A GPU Solver for Sparse Generalized Eigenvalue Problems With Symmetric Complex-Valued Matrices Obtained Using Higher-Order FEM

Abstract

This paper discusses a fast implementation of the stabilized locally optimal block preconditioned conjugate gradient method, using a hierarchical multilevel preconditioner to solve non-Hermitian sparse generalized eigenvalue problems with large symmetric complex-valued matrices obtained using the higher-order finite-element method (FEM), applied to the analysis of a microwave resonator. The resonant frequencies of the low-order modes are the eigenvalues of the smallest real part of a complex symmetric (though non-Hermitian) matrix pencil. These types of pencils arise in the FEM analysis of resonant cavities loaded with a lossy material. To accelerate the computations, graphics processing units (GPU, NVIDIA Pascal P100) were used. Single and dual-GPU variants are considered and a GPU-memory-saving implementation is proposed. An efficient sliced ELLR-T sparse matrix storage format was used and operations were performed on blocks of vectors for best performance on a GPU. As a result, significant speedups (exceeding a factor of six in some computational scenarios) were achieved over the reference parallel implementation using a multicore central processing unit (CPU, Intel Xeon E5-2680 v3, and 12 cores). These results indicate that the solution of generalized eigenproblems needs much more GPU memory than iterative techniques when solving a sparse system of equations, and also requires a second GPU to store some data structures in order to reduce the footprint, even for a moderately large systems.