The Inverse Problem of Identifying Spatial Heat Sources in Biological Heat Transfer Processes From Terminal Data

The inverse problem of reconstructing spatial heat sources in the thermal wave diffusion model for biological organisms, was Investigated. Unlike traditional parabolic models for biological heat transfer, this work was focused on a more complex and practical hyperbolic model, particularly suitable for biomedical engineering applications. Firstly, the optimal control theory was employed, and the inverse problem was formulated as an optimal control problem. To address the challenge of non-uniqueness in the optimal solution due to the non-differentiability of total variation functions, a carefully designed and polished total variation regularization term was introduced. Subsequently, the existence of the optimal solution and its necessary conditions were thoroughly discussed. Secondly, under the assumption of a small terminal time, the uniqueness and stability of the optimal solution were proven with the Sobolev embedding theory. Last, a gradient-based optimization algorithm was developed based on these necessary conditions, and its effectiveness was demonstrated through several numerical examples.