Norm-dependent convergence and stability of the inverse scattering series for diffuse and scalar waves

TL;DR

This paper investigates the convergence and stability of inverse scattering series for scalar and diffuse waves within Sobolev $H^s$ norms, revealing how spectral bias influences inversion stability and convergence in different function spaces.

Contribution

It introduces an analysis of inverse scattering series stability and convergence under Sobolev $H^s$ norms, extending previous $L^p$ norm studies, and provides theoretical and numerical insights into frequency weighting effects.

Findings

Stability estimates vary with different $H^s$ norms for parameter and data.

Frequency bias in $H^s$ norms affects convergence radius.

Numerical examples illustrate differences in convergence under various metrics.

Abstract

This work analyzes the forward and inverse scattering series for scalar waves based on the Helmholtz equation and the diffuse waves from the time-independent diffusion equation, which are important PDEs in various applications. Different from previous works, which study the radius of convergence for the forward and inverse scattering series, the stability, and the approximation error of the series under the $L^p$ norms, we study these quantities under the Sobolev $H^s$ norm, which associates with a general class of $L^2$-based function spaces. The $H^s$ norm has a natural spectral bias based on its definition in the Fourier domain: the case $s<0$ biases towards the lower frequencies, while the case $s>0$ biases towards the higher frequencies. We compare the stability estimates using different $H^s$ norms for both the parameter and data domains and provide a theoretical justification for the frequency weighting techniques in practical inversion procedures. We also provide numerical inversion examples to demonstrate the differences in the inverse scattering radius of convergence under different metric spaces.