Integrating the probe and singular sources methods

TL;DR

This paper develops an integrated theoretical framework combining the probe and singular sources methods for inverse obstacle problems governed by PDEs, enhancing the understanding and capabilities of classical reconstruction techniques.

Contribution

It introduces a unified theory that merges the probe and singular sources methods, including a new indicator function and a fully integrated approach.

Findings

Unified indicator function combining both methods

New decomposition of indicator functions

Complete integration of probe and singular sources techniques

Abstract

The probe and singular sources methods are two well-known classical direct reconstruction methods in inverse obstacle problems governed by partial differential equations. In this paper, by considering an inverse obstacle problem governed by the Laplace equation in a bounded domain as a prototype case, an integrated theory of the probe and singular sources methods is proposed. The theory consists of three parts: (i) introducing the singular sources method combined with the notion of the probe method; (ii) finding a third indicator function whose two ways decomposition yields the indicator functions in the probe and singular sources methods; (iii) finding the completely integrated version of the probe and singular sources methods.