Regula falsi based automatic regularization method for PDE constrained optimization

TL;DR

This paper introduces two innovative methods for PDE-constrained inverse problems that automatically determine regularization parameters during the solution process, reducing computational costs especially for nonlinear problems.

Contribution

The paper presents a novel extension of the Arnoldi Tikhonov method and a new approach tailored for nonlinear inverse PDE problems that jointly solve for parameters and regularization.

Findings

Methods effectively determine regularization parameters automatically.

Approaches reduce computational effort compared to traditional methods.

Applicable to both linear and nonlinear inverse PDE problems.

Abstract

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.