Factorization method for a simply supported obstacle from point source measurements via far--field transformation

TL;DR

This paper develops a factorization method for reconstructing simply supported obstacles in 2D from near-field point-source data by transforming it into a far-field problem, enabling stable numerical reconstruction.

Contribution

It introduces a novel far-field transformation that allows the factorization method to be applied to biharmonic Helmholtz inverse problems, which was not previously possible.

Findings

The method successfully reconstructs obstacles from synthetic data.

The approach demonstrates stability under noisy measurements.

A variant using only scattered field data is also effective.

Abstract

We consider an inverse shape problem for recovering an unknown simply supported obstacle in two dimensions from near--field point--source measurements for the biharmonic Helmholtz equation. The measured data consist of the scattered field and its Laplacian on a closed measurement curve surrounding the obstacle. By exploiting an operator splitting of the biharmonic operator, we decouple the scattered field into propagating and evanescent components. This decoupling allows us to reformulate the measured data in terms of an acoustic near--field operator for a sound--soft scatterer. Since the acoustic near--field operator does not directly admit the symmetric factorization required by the factorization method, we introduce a far--field transformation (defined independently of the obstacle) that augments the near--field operator into a far--field operator with a symmetric factorization. This yields a rigorous factorization method characterization of the obstacle and leads to a practical reconstruction algorithm based on spectral data of the transformed operator. Finally, we present numerical experiments with synthetic data that demonstrate stable reconstructions under noise and illustrate the role of regularization, including a variant that uses only the scattered field data.