Regula falsi based automatic regularization method for PDE constrained optimization
This work addresses the efficiency bottleneck of regularization parameter selection for practitioners solving PDE-constrained inverse problems.
The paper tackles the computational cost of choosing a regularization parameter in PDE-constrained optimization by deriving two methods that simultaneously solve the inverse problem and determine the parameter, with the second method offering advantages for nonlinear problems.
Many inverse problems can be described by a PDE model with unknown parameters that need to be calibrated based on measurements related to its solution. This can be seen as a constrained minimization problem where one wishes to minimize the mismatch between the observed data and the model predictions, including an extra regularization term, and use the PDE as a constraint. Often, a suitable regularization parameter is determined by solving the problem for a whole range of parameters -- e.g. using the L-curve -- which is computationally very expensive. In this paper we derive two methods that simultaneously solve the inverse problem and determine a suitable value for the regularization parameter. The first one is a direct generalization of the Generalized Arnoldi Tikhonov method for linear inverse problems. The second method is a novel method based on similar ideas, but with a number of advantages for nonlinear problems.