Multigrid Methods for an Inverse Potential Problem

Marcus Mohr
Institut für Mathematik, Universität Augsburg, Germany

Constantin Popa
Department of Mathematics, Ovidius University, Constanta, Romania

Ulrich Rüde
Institut für Mathematik, Universität Augsburg, Germany

Abstract

In electrocardiographic diagnostic applications, inverse problems of the following form occur. The electric field is described by a potential equation with spatially varying dielectric constants. Natural boundary conditions are prescribed for the outer boundary of the domain. Additionally, measurements provide values of the potential on the boundary corresponding to extra Dirichlet data. Thus, in a forward problem, the boundary is over-specified. In the inverse problem, however, the right hand side (that is the source term for the potential) is unknown in parts of the domain. Alternatively the problem may have an interior boundary, where no boundary data are given.

This inverse problem is ill-posed and its treatment requires careful use of regularization techniques. The corresponding discrete systems are badly conditioned, even after regularization. Thus the conventional iterative solvers converge very slowly and the current computational solutions are all extremely compute intensive. Clearly, it would be an important progress if multigrid could be used to speed up the solution process.

As a prototype for the general problem we study the initial value problem for Laplace's equation. In simple geometries, this model problem can be analyzed using Fourier techniques and thus it is a first starting point for exploring various alternatives how to incorporate multigrid. Besides the use of multigrid in repeated solutions of the forward problem, as they occur in standard iterative processes for the inverse problem, we will also discuss the possibility to solve the inverse problem directly with a multigrid approach.