A note on several inverse problems with generally random coefficients

TL;DR

This paper investigates what information about random coefficients in elliptic inverse problems can be recovered from various boundary measurements and their statistical properties.

Contribution

It compares the recoverability of the law of the potential from different boundary data and shows that some measurements determine the mean and variance, while others do not.

Findings

Full law of Dirichlet-to-Neumann map determines the law of the potential.

Expected Dirichlet-to-Neumann map does not determine the mean potential.

Averaged Green's operator determines the mean and variance of the potential.

Abstract

We consider several inverse problems for elliptic equations whose coefficients are random, without imposing a special probabilistic structure on the randomness. The main body treats the Schr\"odinger equation. We compare what can be recovered from the full law of the Dirichlet-to-Neumann map, from its expectation, from finitely many joint moments of its boundary bilinear form, and from the averaged interior Green's operator. We obtain both positive and negative results. That the full law of the Dirichlet-to-Neumann map determines the law of the random potential is almost trivial. However, the expected Dirichlet-to-Neumann map and, more generally, any fixed finite hierarchy of its boundary moments need not determine even the mean potential. In contrast, the averaged Schr\"odinger Green's operator determines the pointwise mean and variance of the potential. In a two-atom model it determines all pointwise moments of the two-point law. The appendices contain the corresponding results for the conductivity equation.