An inverse random source problem for the fractional Helmholtz equation

TL;DR

This paper studies an inverse problem for the stochastic fractional Helmholtz equation, showing that the principal symbols of the source's covariance and relation operators can be uniquely identified from far-field data at high frequencies.

Contribution

It introduces a novel approach combining microlocal analysis, asymptotic expansions, and linearization to solve the inverse problem for a stochastic fractional Helmholtz equation.

Findings

Unique determination of principal symbols from far-field patterns

Well-posedness of the direct problem at large wavenumbers

Application of microlocal analysis to stochastic inverse problems

Abstract

This paper investigates an inverse random source problem for the stochastic fractional Helmholtz equation. The source is modeled as a centered, complex-valued, microlocally isotropic generalized Gaussian random field whose covariance and relation operators are described by classical pseudo-differential operators. For sufficiently large wavenumbers, we first establish the well-posedness of the direct problem in the distributional sense by analyzing the corresponding Lippmann--Schwinger integral equation. For the inverse problem, we show that the principal symbols of both the covariance and relation operators can be uniquely determined, with probability one, from the far-field patterns generated by a single realization of the random source. The approach employs a combination of the Born linearization, asymptotic expansions of the fractional Helmholtz Green kernel at high wavenumbers, and microlocal analysis of associated Fourier integral operators.