Regularization of inverse problems via box constrained minimization

Abstract

In the present paper we consider minimization based formulations of inverse problems $ (x, \Phi)\in \mbox{argmin}\left\{{ \mathcal{J}(x, \Phi;y)}:{(x, \Phi)\in M_{ad}(y)}\right\} $ for the specific but highly relevant case that the admissible set $ M_{ad}^\delta(y^\delta) $ is defined by pointwise bounds, which is the case, e.g., if $ L^\infty $ constraints on the parameter are imposed in the sense of Ivanov regularization, and the $ L^\infty $ noise level in the observations is prescribed in the sense of Morozov regularization. As application examples for this setting we consider three coefficient identification problems in elliptic boundary value problems.Discretization of $ (x, \Phi) $ with piecewise constant and piecewise linear finite elements, respectively, leads to finite dimensional nonlinear box constrained minimization problems that can numerically be solved via Gauss-Newton type SQP methods. In our computational experiments we revisit the suggested application examples. In order to speed up the computations and obtain exact numerical solutions we use recently developed active set methods for solving strictly convex quadratic programs with box constraints as subroutines within our Gauss-Newton type SQP approach.