Non-zero constraints in elliptic PDE with random boundary values and applications to hybrid inverse problems

TL;DR

This paper demonstrates that for a second-order elliptic PDE, random boundary conditions can almost surely ensure solutions meet non-zero constraints, aiding hybrid inverse problem reconstruction.

Contribution

It introduces a probabilistic method using random boundary values to satisfy non-zero solution constraints in elliptic PDEs, supported by a new quantitative Runge approximation estimate.

Findings

Random boundary data can satisfy non-zero constraints with high probability.

The approach applies to hybrid inverse problems requiring internal measurements.

A new quantitative Runge approximation estimate is developed.

Abstract

Hybrid inverse problems are based on the interplay of two types of waves, in order to allow for imaging with both high resolution and high contrast. The inversion procedure often consists of two steps: first, internal measurements involving the unknown parameters and some related quantities are obtained, and, second, the unknown parameters have to be reconstructed from the internal data. The reconstruction in the second step requires the solutions of certain PDE to satisfy some non-zero constraints, such as the absence of nodal or critical points, or a non-vanishing Jacobian. In this work, we consider a second-order elliptic PDE and show that it is possible to satisfy these constraints with overwhelming probability by choosing the boundary values randomly, following a sub-Gaussian distribution. The proof is based on a new quantitative estimate for the Runge approximation, a result of independent interest.