Gradient Method in Hilbert-Besov Spaces for the Optimal Control of Parabolic Free Boundary Problems

TL;DR

This paper develops a gradient-based computational method in Hilbert-Besov spaces for solving inverse free boundary problems in heat transfer, with a focus on optimal control and regularization techniques.

Contribution

It introduces a novel gradient descent algorithm utilizing Frechet differentiability in Hilbert-Besov spaces for inverse Stefan problems, including preconditioning and regularization strategies.

Findings

Effective gradient descent algorithm for inverse Stefan problems.

Preconditioning and Tikhonov regularization improve solution stability.

Comparison of control parameter identification methods enhances understanding.

Abstract

This paper presents computational analysis of the inverse Stefan type free boundary problem, where information on the boundary heat flux is missing and must be found along with the temperature and the free boundary. We pursue optimal control framework introduced in {\it U.G. Abdulla, Inverse Problems and Imaging, 7, 2(2013), 307-340;\ 10, 4(2016), 869--898}, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consist of the minimization of the quadratic declinations from the available measurements of the temperature distribution at the final moment, phase transition temperature on the free boundary, and the final position of the free boundary. We develop gradient descent algorithm based on Frechet differentiability in Hilbert-Besov spaces complemented with preconditioning or increase of regularity of the Frechet gradient through implementation of the Riesz representation theorem. Three model examples with various levels of complexity are considered. Extensive comparative analysis through implementation of preconditioning and Tikhonov regularization, calibration of preconditioning and regularization parameters, effect of noisy data, comparison of simultaneous identification of control parameters vs. nested optimization is pursued.