Abstract
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.
Highlights
Inverse elliptic coefficient problems arise in a number of applications in medical imaging and non-destructive material testing
Harrach uniqueness questions for inverse elliptic coefficient problems have mostly been studied in the idealized infinite-dimensional setting where the unknown coefficient function is to be determined with infinite resolution from infinitely many measurements, cf., e.g., [7,12,15]
Lipschitz stability results have been obtained for finitely many unknowns and infinitely many measurements in, e.g., [3,4,11]
Summary
Inverse elliptic coefficient problems arise in a number of applications in medical imaging and non-destructive material testing. This work is closely related to [10] that gives an explicit construction of special measurements that uniquely determine the same number of unknowns in an inverse elliptic coefficient problem by a globally convergent Newton root-finding method. Our main advance in this work is the step from Newton root-finding to a convex semidefinite program This allows utilizing a redundant set of given measurements, and eliminates the need of specially constructed measurements. It simplifies the underlying theory as it no longer requires simultaneously localized potentials, and allows the criterion to be written using the Loewner order, which very naturally arises in elliptic inverse coefficient problems with finite resolution and finitely many measurements [9].
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