Fast Direct Solvers for Integral Equations in High-Frequency Electromagnetics

This thesis presents the development and analysis of a fast direct solver for the computational analysis of electromagnetic scattering from electrically large structures. The proposed solver combines the data-sparse hierarchical H²-matrix representation with an approximate LU factorization, providing an asymptotically fast framework for solving large-scale linear systems efficiently. By leveraging higher-order basis functions and advanced numerical techniques, the solver achieves significant improvements in speed, accuracy, and versatility compared to existing methods.

A novel algorithm is introduced for constructing an H²-matrix representation of rank structured matrices arising from the discretization of Helmholtz integral equations. This algorithm efficiently exploits the accelerated matrix-vector products of the Multi-Level Fast Multipole Method (MLFMM) to compute low-rank approximations. Unlike other attempts at utilizing MLFMM to construct hierarchical matrices, we overcome the large constants associated with performing individual block products, resulting in an error-controllable and very fast algorithm.

To transform this hierarchical representation into a direct solver with controllable accuracy, several stabilization techniques are proposed. The approximate LU factorization is stabilized through block diagonal equilibration, a regularization scheme based on block SVD, and iterative refinement, ensuring robust numerical performance.

Numerical results demonstrate that the fast direct solver achieves high-order convergence O(h^p), as predicted by theory, and significantly improves computational efficiency for problems involving many right-hand sides. While the solver’s asymptotic scaling is less favorable for volumetric geometries, it exhibits strong practical performance for phased array antennas and monostatic problems. In particular, the solver excels at simulating phased array antennas and enables solutions to problems that are infeasible with current state-of-the-art indirect solvers.