Shape and parameter identification by the linear sampling method for a restricted Fourier integral operator

TL;DR

This paper introduces a novel linear sampling method for inverse problems involving Fourier integral operators, enabling simultaneous shape and parameter reconstruction with proven convergence and practical implementation.

Contribution

It develops a new data operator-based linear sampling approach for inverse source and scattering problems, with theoretical analysis and a prolate spheroidal wave function-based implementation.

Findings

Converges to the average of the unknown in small neighborhoods.

Successfully reconstructs shape and parameters in numerical experiments.

Provides a new theoretical framework for inverse problem solving.

Abstract

In this paper we provide a new linear sampling method based on the same data but a different definition of the data operator for two inverse problems: the multi-frequency inverse source problem for a fixed observation direction and the Born inverse scattering problems. We show that the associated regularized linear sampling indicator converges to the average of the unknown in a small neighborhood as the regularization parameter approaches to zero. We develop both a shape identification theory and a parameter identification theory which are stimulated, analyzed, and implemented with the help of the prolate spheroidal wave functions and their generalizations. We further propose a prolate-based implementation of the linear sampling method and provide numerical experiments to demonstrate how this linear sampling method is capable of reconstructing both the shape and the parameter.