Direct Algorithms for Reconstructing Small Conductivity Inclusions in Subdiffusion

TL;DR

This paper introduces novel, computationally inexpensive direct algorithms for reconstructing small conductivity inclusions in subdiffusion models, supported by theoretical analysis and numerical validation.

Contribution

It develops the first direct algorithms for inverse conductivity problems within the subdiffusion framework, utilizing asymptotic expansions and approximate fundamental solutions.

Findings

Algorithms are efficient and robust across various scenarios.

Numerical results demonstrate accurate reconstruction under noise.

Theoretical foundation supports the algorithm's validity.

Abstract

The subdiffusion model that involves a Caputo fractional derivative in time is widely used to describe anomalously slow diffusion processes. In this work we aim at recovering the locations of small conductivity inclusions in the model from boundary measurement, and develop novel direct algorithms based on the asymptotic expansion of the boundary measurement with respect to the size of the inclusions and approximate fundamental solutions. These algorithms involve only algebraic manipulations and are computationally cheap. To the best of our knowledge, they are first direct algorithms for the inverse conductivity problem in the context of the subdiffusion model. Moreover, we provide relevant theoretical underpinnings for the algorithms. Also we present numerical results to illustrate their performance under various scenarios, e.g., the size of inclusions, noise level of the data, and the number of inclusions, showing that the algorithms are efficient and robust.