A Monotonicity-Based Regularization Approach to Shape Reconstruction for the Helmholtz Equation

TL;DR

This paper introduces a convex, monotonicity-based regularization method for reconstructing scatterer shapes in the Helmholtz equation, avoiding additional PDE solutions and ensuring global convergence.

Contribution

The paper presents a novel convex formulation for shape reconstruction that leverages monotonicity relations, providing theoretical guarantees and numerical validation.

Findings

Convex data-fitting formulation successfully reconstructs shapes.

Method does not require solving additional PDEs.

Numerical experiments confirm effectiveness and stability.

Abstract

We consider an inverse boundary value problem for determining unknown scatterers, which is governed by the Helmholtz equation in a bounded domain. To address this, we develop a novel convex data-fitting formulation that is capable of reconstructing the shape of the unknown scatterers.Our formulation is based on a monotonicity relation between the scattering index and boundary measurements. We use this relation to obtain a pixel-wise constraint on the unknown scattering index, and then minimize a data-fitting functional defined as the sum of all positive eigenvalues of a linearized residual operator. The main advantages of our new approach are that this is a convex data-fitting problem that does not require additional PDE solutions. The global convergence and stability of the method are rigorously established to demonstrate the theoretical soundness. In addition, several numerical experiments are conducted to verify the effectiveness of the proposed approach in shape reconstruction.