Solving Schiffer's problem in inverse scattering theory almost surely

TL;DR

This paper introduces a probabilistic approach to Schiffer's inverse scattering problem, demonstrating that obstacle shape can be uniquely identified with high probability from a single measurement, broadening understanding in inverse problems.

Contribution

It provides a new probabilistic framework for Schiffer's problem, extending the conditions under which uniqueness can be almost surely achieved.

Findings

Schiffer's conjecture holds true in a probabilistic sense

The approach broadens the applicability of inverse scattering solutions

Implications for practical inverse problem solving

Abstract

In this short note, we present a probabilistic perspective on the Schiffer's problem in the inverse scattering theory, which asks whether one can uniquely determine the shape of an unknown obstacle by a single far-field measurement. It is a longstanding problem and has received considerable studies in the literature. We show that this conjecture holds true in more general settings in the probability sense. Our new perspective has important implications from the practical viewpoint and also points an interesting direction of research for broader inverse problems.