Multilevel Methods for Inverse Bioelelectric Field Problems

The reconstruction of bioelectric fields from non-invasive measurements can be used as a powerful new diagnostic tool in cardiology and neurology. Mathematically, the reconstruction of a bioelectric field can be modeled as an inverse problem for a potential equation. This problem is ill-posed and requires special treatment, in particular either regularization or an otherwise suitable restriction of the solution space. The differential equation itself can be discretized by finite differences or finite elements and thus gives rise to a large sparse linear systems for which multigrid is one of the most efficient solvers, but regularization, adaptive mesh refinement, and efficient solution techniques must be combined to solve the inverse bioelectric field problem efficiently. While multigrid algorithms can reduce the compute times substantially, new local regularization techniques can be used to improve the quality of the reconstruction. Local mesh refinement can be used to increase the resolution in domains of increased activity, but must be used with care because refined meshes worsen the ill-conditioning of the inverse problem.