Regularization and Discretization of Inverse Problems for PDEs in Banach spaces
Regularization and Discretization of Inverse Problems for PDEs in Banach spaces
DACH: Österreich - Deutschland - Schweiz
Disciplines
Mathematics (100%)
Keywords
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Inverse Problems,
Parameter Identification,
Regularization In Banach Spaces,
Partial Differential Equations,
Adaptive Discretization,
Parameter Choice Strategies
The aim of this project is a combined analysis of regularization and discretization of ill-posed problems in Banach spaces specifically in the context of partial differential equations. Such problems play a crucial role in numerous applications ranging from medical imaging via nondestructive testing to geophysical prospecting, with the Banach space setting mandated by the inherent regularity of the sought coefficients as well as structural features such as sparsity. Our goal is to fill the gap between the existing abstract regularization theory in general Banach spaces and the adaptive discretization of well-posed optimization problems in Hilbert spaces with pointwise constraints to derive explicit source conditions and practical parameter choice rules and to develop adaptive discretization methods based on functional and goal-oriented error estimates that take into account the interdependence of regularization parameter, data noise level and discretization error. This will lead to an integrated approach for the stable and efficient numerical solution method of parameter identification problems in Banach spaces.
The aim of this project is a combined analysis of regularization and discretization of ill-posed problems in Banach spaces specifically in the context of partial differential equations. Such problems play a crucial role in numerous applications ranging from medical imaging via nondestructive testing to geophysical prospecting, with the Banach space setting mandated by the inherent regularity of the sought coefficients as well as structural features such as sparsity. Our goal is to fill the gap between the existing abstract regularization theory in general Banach spaces and the adaptive discretization of well-posed optimization problems in Hilbert spaces with pointwise constraints to derive explicit source conditions and practical parameter choice rules and to develop adaptive discretization methods based on functional and goal-oriented error estimates that take into account the interdependence of regularization parameter, data noise level and discretization error. This will lead to an integrated approach for the stable and efficient numerical solution method of parameter identification problems in Banach spaces.
- Universität Klagenfurt - 100%
- Uno Hämarik, University of Tartu - Estonia
- Martin Burger, Friedrich-Alexander-Universität Erlangen-Nürnberg - Germany
- Daniel Wachsmuth, Julius-Maximilians-Universität Würzburg - Germany
- Boris Vexler, Technische Universität München - Germany
- Christian Clason, Universität Duisburg-Essen - Germany
- Michael Hintermüller, Weierstraß-Institut für Angewandte Analysis und Stochastik - Germany
Research Output
- 306 Citations
- 29 Publications
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2020
Title Regularization of inverse problems via box constrained minimization DOI 10.3934/ipi.2020021 Type Journal Article Author Hungerländer P Journal Inverse Problems and Imaging Pages 437-461 Link Publication -
2019
Title The Ivanov regularized Gauss–Newton method in Banach space with an a posteriori choice of the regularization radius DOI 10.1515/jiip-2018-0093 Type Journal Article Author Kaltenbacher B Journal Journal of Inverse and Ill-posed Problems Pages 539-557 Link Publication -
2019
Title On an inverse potential problem for a fractional reaction–diffusion equation DOI 10.1088/1361-6420/ab109e Type Journal Article Author Kaltenbacher B Journal Inverse Problems Pages 065004 Link Publication -
2019
Title Regularization of a backwards parabolic equation by fractional operators DOI 10.3934/ipi.2019020 Type Journal Article Author Kaltenbacher B Journal Inverse Problems and Imaging Pages 401-430 Link Publication -
2019
Title On the identification of a nonlinear term in a reaction–diffusion equation DOI 10.1088/1361-6420/ab2aab Type Journal Article Author Kaltenbacher B Journal Inverse Problems Pages 115007 Link Publication -
2019
Title The Ivanov regularized Gauss-Newton method in Banach space with an a posteriori choice of the regularization radius DOI 10.48550/arxiv.1910.01811 Type Preprint Author Kaltenbacher B -
2019
Title Minimization based formulations of inverse problems and their regularization DOI 10.48550/arxiv.1910.01813 Type Preprint Author Kaltenbacher B -
2019
Title All-at-once versus reduced iterative methods for time dependent inverse problems DOI 10.48550/arxiv.1910.02857 Type Preprint Author Kaltenbacher B -
2019
Title Regularization of a backwards parabolic equation by fractional operators DOI 10.48550/arxiv.1910.02232 Type Preprint Author Kaltenbacher B -
2019
Title Recovery of multiple coefficients in a reaction-diffusion equation DOI 10.48550/arxiv.1905.12232 Type Preprint Author Kaltenbacher B -
2019
Title On the identification of a nonlinear term in a reaction-diffusion equation DOI 10.48550/arxiv.1905.12067 Type Preprint Author Kaltenbacher B -
2016
Title Functional error estimators for the adaptive discretization of inverse problems DOI 10.1088/0266-5611/32/10/104004 Type Journal Article Author Clason C Journal Inverse Problems Pages 104004 Link Publication -
2016
Title The least error method for sparse solution reconstruction DOI 10.1088/0266-5611/32/9/094001 Type Journal Article Author Bredies K Journal Inverse Problems Pages 094001 Link Publication -
2016
Title Integration based profile likelihood calculation for PDE constrained parameter estimation problems DOI 10.1088/0266-5611/32/12/125009 Type Journal Article Author Boiger R Journal Inverse Problems Pages 125009 Link Publication -
2016
Title Regularization based on all-at-once formulations for inverse problems DOI 10.48550/arxiv.1603.05332 Type Preprint Author Kaltenbacher B -
2016
Title The least error method for sparse solution reconstruction DOI 10.48550/arxiv.1602.04429 Type Preprint Author Bredies K -
2016
Title Lavrentiev's regularization method in Hilbert spaces revisited DOI 10.3934/ipi.2016019 Type Journal Article Author Hofmann B Journal Inverse Problems and Imaging Pages 741-764 Link Publication -
2018
Title Continuous analogue to iterative optimization for PDE-constrained inverse problems DOI 10.1080/17415977.2018.1494167 Type Journal Article Author Boiger R Journal Inverse Problems in Science and Engineering Pages 710-734 Link Publication -
2018
Title On convergence and convergence rates for Ivanov and Morozov regularization and application to some parameter identification problems in elliptic PDEs DOI 10.1088/1361-6420/aab739 Type Journal Article Author Kaltenbacher B Journal Inverse Problems Pages 055008 Link Publication -
2018
Title Minimization Based Formulations of Inverse Problems and Their Regularization DOI 10.1137/17m1124036 Type Journal Article Author Kaltenbacher B Journal SIAM Journal on Optimization Pages 620-645 Link Publication -
2017
Title Convergence and adaptive discretization of the IRGNM Tikhonov and the IRGNM Ivanov method under a tangential cone condition in Banach space DOI 10.48550/arxiv.1707.07589 Type Preprint Author Kaltenbacher B -
2016
Title Identifying conductivity in electrical impedance tomography with total variation regularization DOI 10.48550/arxiv.1609.03714 Type Preprint Author Hinze M -
2016
Title Regularization Based on All-At-Once Formulations for Inverse Problems DOI 10.1137/16m1060984 Type Journal Article Author Kaltenbacher B Journal SIAM Journal on Numerical Analysis Pages 2594-2618 Link Publication -
2017
Title Identifying conductivity in electrical impedance tomography with total variation regularization DOI 10.1007/s00211-017-0920-8 Type Journal Article Author Hinze M Journal Numerische Mathematik Pages 723-765 -
2017
Title All-at-once versus reduced iterative methods for time dependent inverse problems DOI 10.1088/1361-6420/aa6f34 Type Journal Article Author Kaltenbacher B Journal Inverse Problems Pages 064002 Link Publication -
2018
Title Convergence and adaptive discretization of the IRGNM Tikhonov and the IRGNM Ivanov method under a tangential cone condition in Banach space DOI 10.1007/s00211-018-0971-5 Type Journal Article Author Kaltenbacher B Journal Numerische Mathematik Pages 449-478 Link Publication -
2018
Title On convergence and convergence rates for Ivanov and Morozov regularization and application to some parameter identification problems in elliptic PDEs DOI 10.48550/arxiv.1801.10181 Type Preprint Author Kaltenbacher B -
2018
Title Regularization of inverse problems via box constrained minimization DOI 10.48550/arxiv.1807.11316 Type Preprint Author Hungerländer P -
2020
Title Recovery of multiple coefficients in a reaction-diffusion equation DOI 10.1016/j.jmaa.2019.123475 Type Journal Article Author Kaltenbacher B Journal Journal of Mathematical Analysis and Applications Pages 123475 Link Publication