Lipschitz dependence of the coefficients on the resolvent and greedy approximation for scalar elliptic problems

TL;DR

This paper proves that the diffusivity coefficient of a scalar elliptic equation can be uniquely and Lipschitz continuously determined from its resolvent operator, enabling robust greedy algorithms for parameter-dependent problems.

Contribution

It establishes the Lipschitz dependence of the diffusivity coefficient on the resolvent, extending results to multi-dimensional cases and improving understanding of inverse problems in elliptic PDEs.

Findings

Unique determination of diffusivity from resolvent

Lipschitz continuity of the inverse mapping

Extension to multi-dimensional elliptic equations

Abstract

We analyze the inverse problem of identifying the diffusivity coefficient of a scalar elliptic equation as a function of the resolvent operator. We prove that, within the class of measurable coefficients, bounded above and below by positive constants, the resolvent determines the diffusivity in an unique manner. Furthermore we prove that the inverse mapping from resolvent to the coefficient is Lipschitz in suitable topologies. This result plays a key role when applying greedy algorithms to the approximation of parameter-dependent elliptic problems in an uniform and robust manner, independent of the given source terms. In one space dimension the results can be improved using the explicit expression of solutions, which allows to link distances between one resolvent and a linear combination of finitely many others and the corresponding distances on coefficients. These results are also extended to multi-dimensional elliptic equations with variable density coefficients. We also point out towards some possible extensions and open problems.