Preconditioning inverse problems for hyperbolic equations with applications to photoacoustic tomography

Abstract

Abstract This paper is concerned with the preconditioning of wave equation-constrained linear inverse problems from boundary observation data, such as, for instance, photoacoustic tomography, which is considered as an application. The main result of this paper is a concept for regularization parameter robust preconditioning. That is, we propose a preconditioner for the PDE-constrained optimization problem such that the condition number of the preconditioned optimality system is uniformly bounded with respect to the regularization parameter. Using an augmented Lagrangian formulation for the discretized optimality system, we employ a discretization strategy which renders the discretized preconditioned system robust with respect to both the regularization parameter and the mesh size. Numerical experiments complement our analysis.