ECS-9521001 Levin The objective of this project is to create sparse approximations of the fully populated boundary element matrices that one obtains using integral formulations like charge simulation or surface charge simulation. These well known numerical schemes are used to find the electric field on and near high voltage devices like guard rings, insulator strings, and transformers. The sparse approximations are formed by a similarity transform of the coefficient matrix. The cost of the one employed here is of optimal order N . Nonetheless, the benefits of computing with a sparse matrix typically do not justify the costs of creating it unless the problem has multiple right hand sides (that is, one wants to simulate multiple excitation modes). This project will attempt to quantify the advantage of wavelet-based similarity transforms and suggest practical ways in which they can be employed. The proposal suggests that for large problems with thousands of unknowns there may be an immediate advantage in working in the wavelet basis because of the dramatic reduction in disk access; the costs of the transform might be recovered more quickly than previously thought. Additionally, there are attractive ways of economically creating sparse approximations of the inverse matrix. These are the best kinds preconditioners for iterative solvers like generalized conjugate gradients and GMRES. Preconditioners can dramatically reduce the number of iterations required to achieve a particular tolerance. So, not only do sparse approximations reduce the cost of each iteration (linearly by the sparsification), they can be used to reduce the number of iterations as well. Investigating different preconditioning alternatives constitutes the second objective of the proposal. Of course the most interesting possibility offered by wavelet bases is that of creating a sparse coefficient matrix without suffering the expense of the similarity transformation. The third part of the project will be devot ed to integrating these bases into a previously existing boundary element package and producing the sparse systems directly. 3
