Restoring the Discontinuous Heat Equation Source Using Sparse Boundary Data and Dynamic Sensors

TL;DR

This paper presents a novel sampling algorithm with dynamic sensors for solving the inverse source problem in heat equations using sparse boundary data, enabling accurate reconstruction with minimal sensors.

Contribution

Introduces a Langevin dynamics-based sampling method with sensor migration strategies to reconstruct high-dimensional sources from limited boundary measurements.

Findings

Successful source reconstruction with only two sensors

Effective sensor migration strategies improve accuracy

Method applicable to high-dimensional inverse problems

Abstract

This study focuses on addressing the inverse source problem associated with the parabolic equation. We rely on sparse boundary flux data as our measurements, which are acquired from a restricted section of the boundary. While it has been established that utilizing sparse boundary flux data can enable source recovery, the presence of a limited number of observation sensors poses a challenge for accurately tracing the inverse quantity of interest. To overcome this limitation, we introduce a sampling algorithm grounded in Langevin dynamics that incorporates dynamic sensors to capture the flux information. Furthermore, we propose and discuss two distinct sensor migration strategies. Remarkably, our findings demonstrate that even with only two observation sensors at our disposal, it remains feasible to successfully reconstruct the high-dimensional unknown parameters.