High-performance SVD partial spectrum computation

We introduce a new singular value decomposition (SVD) solver based on the QR-based Dynamically Weighted Halley (QDWH) algorithm for computing the partial spectrum SVD (QDWHpartial-SVD) problems. By optimizing the rational function underlying the algorithms in the desired part of the spectrum only, the QDWHpartial-SVD algorithm efficiently computes a fraction (say 1-20%) of the leading singular values/vectors. We develop a high-performance implementation of QDWHpartial-SVD 1 on distributed-memory manycore systems and demonstrate its numerical robustness. We perform a benchmarking campaign against counterparts from the state-of-theart numerical libraries across various matrix sizes using up to 36K MPI processes. Experimental results show performance speedups for QDWHpartial-SVD up to 6X and 2X against vendor-optimized PDGESVD from ScaLAPACK and KSVD on a Cray XC40 system using 1152 nodes based on two-socket 16-core Intel Haswell CPU, respectively. We also port our QDWHpartial-SVD software library to a system composed of 256 nodes with two-socket 64-Core AMD EPYC Milan CPU and achieve performance speedup up to 4X compared to vendor-optimized PDGESVD from ScaLAPACK. We also compare energy consumption for the two algorithms and demonstrate how QDWHpartial-SVD can further outperform PDGESVD in that regard by performing fewer memory-bound operations

Asynchronous Task-Based Polar Decomposition on Manycore Architectures

This paper introduces the first asynchronous, task-based implementation of the polar decomposition on manycore architectures. Based on a new formulation of the iterative QR dynamically-weighted Halley algorithm (QDWH) for the calculation of the polar decomposition, the proposed implementation replaces the original and hostile LU factorization for the condition number estimator by the more adequate QR factorization to enable software portability across various architectures. Relying on fine-grained computations, the novel task-based implementation is also capable of taking advantage of the identity structure of the matrix involved during the QDWH iterations, which decreases the overall algorithmic complexity. Furthermore, the artifactual synchronization points have been severely weakened compared to previous implementations, unveiling look-ahead opportunities for better hardware occupancy. The overall QDWH-based polar decomposition can then be represented as a directed acyclic graph (DAG), where nodes represent computational tasks and edges define the inter-task data dependencies. The StarPU dynamic runtime system is employed to traverse the DAG, to track the various data dependencies and to asynchronously schedule the computational tasks on the underlying hardware resources, resulting in an out-of-order task scheduling. Benchmarking experiments show significant improvements against existing state-of-the-art high performance implementations (i.e., Intel MKL and Elemental) for the polar decomposition on latest shared-memory vendors' systems (i.e., Intel Haswell/Broadwell/Knights Landing, NVIDIA K80/P100 GPUs and IBM Power8), while maintaining high numerical accuracy

Leveraging Task-Based Polar Decomposition Using PARSEC on Massively Parallel Systems

International audienceThis paper describes how to leverage a task-based implementation of the polar decomposition on massively parallel systems using the PARSEC dynamic runtime system. Based on a formulation of the iterative QR Dynamically-Weighted Halley (QDWH) algorithm, our novel implementation reduces data traffic while exploiting high concurrency from the underlying hardware architecture. First, we replace the most time-consuming classical QR factorization phase with a new hierarchical variant, customized for the specific structure of the matrix during the QDWH iterations. The newly developed hierarchical QR for QDWH exploits not only the matrix structure, but also shortens the length of the critical path to maximize hardware occupancy. We then deploy PARSEC to seamlessly orchestrate, pipeline, and track the data dependencies of the various linear algebra building blocks involved during the iterative QDWH algorithm. PARSEC enables to overlap communications with computations thanks to its asynchronous scheduling of fine-grained computational tasks. It employs look-ahead techniques to further expose parallelism, while actively pursuing the critical path. In addition, we identify synergistic opportunities between the task-based QDWH algorithm and the PARSEC framework. We exploit them during the hierarchical QR factorization to enforce a locality-aware task execution. The latter feature permits to minimize the expensive inter-node communication, which represents one of the main bottlenecks for scaling up applications on challenging distributed-memory systems. We report numerical accuracy and performance results using well and ill-conditioned matrices. The benchmarking campaign reveals up to 2X performance speedup against the existing state-of-the-art implementation for the polar decomposition on 36,864 cores

Out-of-core solution of eigenproblems for macromolecular simulations

We consider the solution of large-scale eigenvalue problems that appear in the motion simulation of complex macromolecules on desktop platforms. To tackle the dimension of the matrices that are involved in these problems, we formulate out-of-core (OOC) variants of the two selected eigensolvers, that basically decouple the performance of the solver from the storage capacity. Furthermore, we contend with the high computational complexity of the solvers by off-loading the arithmetically-intensive parts of the algorithms to a hardware graphics accelerator

Solving Large Dense Symmetric Eigenproblem on Hybrid Architectures

Dense symmetric eigenproblem is one of the most significant problems in the numerical linear algebra that arises in numerous research fields such as bioinformatics, computational chemistry, and meteorology. In the past years, the problems arising in these fields become bigger than ever resulting in growing demands in both computational power as well as the storage capacities. In such problems, the eigenproblem becomes the main computational bottleneck for which solution is required an extremely high computational power. Modern computing architectures that can meet these growing demands are those that combine the power of the traditional multi-core processors and the general-purpose GPUs and are called hybrid systems. These systems exhibit very high performance when the data fits into the GPU memory ; however, if the volume of the data exceeds the total GPU memory, i.e. the data is out-of-core from the GPU perspective, the performance rapidly decreases. This dissertation is focused on the development of the algorithms that solve dense symmetric eigenproblems on the hybrid GPU-based architectures. In particular, it aims at developing the eigensolvers that exhibit very high performance even if a problem is out- of-core for the GPU. The developed out-of-core eigensolvers are evaluated and compared on real problems that arise in the simulation of molecular motions. In such problems the data, usually too large to fit into the GPU memory, are stored in the main memory and copied to the GPU memory in pieces. That approach results in the performance drop due to a slow interconnection and a high memory latency. To overcome this problem an approach that applies blocking strategy and re- designs the existing eigensolvers, in order to decrease the volume of data transferred and the number of memory transfers, is presented. This approach designs and implements a set of the block- oriented, communication-avoiding BLAS routines that overlap the data transfers with the number of computations performed. Next, these routines are applied to speed-up the following eigensolvers: the solver based on the multi-stage reduction to a tridiagonal form, the Krylov subspace-based method, and the spectral divide-and-conquer method. Although the out-of-core BLAS routines significantly improve the performance of these three eigensolvers, a careful re-design is required in order to tackle the solution of the large eigenproblems on the hybrid CPU-GPU systems. In the out-of-core multi-stage reduction approach, the factor that mostly influences the performance is the band size of the obtained band matrix. On the other hand, the Krylov subspace- based method, although it is based on the memory- bound BLAS-2 operations, is the fastest method if only a small subset of the eigenpairs is required. Finally, the spectral divide-and- conquer algorithm, which exhibits significantly higher arithmetic cost than the other two eigensolvers, achieves extremely high performance since it can be performed completely in terms of the compute-bound BLAS-3 operations. Furthermore, its high arithmetic cost is further reduced by exploiting the special structure of a matrix. Finally, the results presented in the dissertation show that the three out-of-core eigen- solvers, for a set of the specific macromolecular problems, significantly overcome the multi-core variants and attain high flops rate even if data do not fit into the GPU memory. This proves that it is possible to solve large eigenproblems on modest computing systems equipped with a single GPU

Lecture 11: The Road to Exascale and Legacy Software for Dense Linear Algebra

In this talk, we will look at the current state of high performance computing and look at the next stage of extreme computing. With extreme computing, there will be fundamental changes in the character of floating point arithmetic and data movement. In this talk, we will look at how extreme-scale computing has caused algorithm and software developers to change their way of thinking on implementing and program-specific applications

Efficient Algorithms for Solving Structured Eigenvalue Problems Arising in the Description of Electronic Excitations

Matrices arising in linear-response time-dependent density functional theory and many-body perturbation theory, in particular in the Bethe-Salpeter approach, show a 2 × 2 block structure. The motivation to devise new algorithms, instead of using general purpose eigenvalue solvers, comes from the need to solve large problems on high performance computers. This requires parallelizable and communication-avoiding algorithms and implementations. We point out various novel directions for diagonalizing structured matrices. These include the solution of skew-symmetric eigenvalue problems in ELPA, as well as structure preserving spectral divide-and-conquer schemes employing generalized polar decompostions