An analysis of a multi-level projected steepest descent iteration for nonlinear inverse problems in Banach spaces subject to stability constraints
We consider nonlinear inverse problems described by operator equations in Banach spaces. Assuming conditional stability of the inverse problem, that is, assuming that stability holds on a compact, convex subset of the domain of the operator, we introduce a novel nonlinear projected steepest descent...
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| Published in | Numerische Mathematik Vol. 129; no. 1; pp. 127 - 148 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Springer Berlin Heidelberg
01.01.2015
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| Abstract | We consider nonlinear inverse problems described by operator equations in Banach spaces. Assuming conditional stability of the inverse problem, that is, assuming that stability holds on a compact, convex subset of the domain of the operator, we introduce a novel nonlinear projected steepest descent iteration and analyze its convergence to an approximate solution given limited accuracy data. We proceed with developing a multi-level algorithm based on a nested family of compact, convex subsets on which stability holds and the stability constants are ordered. Growth of the stability constants is coupled to the increase in accuracy of approximation between neighboring levels to ensure that the algorithm can continue from level to level until the iterate satisfies a desired discrepancy criterion, after a finite number of steps. |
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| AbstractList | We consider nonlinear inverse problems described by operator equations in Banach spaces. Assuming conditional stability of the inverse problem, that is, assuming that stability holds on a compact, convex subset of the domain of the operator, we introduce a novel nonlinear projected steepest descent iteration and analyze its convergence to an approximate solution given limited accuracy data. We proceed with developing a multi-level algorithm based on a nested family of compact, convex subsets on which stability holds and the stability constants are ordered. Growth of the stability constants is coupled to the increase in accuracy of approximation between neighboring levels to ensure that the algorithm can continue from level to level until the iterate satisfies a desired discrepancy criterion, after a finite number of steps. |
| Author | Scherzer, Otmar de Hoop, Maarten V. Qiu, Lingyun |
| Author_xml | – sequence: 1 givenname: Maarten V. surname: de Hoop fullname: de Hoop, Maarten V. organization: Center for Computational and Applied Mathemematics, Purdue University – sequence: 2 givenname: Lingyun surname: Qiu fullname: Qiu, Lingyun email: qiu.lingyun@ima.umn.edu, qiu.lingyun@gmail.com organization: Institute for Mathematics and its Applications, University of Minnesota – sequence: 3 givenname: Otmar surname: Scherzer fullname: Scherzer, Otmar organization: Computational Science Center, University of Vienna |
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| Cites_doi | 10.1007/BF01385727 10.1080/01630569208816489 10.1016/0022-247X(91)90144-O 10.1016/j.aam.2004.12.002 10.1088/0266-5611/17/5/313 10.1137/S0036139994267444 10.1023/A:1022631928592 10.1137/120869201 10.1216/JIE-2008-20-2-201 10.1155/S1085337596000024 10.1080/03605302.2011.552930 10.1007/s00041-008-9039-8 10.1007/s002110050379 10.4171/ZAA/679 10.1088/0266-5611/28/4/045001 10.1080/01630560600790835 10.1016/0041-5553(67)90040-7 10.1016/j.jmaa.2012.10.066 10.1007/978-94-009-2121-4 10.1088/0266-5611/26/2/025007 10.1515/9783110255720 10.1515/9783110208276 |
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| References | Butnariu, Iusem, Resmerita (CR8) 2000; 7 Scherzer (CR21) 1996; 17 CR18 Teschke, Borries (CR24) 2010; 26 CR10 Neubauer, Scherzer (CR20) 1995; 14 de Hoop, Qiu, Scherzer (CR12) 2012; 28 Xu, Roach (CR25) 1991; 157 Beretta, de Hoop, Qiu (CR5) 2013; 45 Alber, Butnariu (CR1) 1997; 92 Kaltenbacher (CR17) 2008; 20 CR4 Daubechies, Fornasier, Loris (CR11) 2008; 14 Alessandrini, Vessella (CR3) 2005; 35 Chavent, Kunisch (CR9) 1996; 56 Beretta, Francini (CR6) 2011; 36 Kaltenbacher (CR16) 2006; 27 Hanke (CR15) 1991; 60 Scherzer (CR22) 1998; 80 CR23 Mandache (CR19) 2001; 17 Bregman (CR7) 1967; 7 Eicke (CR13) 1992; 13 Gilyazov (CR14) 1977; 13 Alber, Kartsatos, Litsyn (CR2) 1996; 1 YI Alber (629_CR1) 1997; 92 629_CR10 A Neubauer (629_CR20) 1995; 14 SF Gilyazov (629_CR14) 1977; 13 629_CR18 E Beretta (629_CR5) 2013; 45 B Eicke (629_CR13) 1992; 13 N Mandache (629_CR19) 2001; 17 O Scherzer (629_CR22) 1998; 80 629_CR4 ZB Xu (629_CR25) 1991; 157 MV Hoop de (629_CR12) 2012; 28 G Chavent (629_CR9) 1996; 56 E Beretta (629_CR6) 2011; 36 629_CR24 LM Bregman (629_CR7) 1967; 7 629_CR23 G Alessandrini (629_CR3) 2005; 35 I Daubechies (629_CR11) 2008; 14 B Kaltenbacher (629_CR17) 2008; 20 O Scherzer (629_CR21) 1996; 17 YI Alber (629_CR2) 1996; 1 M Hanke (629_CR15) 1991; 60 B Kaltenbacher (629_CR16) 2006; 27 D Butnariu (629_CR8) 2000; 7 |
| References_xml | – volume: 60 start-page: 341 issue: 3 year: 1991 end-page: 373 ident: CR15 article-title: Accelerated Landweber iterations for the solution of ill-posed equations publication-title: Numer. Math. doi: 10.1007/BF01385727 – volume: 13 start-page: 413 issue: 5–6 year: 1992 end-page: 429 ident: CR13 article-title: Iteration methods for convexly constrained ill-posed problems in Hilbert space publication-title: Numer. Funct. Anal. Optim. doi: 10.1080/01630569208816489 – ident: CR18 – volume: 26 start-page: 025007 issue: 2 year: 2010 ident: CR24 article-title: Accelerated projected steepest descent method for nonlinear inverse problems with sparsity constraints publication-title: Inverse Probl. – volume: 7 start-page: 319 issue: 2 year: 2000 end-page: 334 ident: CR8 article-title: Total convexity for powers of the norm in uniformly convex Banach spaces publication-title: J. Convex Anal. – volume: 157 start-page: 189 issue: 1 year: 1991 end-page: 210 ident: CR25 article-title: Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces publication-title: J. Math. Anal. Appl. doi: 10.1016/0022-247X(91)90144-O – ident: CR4 – ident: CR10 – volume: 35 start-page: 207 issue: 2 year: 2005 end-page: 241 ident: CR3 article-title: Lipschitz stability for the inverse conductivity problem publication-title: Adv. Appl. Math. doi: 10.1016/j.aam.2004.12.002 – volume: 17 start-page: 1435 issue: 5 year: 2001 end-page: 1444 ident: CR19 article-title: Exponential instability in an inverse problem for the Schrödinger equation publication-title: Inverse Probl. doi: 10.1088/0266-5611/17/5/313 – ident: CR23 – volume: 56 start-page: 542 issue: 2 year: 1996 end-page: 572 ident: CR9 article-title: On weakly nonlinear inverse problems publication-title: SIAM. J. Appl. Math. doi: 10.1137/S0036139994267444 – volume: 92 start-page: 33 year: 1997 end-page: 61 ident: CR1 article-title: Convergence of Bregman projection methods for solving consistent convex feasibility problems in reflexive Banach spaces publication-title: J. Optim. Theory Appl. doi: 10.1023/A:1022631928592 – volume: 45 start-page: 679 issue: 2 year: 2013 end-page: 699 ident: CR5 article-title: Lipschitz stability of an inverse boundary value problem for a Schrödinger-type equation publication-title: SIAM J. Math. Anal. doi: 10.1137/120869201 – volume: 13 start-page: 8 year: 1977 end-page: 13 ident: CR14 article-title: Iterative solution methods for inconsistent linear equations with nonself-adjoint operator publication-title: Moscow Univ. Comput. Math. Cybernet. – volume: 20 start-page: 201 issue: 2 year: 2008 end-page: 228 ident: CR17 article-title: Convergence rates of a multilevel method for the regularization of nonlinear ill-posed problems publication-title: J. Integral Equ. Appl. doi: 10.1216/JIE-2008-20-2-201 – volume: 1 start-page: 45 issue: 1 year: 1996 end-page: 64 ident: CR2 article-title: Iterative solution of unstable variational inequalities on approximately given sets publication-title: Abstr. Appl. Anal. doi: 10.1155/S1085337596000024 – volume: 36 start-page: 1723 issue: 10 year: 2011 end-page: 1749 ident: CR6 article-title: Lipschitz stability for the electrical impedance tomography problem: the complex case publication-title: Commun. Partial Differ. Equ. doi: 10.1080/03605302.2011.552930 – volume: 14 start-page: 764 issue: 5–6 year: 2008 end-page: 792 ident: CR11 article-title: Accelerated projected gradient method for linear inverse problems with sparsity constraints publication-title: J. Fourier Anal. Appl. doi: 10.1007/s00041-008-9039-8 – volume: 80 start-page: 579 issue: 4 year: 1998 end-page: 600 ident: CR22 article-title: An iterative multi-level algorithm for solving nonlinear ill-posed problems publication-title: Numer. Math. doi: 10.1007/s002110050379 – volume: 14 start-page: 369 issue: 2 year: 1995 end-page: 377 ident: CR20 article-title: A convergence rate result for a steepest descent method and a minimal error method for the solution of nonlinear ill-posed problems publication-title: Z. Anal. Anwend. doi: 10.4171/ZAA/679 – volume: 28 start-page: 045001 issue: 4 year: 2012 ident: CR12 article-title: Local analysis of inverse problems: Hölder stability and iterative reconstruction publication-title: Inverse Probl. doi: 10.1088/0266-5611/28/4/045001 – volume: 27 start-page: 637 issue: 5–6 year: 2006 end-page: 665 ident: CR16 article-title: Toward global convergence for strongly nonlinear ill-posed problems via a regularizing multilevel approach publication-title: Numer. Funct. Anal. Optim. doi: 10.1080/01630560600790835 – volume: 7 start-page: 200 issue: 3 year: 1967 end-page: 217 ident: CR7 article-title: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming publication-title: USSR Comput. Math. Math. Phys. doi: 10.1016/0041-5553(67)90040-7 – volume: 17 start-page: 197 issue: 1–2 year: 1996 end-page: 214 ident: CR21 article-title: A convergence analysis of a method of steepest descent and a two-step algorithm for nonlinear ill-posed problems publication-title: Numer. Funct. Anal. Optim. – volume: 17 start-page: 197 issue: 1–2 year: 1996 ident: 629_CR21 publication-title: Numer. Funct. Anal. Optim. – volume: 157 start-page: 189 issue: 1 year: 1991 ident: 629_CR25 publication-title: J. Math. Anal. Appl. doi: 10.1016/0022-247X(91)90144-O – volume: 17 start-page: 1435 issue: 5 year: 2001 ident: 629_CR19 publication-title: Inverse Probl. doi: 10.1088/0266-5611/17/5/313 – ident: 629_CR4 doi: 10.1016/j.jmaa.2012.10.066 – volume: 7 start-page: 200 issue: 3 year: 1967 ident: 629_CR7 publication-title: USSR Comput. Math. Math. Phys. doi: 10.1016/0041-5553(67)90040-7 – ident: 629_CR10 doi: 10.1007/978-94-009-2121-4 – ident: 629_CR24 doi: 10.1088/0266-5611/26/2/025007 – ident: 629_CR23 doi: 10.1515/9783110255720 – volume: 1 start-page: 45 issue: 1 year: 1996 ident: 629_CR2 publication-title: Abstr. Appl. Anal. doi: 10.1155/S1085337596000024 – volume: 28 start-page: 045001 issue: 4 year: 2012 ident: 629_CR12 publication-title: Inverse Probl. doi: 10.1088/0266-5611/28/4/045001 – volume: 14 start-page: 369 issue: 2 year: 1995 ident: 629_CR20 publication-title: Z. Anal. Anwend. doi: 10.4171/ZAA/679 – volume: 13 start-page: 8 year: 1977 ident: 629_CR14 publication-title: Moscow Univ. Comput. Math. Cybernet. – volume: 36 start-page: 1723 issue: 10 year: 2011 ident: 629_CR6 publication-title: Commun. Partial Differ. Equ. doi: 10.1080/03605302.2011.552930 – volume: 60 start-page: 341 issue: 3 year: 1991 ident: 629_CR15 publication-title: Numer. Math. doi: 10.1007/BF01385727 – volume: 7 start-page: 319 issue: 2 year: 2000 ident: 629_CR8 publication-title: J. Convex Anal. – volume: 20 start-page: 201 issue: 2 year: 2008 ident: 629_CR17 publication-title: J. Integral Equ. Appl. doi: 10.1216/JIE-2008-20-2-201 – volume: 35 start-page: 207 issue: 2 year: 2005 ident: 629_CR3 publication-title: Adv. Appl. Math. doi: 10.1016/j.aam.2004.12.002 – volume: 14 start-page: 764 issue: 5–6 year: 2008 ident: 629_CR11 publication-title: J. Fourier Anal. Appl. doi: 10.1007/s00041-008-9039-8 – ident: 629_CR18 doi: 10.1515/9783110208276 – volume: 27 start-page: 637 issue: 5–6 year: 2006 ident: 629_CR16 publication-title: Numer. Funct. Anal. Optim. doi: 10.1080/01630560600790835 – volume: 13 start-page: 413 issue: 5–6 year: 1992 ident: 629_CR13 publication-title: Numer. Funct. Anal. Optim. doi: 10.1080/01630569208816489 – volume: 56 start-page: 542 issue: 2 year: 1996 ident: 629_CR9 publication-title: SIAM. J. Appl. Math. doi: 10.1137/S0036139994267444 – volume: 92 start-page: 33 year: 1997 ident: 629_CR1 publication-title: J. Optim. Theory Appl. doi: 10.1023/A:1022631928592 – volume: 45 start-page: 679 issue: 2 year: 2013 ident: 629_CR5 publication-title: SIAM J. Math. Anal. doi: 10.1137/120869201 – volume: 80 start-page: 579 issue: 4 year: 1998 ident: 629_CR22 publication-title: Numer. Math. doi: 10.1007/s002110050379 |
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| Title | An analysis of a multi-level projected steepest descent iteration for nonlinear inverse problems in Banach spaces subject to stability constraints |
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