MREIT: F : σ σ E 2 where ( µ 1 E ) + σ t E + ε 2 E = J in Ω + bndy.cond. t 2 a-example: F : a u where (a u) = 0 in Ω + boundary conditions

9 Forward operator F EIT: F : σ Λ σ where Λ σ : φ σ n φ and (σ φ) = 0 in Ω. MREIT: F : σ σ E 2 where ( µ 1 E ) + σ t E + ε 2 E = J in Ω + bndy.cond. t 2 a-example: F : a u where (a u) = 0 in Ω + boundary conditions c-example: F : c u where u + c u = 0 in Ω + boundary conditions

10 Forward operator F EIT: F : σ Λ σ where Λ σ : φ σ n φ and (σ φ) = 0 in Ω. MREIT: F : σ σ E 2 where ( µ 1 E ) + σ t E + ε 2 E = J in Ω + bndy.cond. t 2 a-example: F : a u where (a u) = 0 in Ω + boundary conditions c-example: F : c u where u + c u = 0 in Ω + boundary conditions phase retrieval: F : f Ff

11 Forward operator F EIT: F : σ Λ σ where Λ σ : φ σ n φ and (σ φ) = 0 in Ω. MREIT: F : σ σ E 2 where ( µ 1 E ) + σ t E + ε 2 E = J in Ω + bndy.cond. t 2 a-example: F : a u where (a u) = 0 in Ω + boundary conditions c-example: F : c u where u + c u = 0 in Ω + boundary conditions phase retrieval: F : f Ff forward operator F is nonlinear and in the last example nonsmooth

12 Forward operator F EIT: F : σ Λ σ where Λ σ : φ σ n φ and (σ φ) = 0 in Ω. MREIT: F : σ σ E 2 where ( µ 1 E ) + σ t E + ε 2 E = J in Ω + bndy.cond. t 2 a-example: F : a u where (a u) = 0 in Ω + boundary conditions c-example: F : c u where u + c u = 0 in Ω + boundary conditions phase retrieval: F : f Ff forward operator F is nonlinear and in the last example nonsmooth

13 Nonlinear ill-posed problems nonlinear operator equation F (x) = y F : D(F )( X ) Y... nonlinear operator; F not continuously invertible; X, Y... Banach spaces; y δ y... noisy data, y δ y δ... noise level. regularization necessary

14 Motivation for working in Banach space X = L P with P 1 sparse solutions MS08 Optimization in Banach Spaces with Sparsity Constraints X = L P with P ellipticity and boundedness in the context of parameter id. in PDEs (e.g. (a u) = 0 ); avoid artificial increase of ill-posedness, that would result from a Hilbert space choice X = H d/2+ɛ

15 Motivation for working in Banach space X = L P with P 1 sparse solutions MS08 Optimization in Banach Spaces with Sparsity Constraints X = L P with P ellipticity and boundedness in the context of parameter id. in PDEs (e.g. (a u) = 0 ); avoid artificial increase of ill-posedness, that would result from a Hilbert space choice X = H d/2+ɛ Y = L R with R = realistic measurement noise model; avoid artificial increase of ill-posedness, that would result from a Hilbert space choice Y = L 2

16 Motivation for working in Banach space X = L P with P 1 sparse solutions MS08 Optimization in Banach Spaces with Sparsity Constraints X = L P with P ellipticity and boundedness in the context of parameter id. in PDEs (e.g. (a u) = 0 ); avoid artificial increase of ill-posedness, that would result from a Hilbert space choice X = H d/2+ɛ Y = L R with R = realistic measurement noise model; avoid artificial increase of ill-posedness, that would result from a Hilbert space choice Y = L 2

17 c-example in Banach spaces u + c u = 0 in Ω. Identify c from measurements of u in Ω R d. (theoretical) reconstruction formula: c = u u

18 c-example in Banach spaces u + c u = 0 in Ω. Identify c from measurements of u in Ω R d. (theoretical) reconstruction formula: c = u u abstract stability result: Assume u u > 0, use Y = L : c 1 c 2 L P 1 u u(c 1) u(c 2 ) W 2,P + u(c 2) W 2,P u 2 u(c 1 ) u(c 2 ) L,

19 c-example in Banach spaces u + c u = 0 in Ω. Identify c from measurements of u in Ω R d. (theoretical) reconstruction formula: c = u u abstract stability result: Assume u u > 0, use Y = L : c 1 c 2 L P 1 u u(c 1) u(c 2 ) W 2,P + u(c 2) W 2,P u 2 u(c 1 ) u(c 2 ) L, where W 2,d/2 (Ω) L (Ω) in the sense that W 2,d/2 (Ω) L (Ω) and W 2,d/2+ɛ (Ω) L (Ω) for any ɛ > 0;

20 c-example in Banach spaces u + c u = 0 in Ω. Identify c from measurements of u in Ω R d. (theoretical) reconstruction formula: c = u u abstract stability result: Assume u u > 0, use Y = L : c 1 c 2 L P 1 u u(c 1) u(c 2 ) W 2,P + u(c 2) W 2,P u 2 u(c 1 ) u(c 2 ) L, where W 2,d/2 (Ω) L (Ω) in the sense that W 2,d/2 (Ω) L (Ω) and W 2,d/2+ɛ (Ω) L (Ω) for any ɛ > 0; choose P = d/2, X = L d/2 (Ω)

21 c-example in Banach spaces u + c u = 0 in Ω. Identify c from measurements of u in Ω R d. (theoretical) reconstruction formula: c = u u abstract stability result: Assume u u > 0, use Y = L : c 1 c 2 L P 1 u u(c 1) u(c 2 ) W 2,P + u(c 2) W 2,P u 2 u(c 1 ) u(c 2 ) L, where W 2,d/2 (Ω) L (Ω) in the sense that W 2,d/2 (Ω) L (Ω) and W 2,d/2+ɛ (Ω) L (Ω) for any ɛ > 0; choose P = d/2, X = L d/2 (Ω)

22 Phase retrieval in Banach spaces Ff = r Reconstruct the real-valued function f : R R from measurements of the intensity r : R R + of its Fourier transform. natural preimage- and image spaces (Hausdorff-Young Theorem): X = L P R (R), Y = L P P 1 R (R) with P [1, 2]

23 Phase retrieval in Banach spaces Ff = r Reconstruct the real-valued function f : R R from measurements of the intensity r : R R + of its Fourier transform. natural preimage- and image spaces (Hausdorff-Young Theorem): X = L P R (R), Y = L P P 1 R (R) with P [1, 2] sparse signal, L measurement noise X = L 1, Y = L

24 Phase retrieval in Banach spaces Ff = r Reconstruct the real-valued function f : R R from measurements of the intensity r : R R + of its Fourier transform. natural preimage- and image spaces (Hausdorff-Young Theorem): X = L P R (R), Y = L P P 1 R (R) with P [1, 2] sparse signal, L measurement noise X = L 1, Y = L

25 Regularization in Banach space case Y = X : iterative and variational regularization methods for linear and nonlinear problems [Plato 92, 94, 95, Bakushinskii&Kokurin 04] Tikhonov regularization for linear and nonlinear problems S(F (x), y δ ) + αr(x) = min! [Burger&Osher 04, Resmerita&Scherzer 06, Hofmann&BK&Pöschl&Scherzer 07, Grasmair&Haltmeier&Scherzer 08, 10,... ] needs global minimizer, but Tikhonov functional i.g. nonsmooth and nonconvex if F nonlinear

26 Regularization in Banach space case Y = X : iterative and variational regularization methods for linear and nonlinear problems [Plato 92, 94, 95, Bakushinskii&Kokurin 04] Tikhonov regularization for linear and nonlinear problems S(F (x), y δ ) + αr(x) = min! [Burger&Osher 04, Resmerita&Scherzer 06, Hofmann&BK&Pöschl&Scherzer 07, Grasmair&Haltmeier&Scherzer 08, 10,... ] needs global minimizer, but Tikhonov functional i.g. nonsmooth and nonconvex if F nonlinear motivates iterative regularization for nonlinear problems [Schöpfer&Louis&Schuster 06, Hein&Kazimierski 10, BK&Schöpfer&Schuster 09, BK&Hofmann 10,... ]

27 Regularization in Banach space case Y = X : iterative and variational regularization methods for linear and nonlinear problems [Plato 92, 94, 95, Bakushinskii&Kokurin 04] Tikhonov regularization for linear and nonlinear problems S(F (x), y δ ) + αr(x) = min! [Burger&Osher 04, Resmerita&Scherzer 06, Hofmann&BK&Pöschl&Scherzer 07, Grasmair&Haltmeier&Scherzer 08, 10,... ] needs global minimizer, but Tikhonov functional i.g. nonsmooth and nonconvex if F nonlinear motivates iterative regularization for nonlinear problems [Schöpfer&Louis&Schuster 06, Hein&Kazimierski 10, BK&Schöpfer&Schuster 09, BK&Hofmann 10,... ]

28 Outline short review on iterative regularization for nonlinear problems in Hilbert space some Banach space tools Landweber for nonlinear problems in Banach space Newton for nonlinear problems in Banach space numerical tests

29 Iterative regularization for nonlinear problems in Hilbert space gradient method for min X F (x) y δ 2 Landweber iteration x δ k+1 = x δ k µ kf (x δ k ) ( F (x δ k ) y δ) [Hanke&Neubauer&Scherzer 96]

30 Iterative regularization for nonlinear problems in Hilbert space Newton s method for F (x) = y δ Levenberg Marquardt method plus regularization x δ k+1 = x δ k (F (x δ k ) F (x δ k ) + α ki ) 1 F (x δ k ) }{{} F (x δ k ) 1 (F (x δ k ) y δ) [Hanke 97, 10], Newton-CG: [Hanke 97], inexact Newton [Rieder 01] iteratively regularized Gauss-Newton method (IRGN) ( ) xk+1 δ = x k δ (F (xk δ ) F (xk δ ) + α ki ) 1 F (xk δ }{{ ) F (xk δ } ) y δ +α k (xk δ x 0) F (xk δ) 1 [Bakushinskii 92, BK&Neubauer&Scherzer 96, 08, BK 97, Hohage 97]

31 Some Banach space tools (I) Smoothness X... smooth norm Gâteaux differentiable on X \ {0}; X... uniformly smooth norm Fréchet differentiable on unit sphere; Convexity X... strictly convex boundary of unit ball contains no line segment; X... uniformly convex modulus of convexity δ X (ɛ) > 0 ɛ (0, 2]; δ X (ɛ) = inf{1 1 2 (x + y) : x = y = 1, x y ɛ} L P (Ω), P (1, ) is uniformly convex (Hanner s ineq.) and uniformly smooth

32 Some Banach space tools (II) Dual space: X = L(X, R)... bounded linear functionals on X x : x x, x X uniformly smooth X uniformly convex X reflexive: X smooth X strictly convex

33 Some Banach space tools (III) Duality mapping: J p : X 2 X, J p (x) = {x X : x, x = x x = x p } J p set valued; j p... single valued selection of J p ; J p = 1 p p (Asplund) X smooth J p single valued X reflexive, smooth, strictly convex J 1 p = J p p 1 L P (Ω), P (1, ): J P (x) = x p P L P x P 1 sign(x) J P possibly nonlinear and nonsmooth

34 Some Banach space tools (IV) Bregman distance: D p (x, x) = 1 p x p 1 p x p inf{ ξ, x x : ξ J p ( x)}, x, x X ; smooth X : D p (x, x) = p 1 p ( x p x p ) + J p (x) J p ( x), x ; smooth and uniformly convex X : convergence in D p convergence in Hilbert space case: D 2 (x, x) = 1 2 x x 2

35 Assumptions on pre-image and image space X, Y X smooth, uniformly convex X reflexive (Milman-Pettis) and strictly convex J p single valued, norm-to-weak-continuous, bijective Y arbitrary Banach space

36 Further assumptions for convergence proofs closeness to a solution x : x 0 x sufficiently small F Lipschitz continuous and x sufficiently smooth or tangential cone condition: For all x D(F ) there exists F (x) L(X, Y ) (F (x) not necess. Fréchet derivative) s.t. F (x) F ( x) F (x)(x x) c tc F (x) F ( x) x, x B for Landweber: F, F continuous and interior of D(F ) nonempty for IRGN: F (weakly) sequentially closed

37 Stopping rule discrepancy principle k (δ) = min{k N : F (x δ k ) y δ Cdp δ} C dp > 1, y δ y δ... noise level Trade off between stability and approximation: Stop as early as possible (stability) such that the resudual is lower than the noise level (approximation)

38 Landweber for inverse problems in Banach space J p (x δ k+1 ) = J p(x δ k ) µ kf (x δ k ) j r (F (x δ k ) y δ ), xk+1 δ = J p (J p (xk+1 δ )) p 1 p, r (1, ). for comparison: Landweber in Hilbert space: x δ k+1 = x δ k µ kf (x δ k ) ( F (x δ k ) y δ)

39 Convergence results for Landweber (I) Theorem (monotonicity of the error) µ k appropriately chosen (suff. small), c tc suff. small, C dp suff. large. Then for all k k (δ) 1, xk+1 δ D(F ) and D p (x, x δ k+1 ) D p(x, x δ k ) C F (xk ) y δ p F (x δ k ) p < 0

40 Idea of proof recall: D p (x, x) = p 1 p ( x p x p ) + (J p (x) J p ( x), x) D p (x, xk+1 δ ) D p(x, xk δ ) = p 1 ( x δ p x p k+1 δ p) k J p (xk+1 δ ) J p(xk δ }{{} ), x =µ k F (xk δ) j r (F (xk δ) y δ ) = D p (xk δ, x k+1 δ ) µ k j r (F (xk δ ) y δ ), F (xk δ )(x k δ x ) }{{} F (xk δ) y δ

41 Idea of proof recall: D p (x, x) = p 1 p ( x p x p ) + (J p (x) J p ( x), x) D p (x, xk+1 δ ) D p(x, xk δ ) = p 1 ( x δ p x p k+1 δ p) k J p (xk+1 δ ) J p(xk δ }{{} ), x =µ k F (xk δ) j r (F (xk δ) y δ ) = D p (xk δ, x k+1 δ ) µ k j r (F (xk δ ) y δ ), F (xk δ )(x k δ x ) }{{} ) D p (xk δ, x k+1 δ F }{{} ) µ k (1 c(c tc, C dp ) (x δ k ) y δ r =O(µ 1+ɛ k ) F (x δ k ) y δ

42 Idea of proof recall: D p (x, x) = p 1 p ( x p x p ) + (J p (x) J p ( x), x) D p (x, xk+1 δ ) D p(x, xk δ ) = p 1 ( x δ p x p k+1 δ p) k J p (xk+1 δ ) J p(xk δ }{{} ), x =µ k F (xk δ) j r (F (xk δ) y δ ) = D p (xk δ, x k+1 δ ) µ k j r (F (xk δ ) y δ ), F (xk δ )(x k δ x ) }{{} ) D p (xk δ, x k+1 δ F }{{} ) µ k (1 c(c tc, C dp ) (x δ k ) y δ r =O(µ 1+ɛ k ) 0 by the choice of µ k. F (x δ k ) y δ

43 Idea of proof recall: D p (x, x) = p 1 p ( x p x p ) + (J p (x) J p ( x), x) D p (x, xk+1 δ ) D p(x, xk δ ) = p 1 ( x δ p x p k+1 δ p) k J p (xk+1 δ ) J p(xk δ }{{} ), x =µ k F (xk δ) j r (F (xk δ) y δ ) = D p (xk δ, x k+1 δ ) µ k j r (F (xk δ ) y δ ), F (xk δ )(x k δ x ) }{{} ) D p (xk δ, x k+1 δ F }{{} ) µ k (1 c(c tc, C dp ) (x δ k ) y δ r =O(µ 1+ɛ k ) 0 by the choice of µ k. F (x δ k ) y δ

44 Convergence results for Landweber (II) Theorem (convergence with exact data) δ = 0, µ k appropriately chosen (suff. small). Then x k x solution to F (x) = y as k Theorem (stability for δ > 0 and convergence as δ 0) µ k appropriately chosen, c tc suff. small, C dp suff. large. Y uniformly smooth. Then for all k k (δ), x δ k continuously depends on y δ and x δ k (δ) x solution to F (x) = y as δ 0 [Schöpfer&Louis&Schuster 06] linear case, [BK&Schöpfer&Schuster 09] nonlinear case

45 Remark convergence rates can be shown for the iteratively regularized Landweber iteration J p (x δ k+1 x 0) = (1 α k )J p (x δ k x 0) µ k F (x δ k ) j r (F (x δ k ) y δ ) xk+1 δ = x 0 + J p (J p (xk+1 δ x 0)) p 1 p, r (1, ), and x 0... initial guess.

46 IRGN for inverse problems in Banach space x δ k+1 argmin x D(F ) p, r (1, ), and x 0... initial guess; F (x δ k )(x x δ k ) + F (x δ k ) y δ r +αk x x 0 p, convex minimization problem: efficient solution see, e.g., [Bonesky, Kazimierski, Maass, Schöpfer, Schuster 07] for comparison: IRGN in Hilbert space: xk+1 δ = x k δ (F (xk δ ) F (xk δ ) + α ki ) 1( ) F (xk δ ) y δ + α k (xk δ x 0)

47 Choice of α k discrepancy type principle: θ F (x k ) y δ F (xk δ )(x k+1 δ (α) x k δ ) + F (x k δ ) y δ F θ (xk ) y δ 0 < θ < θ < 1 Trade off between stability and approximation: Choose α k as large as possible (stability) such that the predicted residual is smaller than the old one (approximation) see also: inexact Newton (for inverse problems: [Hanke 97, Rieder 99, 01])

48 Convergence of the IRGN Theorem [BK&Schöpfer&Schuster 09] C dp, θ, θ sufficiently large, c tc sufficently small. Additionally, assume that either (a) F (x) : X Y is weakly closed for all x D(F ) and Y reflexive or (b) D(F ) weakly closed. Then for all k k (δ) 1 the iterates xk+1 δ F argmin (xk δ )(x x k δ ) + F (x k δ ) y δ r + αk x x 0 p α k s.t. F (xk δ )(x k+1 δ x k δ ) + F (x k δ ) y δ F θ (xk ) y δ are well-defined and x k (δ) x solution to F (x) = y as δ 0 if x unique, (and along subsequences otherwise).

49 Idea of proof By minimality of x δ k+1 : F (xk δ )(x k+1 δ x k δ ) + F (x k δ ) y δ r x + δ p αk k+1 x 0 F (x δ k )(x x δ k ) + F (x δ k ) y δ r } {{ } δ r C r dp F (xk δ) y δ r +α k x x 0 p. Choice of α k F (x δ k )(x δ k+1 x δ k ) + F (x δ k ) y δ θ F (x δ k ) y δ

50 Idea of proof By minimality of x δ k+1 : F (xk δ )(x k+1 δ x k δ ) + F (x k δ ) y δ r x + δ p αk k+1 x 0 F (x δ k )(x x δ k ) + F (x δ k ) y δ r } {{ } δ r C r dp F (xk δ) y δ r +α k x x 0 p. Choice of α k F (x δ k )(x δ k+1 x δ k ) + F (x δ k ) y δ θ F (x δ k ) y δ α k ( x δ k+1 x 0 p x x 0 p) (c(c tc, C dp ) θ r ) F (x δ k ) y δ r

51 Idea of proof By minimality of x δ k+1 : F (xk δ )(x k+1 δ x k δ ) + F (x k δ ) y δ r x + δ p αk k+1 x 0 F (x δ k )(x x δ k ) + F (x δ k ) y δ r } {{ } δ r C r dp F (xk δ) y δ r +α k x x 0 p. Choice of α k F (x δ k )(x δ k+1 x δ k ) + F (x δ k ) y δ θ F (x δ k ) y δ α k ( x δ k+1 x 0 p x x 0 p) (c(c tc, C dp ) θ r ) F (x δ k ) y δ r Choice of θ r > c(c tc, C dp ) xk+1 δ x p 0 x x 0 p

52 Idea of proof By minimality of x δ k+1 : F (xk δ )(x k+1 δ x k δ ) + F (x k δ ) y δ r x + δ p αk k+1 x 0 F (x δ k )(x x δ k ) + F (x δ k ) y δ r } {{ } δ r C r dp F (xk δ) y δ r +α k x x 0 p. Choice of α k F (x δ k )(x δ k+1 x δ k ) + F (x δ k ) y δ θ F (x δ k ) y δ α k ( x δ k+1 x 0 p x x 0 p) (c(c tc, C dp ) θ r ) F (x δ k ) y δ r Choice of θ r > c(c tc, C dp ) xk+1 δ x p 0 x x 0 p

53 Convergence rates for the IRGN Theorem [BK&Hofmann 10] Let the assumptions of the previous theorem be satisfied. Under the source type condition J p (x x 0 ) R(F (x ) ), i.e., ˆξ J p (x x 0 ), v Y : ˆξ = F (x ) v we obtain optimal convergence rates D p (x k x 0, x x 0 ) = O(δ), where D x 0 p (x, x) = D p (x x 0, x x 0 ). Hilbert space case: x k x = O( δ)

54 Idea of proof recall: D p (x, x) = 1 p x p 1 p x p inf{ ξ, x x : ξ J p ( x)} We have, from the previous proof, xk δ x p 0 x x 0 p for k k.

55 Idea of proof recall: D p (x, x) = 1 p x p 1 p x p inf{ ξ, x x : ξ J p ( x)} We have, from the previous proof, xk δ x p 0 x x 0 p for k k. D p (xk δ x 0, x x 0 ) 1 xk δ p x 0 p 1 p x p x 0 } {{ } 0 ˆξ, x x }{{} k δ =F (x ) v

56 Idea of proof recall: D p (x, x) = 1 p x p 1 p x p inf{ ξ, x x : ξ J p ( x)} We have, from the previous proof, xk δ x p 0 x x 0 p for k k. D p (xk δ x 0, x x 0 ) 1 xk δ p x 0 p 1 p x p x 0 } {{ } 0 v, F (x )(xk δ x ) }{{} F (xk δ) y δ ˆξ, x x }{{} k δ =F (x ) v

57 Idea of proof recall: D p (x, x) = 1 p x p 1 p x p inf{ ξ, x x : ξ J p ( x)} We have, from the previous proof, xk δ x p 0 x x 0 p for k k. D p (xk δ x 0, x x 0 ) 1 xk δ p x 0 p 1 p x p x 0 } {{ } 0 v, F (x )(xk δ x ) }{{} F (xk δ) y δ v (1 + c tc ) F (x δ k ) y δ ˆξ, x x }{{} k δ =F (x ) v

58 Idea of proof recall: D p (x, x) = 1 p x p 1 p x p inf{ ξ, x x : ξ J p ( x)} We have, from the previous proof, xk δ x p 0 x x 0 p for k k. D p (xk δ x 0, x x 0 ) 1 xk δ p x 0 p 1 p x p x 0 } {{ } 0 v, F (x )(xk δ x ) }{{} F (xk δ) y δ v (1 + c tc ) F (x δ k ) y δ ˆξ, x x }{{} k δ =F (x ) v Hence, for k = k : D p (x x 0, x δ k x 0 ) v (1 + c tc )C dp δ.

59 Idea of proof recall: D p (x, x) = 1 p x p 1 p x p inf{ ξ, x x : ξ J p ( x)} We have, from the previous proof, xk δ x p 0 x x 0 p for k k. D p (xk δ x 0, x x 0 ) 1 xk δ p x 0 p 1 p x p x 0 } {{ } 0 v, F (x )(xk δ x ) }{{} F (xk δ) y δ v (1 + c tc ) F (x δ k ) y δ ˆξ, x x }{{} k δ =F (x ) v Hence, for k = k : D p (x x 0, x δ k x 0 ) v (1 + c tc )C dp δ.

60 Remarks rates result can be extended to D x 0 p (x k, x ) = O(δ κ ) with κ (0, 1) or D x 0 p (x k, x ) = O(log(δ) κ ) with κ > 0 under approximate source conditions; rates can be shown alternatively with a priori choice of α k and k instead of the discrepancy principle; needs a priori information on smoothness of x, though

61 Numerical tests c-example: u + c u = 0 in Ω. Identify c from measurements of u in Ω. Ω = (0, 1) 2 R 2, smooth c: c(x, y) = 10 exp ( (x 0.3)2 + (y 0.3) 2 ) 0.04 sparse c: c(x, y) = 40χ [0.19,0.24] 2(x, y)

62 Test examples smooth c: sparse c:

63 Smooth test example, 1% L -noise exact data u: noisy data u δ, u u δ L =1%:

64 Smooth test example, 1% L -noise exact potential c: reconstruction c δ k, X = L 2, Y = L 2, p = 2, r = 2:

65 Smooth test example, 1% L -noise exact potential c: reconstruction c δ k, X = L 2, Y = L 22, p = 2, r = 2:

66 Smooth test example, 10% L -noise exact data u: noisy data u δ, u u δ L =10%:

67 Smooth test example, 10% L -noise exact potential c: reconstruction c δ k, X = L 2, Y = L 2, p = 2, r = 2:

68 Smooth test example, 10% L -noise exact potential c: reconstruction c δ k, X = L 2, Y = L 22, p = 2, r = 2:

69 Smooth test example, 10% L -noise exact potential c: reconstruction c δ k, X = L 1.1, Y = L 22, p = 1.1, r = 2:

70 Sparse test example, 3% L -noise exact potential c: exact data u: noisy data u δ : computations by Frank Schöpfer

71 Sparse test example, 3% L -noise exact potential c: reconstructions c δ k : X = L 2, Y = L 11 X = L 1.1, Y = L 11 p = 2, r = 2: p = 1.1, r = 2:

72 Sparse test example, 3% L -noise

73 Sparse test example, 1% L -noise exact potential c: reconstructions c δ k : X = L 2, Y = L 11 X = L 1.1, Y = L 11 p = 2, r = 2: p = 1.1, r = 2:

74 Sparse test example, 1% L -noise

75 Summary and Outlook motivation for solving inverse problems in Banach spaces: more natural norms, possible reduction of ill-posedness, sparsity use of Banach spaces instead of Hilbert spaces may add nonlinearity and nonsmoothness, but keeps convexity gradient (Landweber) and Gauss-Newton methods for nonlinear inverse problems formulation and convergence analysis in Banach space replace Tikhonov for Newton step by an inner iteration...

76 Thank you for your attention!