On the recovery of two function-valued coefficients in the Helmholtz equation for inverse scattering problems via inverse Born series

TL;DR

This work addresses the inverse scattering problem for the 2D Helmholtz equation with two function-valued contrasts $\gamma$ and $\eta$, using far-field data at two frequencies. It develops a regularized inverse Born framework built on a linearized first-order Born approximation, a two-frequency data relation, and a disk prolate spheroidal wave function (PSWF) based spectral cutoff to regularize the outer inversion, with convergence analysis and explicit error bounds for the inverse Born series (IBS). The authors derive convergence criteria for the general IBS and specialize to PSWF-based regularization, providing explicit radius-of-convergence estimates in terms of the wave numbers and regularization parameters. Numerical experiments on a unit disk validate the approach, showing accurate recovery of small perturbations via the regularized inverse Born approximation and improved reconstructions for larger perturbations using the truncated IBS, with performance influenced by the choice of frequencies.

Abstract

In this work, we construct the Born and inverse Born approximation and series to recover two function-valued coefficients in the Helmholtz equation for inverse scattering problems from the scattering data at two different frequencies. An analysis of the convergence and approximation error of the proposed regularized inverse Born series is provided. The results show that the proposed series converges when the inverse Born approximations of the perturbations are sufficiently small. The preliminary numerical results show the capability of the proposed regularized inverse Born approximation and series for recovering the isotropic inhomogeneous media.

Figures (6)

• Figure 1: Reconstructions of $\gamma$ and $\eta$ derived from the proposed regularized inverse Born approximation for different scattering data (with $2\%$ relative noise).
• Figure 2: Reconstructions of $\gamma$ and $\eta$, i.e., $\gamma_i$ and $\eta_i$ that derived from the truncated IBS for different scattering data (with $2\%$ relative noise) at wave frequencies $k=5$ and $\ell k=10$, generated with the perturbations of the form \ref{['eqn:dataset_pertu2']}.
• Figure 3: Reconstructions of $\gamma$ and $\eta$, i.e., $\gamma_i$ and $\eta_i$ that derived from the truncated IBS for different scattering data (with $2\%$ relative noise) at wave frequencies $k=5$ and $\ell k=10$, generated with the perturbations of the form \ref{['eqn:dataset_s']} ($J=5$).
• Figure 4: Reconstructions of $\gamma$ and $\eta$, i.e., $\gamma_i$ and $\eta_i$ that derived from the truncated IBS for different scattering data (with $2\%$ relative noise) at wave frequencies $k=10$ and $\ell k=15$, generated with the perturbations of Example 1 in Figure \ref{['fig:IBS']}.
• Figure 5: Reconstructions of $\gamma$ and $\eta$, i.e., $\gamma_i$ and $\eta_i$ that derived from the truncated IBS for different scattering data (with $2\%$ relative noise) at wave frequencies $k=10$ and $\ell k=15$, generated with the perturbations of Example 2 in Figure \ref{['fig:IBS']}.

Theorems & Definitions (20)

• Definition 1.1: Scattering Data and the Inverse Problem
• Lemma 2.1
• Lemma 4.1
• Proposition 4.1
• Corollary 4.1
• proof
• Theorem 4.1
• proof
• Corollary 4.2
• proof