Solving an inverse elliptic coefficient problem by convex non-linear semidefinite programming

  • Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its solutions. This is usually a highly non-linear ill-posed inverse problem, for which unique reconstructability results, stability estimates and global convergence of numerical methods are very hard to achieve. The aim of this note is to point out a new connection between inverse coefficient problems and semidefinite programming that may help addressing these challenges. We show that an inverse elliptic Robin transmission problem with finitely many measurements can be equivalently rewritten as a uniquely solvable convex non-linear semidefinite optimization problem. This allows to explicitly estimate the number of measurements that is required to achieve a desired resolution, to derive an error estimate for noisy data, and to overcome the problem of local minima that usually appears in optimization-based approaches for inverse coefficient problems.
Author:Bastian von HarrachORCiDGND
Parent Title (English):Optimization letters
Place of publication:Berlin; Heidelberg
Document Type:Article
Date of Publication (online):2021/09/09
Date of first Publication:2021/09/09
Publishing Institution:Universit├Ątsbibliothek Johann Christian Senckenberg
Release Date:2022/02/09
Tag:Convexity; Finitely many measurements; Inverse Problem; Loewner order; Monotonicity
Issue:Online Version of Record before inclusion in an issue
Page Number:11
Open Access funding enabled and organized by Projekt DEAL.
Early View-Version: Online Version of Record before inclusion in an issue.
Institutes:Informatik und Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
MSC-Classification:35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Rxx Miscellaneous topics (For equations on manifolds, see 58Jxx; for manifolds of solutions, see 58Bxx; for stochastic PDE, see also 60H15) / 35R30 Inverse problems
90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C22 Semidefinite programming
Licence (German):License LogoCreative Commons - Namensnennung 4.0