Parallel Jacobi-type algorithms for the singular and the generalized singular value decomposition
Vedran Novaković · 2017
In this thesis, a hierarchically blocked one-sided Jacobi algorithm for the singular value decomposition (SVD) is presented. The algorithm targets both single and multiple graphics processing units (GPUs). The blocking structure reflects the levels of the GPU’s memory hierarchy. To this end, a family of parallel pivot strategies on the GPU’s shared address space has been developed, but the strategies are applicable to inter-node communication as well, with GPU nodes, CPU nodes, or, in general, any NUMA nodes. Unlike common hybrid approaches, the presented algorithm in a single-GPU setting needs a CPU for the controlling purposes only, while utilizing the GPU’s resources to the fullest extent permitted by the hardware. When required by the problem size, the algorithm, in principle, scales to an arbitrary number of GPU nodes. The scalability is demonstrated by more than twofold speedup for sufficiently large matrices on a four-GPU system vs. a single GPU. The subsequent part of the thesis describes how to modify the two-sided Hari–Zimmermann algorithm for computation of the generalized eigendecomposition of a symmetric matrix pair (A; B), where B is positive definite, to an implicit algorithm that computes the generalized singular value decomposition (GSVD) of a pair (F; G). In addition, blocking and parallelization techniques for accelerating both the CPU and the GPU computation are presented, with the GPU approach following the Jacobi SVD algorithm from the first part of the thesis. For triangular matrix pairs of a moderate size, numerical tests show that the double precision sequential pointwise algorithm is several times faster than the established DTGSJA algorithm in LAPACK, while the accuracy is slightly better, especially for the small generalized singular values. Cache-aware blocking increases the performance even further. As with the one-sided Jacobi-type (G)SVD algorithms in general, the presented algorithm is almost perfectly parallelizable and scalable on the shared memory machines, where the speedup almost solely depends on the number of cores used. A distributed memory variant, intended for huge matrices that do not fit into a single NUMA node, as well as a GPU variant, are also sketched. The thesis concludes with the affirmative answer to a question whether the onesided Jacobi-type algorithms can be an efficient and scalable choice for computing the (G)SVD of dense matrices on the massively parallel CPU and GPU architectures. Unless otherwise noted by the inline citations or implied by the context, this thesis is an overview of the original research results, most of which has already been published in [55, 58]. The author’s contributions are the one-sided Jacobi-type GPU algorithms for the ordinary and the generalized SVD, of which the latter has not yet been published, as well as the parallelization technique and some implementation details of the one-sided Hari–Zimmermann CPU algorithm for the GSVD. The rest is joint work with Sanja and Sasa Singer.