This dissertation presents and analyzes two new algorithms for sparse direct solution methods based on the use of local-global solution (LOGOS) modes. One of the new algorithms is a rigorous error control strategy for LOGOS-based matrix factorizations that utilize overlapped, localizing modes (OL-LOGOS) on a shifted grid. The use of OL-LOGOS modes is critical to obtaining asymptotically efficient factorizations from LOGOS-based methods. Unfortunately, the approach also introduces a non-orthogonal basis function structure. This can cause errors to accumulate across levels of a multilevel implementation, which has previously posed a barrier to rigorous error control for the OL-LOGOS factorization method. This limitation is overcome, and it is shown that it is possible to efficiently decouple the fundamentally non-orthogonal factorization subspaces in a manner that prevents multilevel error propagation. This renders the OL-LOGOS factorization error controllable in a relative RMS sense. The impact of the new, error-controlled OL-LOGOS factorization algorithm on computational resource utilization is discussed and several numerical examples are presented to illustrate the performance of the improved algorithm relative to previously reported results. The second algorithmic development considered is the development of efficient out-of-core (OOC) versions of the OL-LOGOS factorization algorithm that allow associated software tools to take advantage of additional resources for memory management. The proposed OOC algorithm incorporates a memory page definition that is tailored to match the flow of the OL-LOGOS factorization procedure. Efficiency of the function of the part is evaluated using a quantitative approach, because the tested massive storage device performances do not follow analytical results. The performance latency and the memory usage of the resulting OOC tools are compared with in-core performance results. Both the new error control algorithm and the OOC method have been incorporated into previously existing software tools, and the dissertation presents results for real-world simulation problems.

This dissertation proposes a modular fast direct (MFD) analysis method for a class of problems involving a large fixed platform region and a smaller, variable design region. A modular solution algorithm is obtained by first decomposing the problem geometry into platform and design regions. The two regions are effectively detached from one another using basic equivalence concepts. Equivalence principles allow the total system model to be constructed in terms of independent interaction modules associated with the platform and design regions. These modules include interactions with the equivalent surface that bounds the design region. This dissertation discusses how to analyze (fill and factor) each of these modules separately and how to subsequently compose the solution to the original system using the separately analyzed modules. The focus of this effort is on surface integral equation formulations of electromagnetic scattering from conductors and dielectrics. In order to treat large problems, it is necessary to work with sparse representations of the underlying system matrix and other, related matrices. Fortunately, a number of such representations are available. In the following, we will primarily use the adaptive cross approximation (ACA) to fill the multilevel simply sparse method (MLSSM) representation of the system matrix. The MLSSM provides a sparse representation that is similar to the multilevel fast multipole method. Solutions to the linear systems obtained using the modular analysis strategies described above are obtained using direct methods based on the local-global solution (LOGOS) method. In particular, the LOGOS factorization provides a data sparse factorization of the MLSSM representation of the system matrix. In addition, the LOGOS solver also provides an approximate sparse factorization of the inverse of the system matrix. The availability of the inverse eases the development of the MFD method. Because the behavior of the LOGOS factorization is critical to the development of the proposed MFD method, a significant part of this dissertation is devoted to providing additional analyses, improvements, and characterizations of LOGOS-based direct solution methods. These further developments of the LOGOS factorization algorithms and their application to the development of the MFD method comprise the most significant contributions of this dissertation.

This dissertation develops surface integral equations using constraint-based Helmholtz decompositions for electromagnetic modeling. This new approach is applied to the electric field integral equation (EFIE), and it incorporates a Helmholtz decomposition (HD) of the current. For this reason, the new formulation is referred to as the EFIE-hd. The HD of the current is accomplished herein via appropriate surface integral constraints, and leads to a stable linear system. This strategy provides accurate solutions for the electric and magnetic fields at both high and low frequencies, it allows for the use of a locally corrected Nyström (LCN) discretization method for the resulting formulation, it is compatible with the local global solution framework, and it can be used with non-conformal meshes. To address large-scale and complex electromagnetic problems, an overlapped localizing local-global (OL-LOGOS) factorization is used to factorize the system matrix obtained from an LCN discretization of the augmented EFIE (AEFIE). The OL-LOGOS algorithm provides good asymptotic performance and error control when used with the AEFIE. This application is used to demonstrate the importance of using a well-conditioned formulation to obtain efficient performance from the factorization algorithm.

Computational electromagnetic modeling involves generating system matrices by discretizing integral equations and solving the resulting system of linear equations. Many methods of solving the system of linear equations exist and one such method is the factorization of the matrix using the so called local-global solution (LOGOS) modes. Computer codes to perform the discretization of the integral equations, filling of the matrix, and the subsequent LOGOS factorization have previously been developed by others. However, these codes are limited to complex double precision arithmetic only. This thesis extends and expands the existing computer by creating a more general implementation that is able to analyze a problem not only in complex double precision but also in real double precision and both complex and real single precision. The existing code is expanded using "templates" in Fortran 90 and the resulting generic code is used test the performance of the LOGOS (both OL- and NL-LOGOS) factorization on matrices generated by discretization of the volume integral equation. As part of this effort, we demonstrate for the first time that the LOGOS factorization provides an O(N log N) complexity solution to the volume integral equation formulation of low-frequency electromagnetic problems.