Trusted frequency region of convergence for the enclosure method in an inverse heat equation

TL;DR

This paper investigates the convergence properties of a numerical formula in the enclosure method for inverse heat problems, establishing a trusted frequency region for accurate thermal imaging with fixed discretization.

Contribution

The paper provides a theoretical and numerical analysis of the trusted frequency region of convergence for the enclosure method in inverse heat equations, including error estimates and practical implementation.

Findings

Established a theoretical trusted frequency region for convergence.

Analyzed the effect of discretization on the formula's accuracy.

Numerically validated the trusted frequency region for various cases.

Abstract

This paper is concerned with the numerical implementation of a formula in the enclosure method as applied to a prototype inverse initial boundary value problem for thermal imaging in a one-space dimension. A precise error estimate of the formula is given and the effect on the discretization of the used integral of the measured data in the formula is studied. The formula requires a large frequency to converge; however, the number of time interval divisions grows exponetially as the frequency increases. Therefore, for a given number of divisions, we fixed the trusted frequency region of convergence with some given error bound. The trusted frequency region is computed theoretically using theorems provided in this paper and is numerically implemented for various cases.