The Calder\'on problem with finitely many unknowns is equivalent to convex semidefinite optimization

TL;DR

This paper reformulates the Calderón inverse boundary value problem with finitely many unknowns as a convex semidefinite optimization problem, enabling new computational approaches and error analysis for noisy and finite measurement data.

Contribution

It establishes an equivalence between the Calderón problem with piecewise constant coefficients and a convex semidefinite program, extending to noisy and finite measurement scenarios.

Findings

Reformulation of Calderón problem as semidefinite optimization

Error estimates for noisy data

Extension to finite-dimensional Galerkin projections

Abstract

We consider the inverse boundary value problem of determining a coefficient function in an elliptic partial differential equation from knowledge of the associated Neumann-Dirichlet-operator. The unknown coefficient function is assumed to be piecewise constant with respect to a given pixel partition, and upper and lower bounds are assumed to be known a-priori. We will show that this Calder\'on problem with finitely many unknowns can be equivalently formulated as a minimization problem for a linear cost functional with a convex non-linear semidefinite constraint. We also prove error estimates for noisy data, and extend the result to the practically relevant case of finitely many measurements, where the coefficient is to be reconstructed from a finite-dimensional Galerkin projection of the Neumann-Dirichlet-operator. Our result is based on previous works on Loewner monotonicity and convexity of the Neumann-Dirichlet-operator, and the technique of localized potentials. It connects the emerging fields of inverse coefficient problems and semidefinite optimization.