Homological computations in electromagnetic modeling

Abstract

The users of modern design software for electrical appliances can accidentally attempt an analysis of ill-posed design problems, ones with no sensible solution. It is tedious to track down such mistakes manually, but certain topological objects, the homology groups, provide systematic procedures for detection of such mistakes. However, the procedures are not necessarily practical if the time consumed in computation of the homology groups is excessive. This thesis analyzes the electromagnetic modeling problems and different methods to compute homology groups for the models. A strong emphasis is placed on the computational complexity.

The spatial model for electromagnetics, differentiable manifold with boundary, is introduced and some machinery is constructed to define its integer-coefficient homology groups. Their computation is expressed in terms of standard computational problems of Abelian group theory. The problems involve integer matrix computations, where large intermediate results may emerge and require attention in complexity analysis. The problems admit polynomial-time solution, but some of the polynomials are of unacceptably high degree. Particularly, the computation is known to consume considerable time if the homology groups have torsion subgroups, and this possibility in electromagnetic models is investigated in detail. Also, homologies over different coefficient groups are introduced as alternatives and their connection with integer-coefficient homology is characterized.

Numerical computation typically requires tessellations of electromagnetic models into elements --- up to millions, even if the model is homologically rather simple. This is an unnecessary burden for the group theoretic solution schemes, and various methods are introduced to simplify the tessellations into a modest fraction of the original and thus reduce the overall computational complexity. Unfortunately, the methods do not admit rigorous performance bounds, but remain heuristics, leaving the rigorous upper bound for overall complexity very pessimistic: the overall time hardly ever attains the bound in any practical design problem.