Proposal for the Development of a Fast Matrix Inversion Algorithm for EIGER

Statement of Work Robert John Adams of the Department of Electrical & Computer Engineering at the University of Kentucky shall to the best of his ability undertake and carry out research to develop and implement in EIGER boundary integral equation fonnulations for bulk dielectric materials (including PEe) that exhibit stability with respect to the mesh used to discretize the equation for smooth geometries. The implementation will also be perfonned for non-smooth geometries. In the latter case, the conditioning of the fonnulation will be significantly improved relative to the present EIGER implementation, with remaining conditioning problems near the discontinuities in the surface gradient. Methods for resolving this remaining instability will be considered and implemented as warranted. The implementation for smooth and non-smooth geometries will be specifically addressed in the low-frequency limit. These results will be compared with the present EIGER implemen tation, which relies on a basis decomposition in order to achieve an approximate Helmholtz decomposition of the range spaces of the relevant integral operators. An evaluation will be provided indicating any additional effort that may be required in order to replace the present basis decomposition method used by EIGER with an alternative, stable surface integral fonnulation. He will also incorporate an efficient algorithm for computing the inverse of the boundary integral matrix operator. The algorithm will be based on an analytic continuation of the BlE operator from the origin to unity in the complex plane. The origin in the complex plane rep resents the BIE fonnulation of the problem, which we seek to invert. The inverse operator is obtained at unity. In this context, the preceding refonnulation efforts provide a method of moving poles away from the continuation path. An essential component of the inversion scheme will be the incorporation of a fast algo rithm for the computation and storage of matrix-matrix products at various points along the continuation path. Familiar techniques such as the fast multipole method are useful only at the origin where the BlE kernel is the free space Green function. In order to implement the proposed inversion algorithm, we introduce a generalization of the FMM which is applicable to a Green function satisfying arbitrary boundary conditions, including those of the inverse operator. The incorporation of this generalization of the FMM will rely on standard matrix manipulations and will not require the introduction of additional special function routines.