Advanced Algorithmic Strategies and Parallel Computational Models for Low and High Frequency Electromagnetics Scenarios

The increasing demand for computational power in diverse scientific and engineering fields has resulted in a notable rise in parallel computing research. This is particularly evident in disciplines such as Computational Electromagnetics, which strive to develop accurate and efficient solvers for Maxwell’s equations. Given the current technological advancements that necessitate large-scale electromagnetic simulations involving billions of unknowns, distributed- memory systems have emerged as the primary solution due to their improved computation and memory capabilities. This thesis analyzes the way in which parallel computing impacts two widely used algo- rithms for electromagnetic problems: the Adaptive Cross Approximation (ACA) and the Fast Multipole Method (FMM), with a special focus on the computational and parallel execution aspects. Considering the impressive efficiency of these algorithms, in particular for low and high frequency problems respectively, a carefully crafted parallelization strategy has the potential to greatly enhance their performance. Moreover, it enables the resolution of larger problems that would be impractical using a sequential implementation. The initial part of the thesis concentrates on the ACA algorithm and examines the chal- lenges that arise from implementing it in parallel. To enhance the performance of parallel execution, the thesis introduces a scheduling strategy, particularly beneficial when operating on distributed-memory systems. Extensive benchmarking and profiling have been conducted on both shared and distributed memory systems to rigorously test all the implemented code. The second part of the thesis is dedicated to the FMM algorithm. Building upon an existing parallel implementation, thorough analyses have been conducted to evaluate the algorithm’s performance. These analyses aim to provide a comprehensive understanding that can be uti- lized in the final part of the thesis. Additionally, estimations have been made regarding the requirements necessary to successfully apply the solver on larger problems. The third part of the thesis centers around a recently developed Fast Direct Solver (FDS) based on a modified implementation of the FMM algorithm. The purpose of a FDS is to construct an approximation of the inverse system matrix at reduced complexity, thus, unlike an iterative solver, offering the advantage of efficiently solving problems with multiple right- hand sides. Drawing from an existing serial implementation and leveraging insights from the previously analyzed parallel implementation of the FMM, this thesis presents a novel paral- lelization strategy that specifically addresses the challenges stemming from the deviations from the original algorithm. In summary, this thesis offers a comprehensive investigation into the parallel execution of the ACA and FMM algorithms. It proposes parallelization strategies, as required, and analyzes their performance on both shared and distributed memory systems, with the ultimate target of handling extremely large problems in the most efficient way possible.