A Direct Sampling Method for Simultaneously Recovering Inhomogeneous Inclusions of Different Nature

TL;DR

This paper introduces a fast, stable, and parallelizable direct sampling method for simultaneously recovering multiple inhomogeneous inclusions of different physical natures from limited boundary data, with robustness to noise.

Contribution

The paper develops a novel DSM that can distinguish and reconstruct multiple inclusions of different physical types using limited data and introduces a new orthogonality concept for decoupling.

Findings

Method effectively recovers inclusions with noisy data

Numerical experiments demonstrate robustness and efficiency

Decoupling of different physical inclusions achieved

Abstract

In this work, we investigate a class of elliptic inverse problems and aim to simultaneously recover multiple inhomogeneous inclusions arising from two different physical parameters, using very limited boundary Cauchy data collected only at one or two measurement events. We propose a new fast, stable and highly parallelable direct sampling method (DSM) for the simultaneous reconstruction process. Two groups of probing and index functions are constructed, and their desired properties are analyzed. In order to identify and decouple the multiple inhomogeneous inclusions of different physical nature, we introduce a new concept of mutually almost orthogonality property that generalizes the important concept of almost orthogonality property in classical DSMs for inhomogeneous inclusions of same physical nature. With the help of this new concept, we develop a reliable strategy to distinguish two different types of inhomogeneous inclusions with noisy data collected at one or two measurement events. We further improve the decoupling effect by choosing an appropriate boundary influx. Numerical experiments are presented to illustrate the robustness and efficiency of the proposed method.