Automatic selection of regularization parameter in inverse heat conduction problems

Abstract

This paper presents a fast, robust and automatic approach for selecting the regularization parameter in Tikhonov regularization. The methodology is based on the Generalized Cross-Validation (GCV) method, where the minimum of the GCV curve is sought via a numerical optimization algorithm. Typical convergence issues near the global minimum are addressed by changing the variables in the optimization problem in an appropriate fashion. Evidences for the effectiveness of the proposed approach are presented by performing three numerical experiments, consisting of linear and nonlinear inverse heat conduction problems. The obtained estimates are in good agreement with the reference values, showing that the obtained solutions in both problems were appropriately regularized. In this methodology, inverse problem needs not be solved with multiple regularization parameter candidates. Instead, the inverse problem is solved only once, using the optimal parameter. This feature is even more pressing in nonlinear problems, for the optimal regularization parameter may change during the iterative solution of the least squares problem, thus being adjusted at each iteration. Therefore, manual selection of the regularization parameter is not necessary, thus yielding an automatic selection methodology. Finally, as for the “robust” statement, it was made based on the numerical experiments shown on the paper, where the optimal regularization parameters for each problem vary in orders of magnitude.