Optimal Control of Coefficients in Parabolic Free Boundary Problems Modeling Laser Ablation

TL;DR

This paper develops an optimal control approach for inverse Stefan problems in laser ablation, estimating missing coefficients, heat flux, and free boundary position from temperature measurements.

Contribution

It introduces a novel optimal control framework that jointly identifies unknown parameters and free boundary in parabolic free boundary problems, with proven convergence of discretized solutions.

Findings

Finite difference discretization converges to the continuous problem.

The method effectively reconstructs missing data and free boundary.

The approach is applicable to biomedical laser ablation modeling.

Abstract

Inverse Stefan problem arising in modeling of laser ablation of biomedical tissues is analyzed, where information on the coefficients, heat flux on the fixed boundary, and density of heat sources are missing and must be found along with the temperature and free boundary. Optimal control framework is employed, where the missing data and the free boundary are components of the control vector, and optimality criteria are based on the final moment measurement of the temperature and position of the free boundary. Discretization by finite differences is pursued, and convergence of the discrete optimal control problems to the original problem is proven.