Adam Dziekonski

The paper discusses a fast implementation of the stabilized locally optimal block preconditioned conjugate gradient (sLOBPCG) 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 type 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-memorysaving 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, twelve 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.

This paper discusses two fast implementations of the conjugate gradient iterative method using a hierarchical multilevel preconditioner to solve the complex-valued, sparse systems obtained using the higher order finite-element method applied to the solution of the time-harmonic Maxwell equations. In the first implementation, denoted PCG-V, a classical V-cycle is applied and the system of equations on the lowest level is solved exactly. The second variant involves an approximate solution to the system of equations on the lowest level. To this end, auxiliary space preconditioning (ASP) is used instead of a direct solution. In this approach, denoted PCG-V-ASP, the time needed to solve the sparse system of equations is longer, but the memory requirements are smaller. To accelerate the computations, a graphics processing unit (GPU, Pascal P100) was used for both variants of the multilevel preconditioner. As a result, significant speedups were achieved over the reference parallel implementation using a multicore central processing unit (CPU, Intel Xeon E5-2680 v3, twelve cores). The results indicate that the auxiliary space preconditioning does in fact reduce the memory requirements, as compared with the reference PCG-V method, and at the same time performs each iteration faster. However, if symmetry is taken into account and the memory-efficient supernodal $LDL^{T}$ factorization is employed, the savings are less spectacular than anticipated based on previously published results using LU factorization and the multifrontal technique. PCG-V also requires a fewer iterations, so it’s time to solution is ultimately shorter. The difference is more pronounced if both preconditioners are run on a CPU. The use of a GPU as an accelerator for the computations considerably improves the performance of PCG-V-ASP over that of PCG-V.

In this paper a GPU-accelerated block conjugate-gradient solver with multilevel preconditioning is presented for solving large system of sparse equations with multiple right-hand-sides (RHSs) which arise in the finite-element analysis of electromagnetic problems. We demonstrate that blocking reduces the time to solution significantly and allows for better utilization of the computing power of GPUs, especially when the system matrix is complex-valued. We show that, for a complex-valued sparse matrix with over 1.2 million unknowns and sixteen right-hand-sides, the block version of the preconditioned conjugate-gradient method using a single NVIDIA Tesla P100 accelerator is almost three times faster than a nonblocked version. Numerical tests have also shown that a CPU-only nonblocked complex implementation is unlikely to benefit much from blocking. Compared to an optimized CPU-only solver using an Intel Xeon E5-2680 v3 with twelve cores, the nonblocked GPU-accelerated version was 4.5 times faster, while the blocked version was 12.4 times faster.

This letter proposes a stabilized locally optimal block preconditioned conjugate gradient method for computing selected eigenvalues for complex symmetric generalized non-Hermitian eigenproblems. Effectiveness of the presented approach is demonstrated for a moderately lossy dual-mode dielectric resonator, modeled using finite-element method with higher order elements.

This paper presents a GPU-accelerated implementation of the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method with an inexact nullspace filtering approach to find eigenvalues in electromagnetics analysis with higher-order FEM. The performance of the proposed approach is verified using the Kepler (Tesla K40c) graphics accelerator, and is compared to the performance of the implementation based on functions from the Intel MKL on the Intel Xeon (E5-2680 v3, 12 threads) central processing unit (CPU) executed in parallel mode. Compared to the CPU reference implementation based on the Intel MKL functions, the proposed GPU-based LOBPCG method with inexact nullspace filtering allowed us to achieve up to 2.9-fold acceleration.

This paper considers a method for accelerating finite-element simulations of electromagnetic problems on a workstation using graphics processing units (GPUs). The focus is on finite-element formulations using higher order elements and tetrahedral meshes that lead to sparse matrices too large to be dealt with on a typical workstation using direct methods. We discuss the problem of rapid matrix generation and assembly, as well as accelerating preconditioned iterative solvers in the context of limited on-board GPU memory, and we show how to mitigate some of these problems using multiple GPUs. We propose a new fast data-distribution technique for multi-GPU platforms that allows optimal splitting of finite-element method (FEM) matrices between graphics accelerators. The technique draws upon the graph partitioning approach used in nonoverlapping domain-decomposition methods and provides information that drives the FEM matrix-generation and assembly process in such a way that it produces data structures for each GPU; this not only ensures load balancing and minimizes communication between GPUs, but also reflects the hierarchy of the basis functions. The concepts proposed in this paper are illustrated with examples involving sparse matrices of up to 13.9 million rows and over a billion nonzero elements.

This paper presents an approach to performing computations of a block sparse matrix-vector product, a corner-stone operation of the GPU-accelerated block Krylov subspace methods used in the solution phase of finite-element analysis. The results obtained for a high-end test platform (GPU: Tesla K40; CPU: Intel Xeon E5-2680 v3) indicate that the proposed code optimization allows significant time reductions in comparison with the functions available in the MAGMA, cuSPARSE, and Intel MKL libraries.

In this paper the results of the acceleration of computations involved in analysing electromagnetic problems by means of the finite element method (FEM), obtained with graphics processors (GPU), are presented. A 4.7-fold acceleration was achieved thanks to the massive parallelization of the most time-consuming steps of FEM, namely finite-element matrix-generation and the solution of a sparse system of linear equations with the conjugate gradient method and a V-cycle multilevel preconditioner.

In this paper, a task-scheduling approach to efficiently calculating sparse symmetric matrix-vector products and designed to run on graphics processing units (GPUs) is presented. The main premise is that, for many sparse symmetric matrices occurring in common applications, it is possible to obtain significant reductions in memory usage and improvements in performance when the matrix is prepared in certain ways prior to computation. The preprocessing proposed in this paper employs task scheduling to overcome the difficulties that have suppressed the development of methods taking advantage of the symmetry of sparse matrices. The performance of the proposed task-scheduling method is verified using a Kepler (Tesla K40c) graphics accelerator, and is compared to the performance of cuSPARSE library functions on a GPU and to functions from the Intel MKL on central processing units (CPUs) executed in the parallel mode. The obtained results indicate that the proposed approach for sparse symmetric matrix-vector prod...