Two-Stage Block Orthogonalization to Improve Performance of ss-step GMRES

On current computer architectures, GMRES' performance can be limited by its communication cost to generate orthonormal basis vectors of the Krylov subspace. To address this performance bottleneck, its $s$-step variant orthogonalizes a block of $s$ basis vectors at a time, potentially reducing the communication cost by a factor of $s$. Unfortunately, for a large step size $s$, the solver can generate extremely ill-conditioned basis vectors, and to maintain stability in practice, a conservatively small step size is used, which limits the performance of the $s$-step solver. To enhance the performance using a small step size, in this paper, we introduce a two-stage block orthogonalization scheme. Similar to the original scheme, the first stage of the proposed method operates on a block of $s$ basis vectors at a time, but its objective is to maintain the well-conditioning of the generated basis vectors with a lower cost. The orthogonalization of the basis vectors is delayed until the second stage when enough basis vectors are generated to obtain higher performance. Our analysis shows the stability of the proposed two-stage scheme. The performance is improved because while the same amount of computation as the original scheme is required, most of the communication is done at the second stage of the proposed scheme, reducing the overall communication requirements. Our performance results with up to 192 NVIDIA V100 GPUs on the Summit supercomputer demonstrate that when solving a 2D Laplace problem, the two-stage approach can reduce the orthogonalization time and the total time-to-solution by the respective factors of up to $2.6\times$ and $1.6\times$ over the original $s$-step GMRES, which had already obtained the respective speedups of $2.1\times$ and $1.8\times$ over the standard GMRES. Similar speedups were obtained for 3D problems and for matrices from the SuiteSparse Matrix Collection.Comment: Accepted for publication in IPDPS'2

Performance of random sampling for computing low-rank approximations of a dense matrix on GPUs

International audienceA low-rank approximation of a dense matrix plays an important role in many applications. To compute such an approximation , a common approach uses the QR factorization with column pivoting (QRCP). Though the reliability and efficiency of QRCP have been demonstrated, this determin-istic approach requires costly communication at each step of the factorization. Since such communication is becoming increasingly expensive on modern computers, an alternative approach based on random sampling, which can be implemented using communication-optimal kernels, is becoming attractive. To study its potential, in this paper, we compare the performance of random sampling with that of QRCP on an NVIDIA Kepler GPU. Our performance results demonstrate that random sampling can be up to 12.8× faster than the deterministic approach for computing the approximation of the same accuracy. We also present the parallel scaling of the random sampling over multiple GPUs on a single compute node, showing a speedup of 3.8× over three Kepler GPUs. These results demonstrate the potential of the random sampling as an excellent computational tool for many applications, and its potential is likely to grow on the emerging computers with the increasing communication costs

An overview of block Gram-Schmidt methods and their stability properties

Block Gram-Schmidt algorithms serve as essential kernels in many scientific computing applications, but for many commonly used variants, a rigorous treatment of their stability properties remains open. This survey provides a comprehensive categorization of block Gram-Schmidt algorithms, particularly those used in Krylov subspace methods to build orthonormal bases one block vector at a time. All known stability results are assembled, and new results are summarized or conjectured for important communication-reducing variants. Additionally, new block versions of low-synchronization variants are derived, and their efficacy and stability are demonstrated for a wide range of challenging examples. Low-synchronization variants appear remarkably stable for s-step-like matrices built with Newton polynomials, pointing towards a new stable and efficient backbone for Krylov subspace methods. Numerical examples are computed with a versatile MATLAB package hosted at https://github.com/katlund/BlockStab, and scripts for reproducing all results in the paper are provided. Block Gram-Schmidt implementations in popular software packages are discussed, along with a number of open problems. An appendix containing all algorithms type-set in a uniform fashion is provided.Comment: 42 pages, 5 tables, 17 figures, 20 algorithm