The principal investigator previously developed an adaptive algorithm for solving the Poisson problem in complex three-dimensional domains. This algorithm found immediate application in modeling the electric field produced by surgically implanted defibrillators as a guide to electrode placement for the cardiologist, where the algorithm reduced solution times from 3 hours to less than 10 minutes.
A logical next step is the pursuit of more demanding applications, such as those with an inverse problem formulation. The inverse problem seeks to identify volume energy sources on the basis of electric field measurements on the surface or magnetic field measurements outside of the modeled domain. In general, there is no unique solution unless some a-priori knowledge of the sources is exploited. Inverse problems can be formulated for many applications with domains governed by partial differential equations. They are computationally intensive because many forward computations are typically required for a solution to the inverse problem.
The primary aim of this work is to develop an efficient and practical inverse algorithm based on a hierarchical spatial decomposition of the domain, integrate it with the existing forward algorithm, and disseminate it to researchers with a need to solve such problems. Cardiology and neurology are examples of two fields that can benefit from the availability of such tools. In clinical applications, these tools can prove invaluable in the diagnosis and treatment of disease. In research applications, these tools can improve our knowledge of the fundamental behavior of the system.
