Regula falsi based automatic regularization method for PDE constrained optimization

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. (C) 2018 Elsevier B.V. All rights reserved.