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.
In this talk we will consider an inverse coefficient problem with finitely many measurements and a finite desired resolution. We will present a criterion based on monotonicity, convexity and localized potentials arguments that allows us to explicitly estimate the number of measurements that is required to achieve the desired resolution. We also obtain an error estimate for noisy data, and overcome the problem of local minima by rewriting the problem as an equivalent uniquely solvable convex non-linear semidefinite optimization problem.
1. B. Harrach, Uniqueness, stability and global convergence for a discrete inverse elliptic Robin transmission problem, Numer. Math. 147 (2021), pp. 29-70, https://doi.org/10.1007/s00211-020-01162-8
2. B. Harrach, Solving an inverse elliptic coefficient problem by convex non-linear semidefinite programming, arXiv preprint (2021), arXiv:2105.11440