Stability for an inverse random source problem of the biharmonic Schrodinger equation

TL;DR

This paper investigates the stability of reconstructing a random source in a biharmonic Schrödinger equation using multi-frequency scattering data, establishing well-posedness and quantitative stability estimates.

Contribution

It introduces a novel stability analysis for an inverse random source problem in biharmonic Schrödinger equations, utilizing scattering theory and analytic continuation techniques.

Findings

Quantitative stability estimates for the inverse problem.

Well-posedness of the direct scattering problem.

Effective reconstruction using finite frequency data.

Abstract

In this paper, we study the inverse random source scattering problem for the biharmonic Schrodinger equation in two and three dimensions. The driven source is assumed to be a generalized microlocally isotropic Gaussian random function whose covariance operator is a classical pseudodifferential operator. We examine the meromorphic continuation and estimates for the resolvent of the biharmonic Schrodinger operator with respect to complex frequencies, which yields the well-posedness of the direct problem and a Born series expansion for the solution. For the inverse problem, we present quantitative stability estimates in determining the micro-correlation strength of the random source by using the correlation far-field data at multiple frequencies. The key ingredient in the analysis is employing scattering theory to obtain the analytic domain and an upper bound for the resolvent of the biharmonic Schrodinger operator and applying a quantitative analytic continuation principle in complex theory. The analysis only requires data in a finite interval of frequencies.