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 prob...

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Bibliographic Details
Published in:Optimization letters Vol. 16; no. 5; pp. 1599 - 1609
Main Author: Harrach, Bastian
Format: Journal Article
Language:English
Published: Berlin, Heidelberg Springer 01.06.2022
Springer Berlin Heidelberg
Subjects:
Computational Intelligence
Convexity
Finitely many measurements
Inverse Problem
Loewner order
Mathematics
Mathematics and Statistics
Monotonicity
Numerical and Computational Physics
Operations Research/Decision Theory
Optimization
Original Paper
Simulation
35R30
Inverse Problem
90C22
Loewner order
Convexity
Finitely many measurements
Monotonicity
ISSN:1862-4480, 1862-4472, 1862-4480
Online Access:Get full text
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Summary: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.
ISSN:1862-4480
1862-4472
1862-4480
DOI:10.1007/s11590-021-01802-4