An inverse problem in estimating the time dependent source term and initial temperature simultaneously by the polynomial regression and conjugate gradient method

TL;DR

This paper presents a method combining polynomial regression and conjugate gradient techniques within an optimal control framework to simultaneously estimate time-dependent source terms and initial temperatures in an ill-posed inverse heat conduction problem.

Contribution

It introduces a novel approach that derives the gradient of the Tikhonov functional and demonstrates the stability and effectiveness of the evolutionary algorithm for this inverse problem.

Findings

Stable and effective estimation demonstrated on test examples

Gradient derivation improves the optimization process

Method addresses ill-posedness in inverse heat conduction

Abstract

From the final and interior temperature measurements identifying the source term with initial temperature simultaneously is an inverse heat conduction problem which is a kind of ill-posed. The optimal control framework has been found to be effective in dealing with these problems. However, they require to find the gradient information. This idea has been employed in this research. We derive the gradient of Tikhonov functional and establish the stability of the minimizer from the necessary condition. The stability and effectiveness of the evolutionary algorithm are presented for various test examples.