H\"older-Logarithmic Stability and Convergence Rates for an Inverse Random Source Problem

TL;DR

This paper establishes stability estimates and convergence rates for an inverse problem involving recovering a random acoustic source from measurements, with theoretical analysis supported by numerical experiments.

Contribution

We derive H"older-logarithmic stability and convergence rate estimates for an inverse random source problem using complex geometrical optics solutions and variational source conditions.

Findings

Stability estimates depend on Sobolev regularity of the source.

Spectral regularization methods achieve H"older-logarithmic convergence rates.

Numerical experiments confirm theoretical predictions.

Abstract

In this paper, we investigate an inverse random source problem concerned with recovering the strength of a random, uncorrelated acoustic source from correlation measurements of emitted time-harmonic acoustic waves. Such problems arise in applications including aeroacoustics and seismic imaging. Unlike their deterministic counterparts, inverse random source problems are known to be uniquely solvable in the absence of noise. Nevertheless, due to their inherent ill-posedness, regularization is required to stably reconstruct the source strength. We derive conditional H\"older-logarithmic stability estimates under Sobolev smoothness assumptions by employing complex geometrical optics solutions. Moreover, by establishing a variational source condition, we obtain H\"older-logarithmic convergence rates for spectral regularization methods. At fixed frequency, the exponents in the logarithmic stability and convergence estimates grow unboundedly as the Sobolev regularity of the source increases. Finally, we present numerical experiments supporting our theoretical findings.