A GSVD-based methodology for automatic selection of high-order regularization parameters in inverse heat conduction problems

Abstract

Regularization is a powerful tool for solving inverse problems, frequently affected by ill-posedness. Through this process, it is possible to obtain stable solutions by penalizing solution candidates with little or no physical significance. Yet most techniques rely on some sort of tuning, which oftentimes is performed manually or even visually. Tikhonov Regularization ranks among the most versatile and employed techniques in literature, requiring the user to select the regularization parameter, weighing between the norms of the residuals (or equivalent) and of the solution (or its derivatives). As in other techniques, the selection of this parameter poses specific challenges to its automation. Past research has shown that such automation could be achieved by employing Singular Value Decomposition, while restricted to 0th-order regularization. This publication extended this methodology via the Generalized Singular Value Decomposition to render it possible to employ any regularization order. Numerical examples based on previous research were explored, with their results being compared. It was shown that the same methodology and calculations can be used by simply replacing the singular values with their generalized versions. The proposed extension increased the robustness of this approach, improved its versatility and greatly simplified the solution of function estimation problems. As in its predecessor, automatic regularization is herein achieved without requiring solving the inverse problem multiple times. The computational overhead is negligible, mainly due to the GSVD properties and the customized optimization functional. This feature grants a significant advantage in comparison with alternative methods, who have no such capability so far.