Solving an inverse elliptic coefficient problem by convex nonlinear semidefinite programming
Abstract
Several applications in medical imaging and nondestructive 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 nonlinear illposed 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 nonlinear 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 optimizationbased approaches for inverse coefficient problems.
 Publication:

arXiv eprints
 Pub Date:
 May 2021
 arXiv:
 arXiv:2105.11440
 Bibcode:
 2021arXiv210511440H
 Keywords:

 Mathematics  Optimization and Control;
 Mathematics  Analysis of PDEs;
 Mathematics  Numerical Analysis;
 35R30;
 90C22
 EPrint:
 Optim. Lett. 2021