Fast computational techniques for multiscale electromagnetic simulations

Multiscale electromagnetic simulations contain features with multiple length or frequency scales or both. Multiscale features are characteristic of realitic simulations as large degrees of freedom (N) are required to capture the minute physical details. Though integral equation (IE) approaches are well-suited for electromagnetic simulations, they require repeated evaluation of pair-wise potentials - also referred to as N-body problems. It is well known that the direct computation of these potentials scales as O (N2) both in terms of computer memory and time. Even with the rapid advancements in computer technology, this places severe limitation on the size of the problem (N) that can be analyzed in a realistic time frame. Further, multiscale simulations produce badly-conditioned systems of equations that require large number of iterations when using Krylov-subspace solvers. The main goal of this thesis is to develop a suite of computational techniques that enables multiscale electromagnetic simulations in a fast, efficient and stable fashion. In this work, the accelerated Cartesian expansion (ACE) algorithm is used to overcome the quadratic cost-scaling of N-body problems. ACE was intially developed for the fast evaluation of polynomial potentials and here it is extended to the fast computation of retarded and Helmholtz potentials. These algorithms are shown to be stable and efficient for computation of electromagnetic potentials at sub-wavelength or low-frequency scales. Hybrid combination of these algorithms with existing fast methods leads to the development of multiscale electromagnetic solvers that are stable and efficient across length and frequency scales. Since the fast algorithms only reduce the time spent in each iteration, a new integral equation formulation is developed that yields better conditioned systems of equations. This is achieved by reformulating the augmented field integral equations such that the resulting operators are bounded and compact. Further, the widespread availability of parallel distributed or cluster computers combined with the memory and speed restriction of single processor computers necessitates the development of efficient parallel implementation of the sophisticated fast algorithms. The parallel algorithms developed in this work are provably scalabale and enables simulation of problems with several millions of unknowns on large scale clusters, with hundreds of processors and beyond. In this thesis, ACE algorithm is also extended to rapid computation of time domain diffusion potentials.