Analysis of the factorization method for a general class of boundary conditions

TL;DR

This paper extends the factorization method for inverse scattering problems to a broader class of boundary conditions, including pseudo-differential operators, and provides theoretical analysis and numerical validation.

Contribution

It generalizes the factorization method to boundary conditions with pseudo-differential operators of various orders, beyond impedance conditions.

Findings

Method works if the boundary operator is Fredholm of index zero with non-negative imaginary part.

Validates the approach with numerical examples for second-order boundary operators.

Provides criteria for the applicability of the factorization method to generalized boundary conditions.

Abstract

We analyze the factorization method (introduced by Kirsch in 1998 to solve inverse scattering problems at fixed frequency from the far field operator) for a general class of boundary conditions that generalizes impedance boundary conditions. For instance, when the surface impedance operator is of pseudo-differential type, our main result stipulates that the factorization method works if the order of this operator is different from one and the operator is Fredholm of index zero with non negative imaginary part. We also provide some validating numerical examples for boundary operators of second order with discussion on the choice of the test function.