An Iterative Direct Sampling Method for Reconstructing Moving Inhomogeneities in Parabolic Problems

TL;DR

This paper introduces an iterative direct sampling method for efficiently reconstructing moving inhomogeneities in parabolic problems using minimal boundary data, demonstrating robustness and stability even with noisy measurements.

Contribution

The paper presents a novel iterative sampling approach applicable to both linear and nonlinear parabolic problems, capable of reconstructing moving inhomogeneities with limited data and high stability.

Findings

Effective reconstruction with limited boundary measurements

High stability against noisy data

Applicable to diverse parabolic problem types

Abstract

We propose in this work a novel iterative direct sampling method for imaging moving inhomogeneities in parabolic problems using boundary measurements. It can efficiently identify the locations and shapes of moving inhomogeneities when very limited data are available, even with only one pair of lateral Cauchy data, and enjoys remarkable numerical stability for noisy data and over an extended time horizon. The method is formulated in an abstract framework, and is applicable to linear and nonlinear parabolic problems, including linear, nonlinear, and mixed-type inhomogeneities. Numerical experiments across diverse scenarios show its effectiveness and robustness against the data noise.