Divide and Conquer Symmetric Tridiagonal Eigensolver for Multicore Architectures

International audienceComputing eigenpairs of a symmetric matrix is a problem arising in many industrial applications, including quantum physics and finite-elements computation for automo-biles. A classical approach is to reduce the matrix to tridiagonal form before computing eigenpairs of the tridiagonal matrix. Then, a back-transformation allows one to obtain the final solution. Parallelism issues of the reduction stage have already been tackled in different shared-memory libraries. In this article, we focus on solving the tridiagonal eigenproblem, and we describe a novel implementation of the Divide and Conquer algorithm. The algorithm is expressed as a sequential task-flow, scheduled in an out-of-order fashion by a dynamic runtime which allows the programmer to play with tasks granularity. The resulting implementation is between two and five times faster than the equivalent routine from the INTEL MKL library, and outperforms the best MRRR implementation for many matrices

Out-of-core macromolecular simulations on multithreaded architectures

We address the solution of large-scale eigenvalue problems that appear in the motion simulation of complex macromolecules on multithreaded platforms, consisting of multicore processors and possibly a graphics processor (GPU). In particular, we compare specialized implementations of several high- performance eigensolvers that, by relying on disk storage and out-of-core (OOC) techniques, can in principle tackle the large memory requirements of these biological problems, which in general do not fit into the main memory of current desktop machines. All these OOC eigensolvers, except for one, are composed of compute-bound (i.e., arithmetically-intensive) operations, which we accelerate by exploiting the performance of current multicore processors and, in some cases, by additionally off-loading certain parts of the computation to a GPU accelerator. One of the eigensolvers is a memory-bound algorithm, which strongly constrains its performance when the data is on disk. However, this method exhibits a much lower arithmetic cost compared with its compute- bound alternatives for this particular application. Experimental results on a desktop platform, representative of current server technology, illustrate the potential of these methods to address the simulation of biological activity

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

Mixed-Precision Numerical Linear Algebra Algorithms: Integer Arithmetic Based LU Factorization and Iterative Refinement for Hermitian Eigenvalue Problem

Mixed-precision algorithms are a class of algorithms that uses low precision in part of the algorithm in order to save time and energy with less accurate computation and communication. These algorithms usually utilize iterative refinement processes to improve the approximate solution obtained from low precision to the accuracy we desire from doing all the computation in high precision. Due to the demand of deep learning applications, there are hardware developments offering different low-precision formats including half precision (FP16), Bfloat16 and integer operations for quantized integers, which uses integers with a shared scalar to represent a set of equally spaced numbers. As new hardware architectures focus on bringing performance in these formats, the mixed-precision algorithms have more potential leverage on them and outmatch traditional fixed-precision algorithms. This dissertation consists of two articles. In the first article, we adapt one of the most fundamental algorithms in numerical linear algebra---LU factorization with partial pivoting--- to use integer arithmetic. With the goal of obtaining a low accuracy factorization as the preconditioner of generalized minimal residual (GMRES) to solve systems of linear equations, the LU factorization is adapted to use two different fixed-point formats for matrices L and U. A left-looking variant is also proposed for matrices with unbounded column growth. Finally, GMRES iterative refinement has shown that it can work on matrices with condition numbers up to 10000 with the algorithm that uses int16 as input and int32 accumulator for the update step. The second article targets symmetric and Hermitian eigenvalue problems. In this section we revisit the SICE algorithm from Dongarra et al. By applying the Sherman-Morrison formula on the diagonally-shifted tridiagonal systems, we propose an updated SICE-SM algorithm. By incorporating the latest two-stage algorithms from the PLASMA and MAGMA software libraries for numerical linear algebra, we achieved up to 3.6x speedup using the mixed-precision eigensolver with the blocked SICE-SM algorithm for iterative refinement when compared with full double complex precision solvers for the cases with a portion of eigenvalues and eigenvectors requested

MRRR-based Eigensolvers for Multi-core Processors and Supercomputers

The real symmetric tridiagonal eigenproblem is of outstanding importance in numerical computations; it arises frequently as part of eigensolvers for standard and generalized dense Hermitian eigenproblems that are based on a reduction to tridiagonal form. For its solution, the algorithm of Multiple Relatively Robust Representations (MRRR or MR3 in short) - introduced in the late 1990s - is among the fastest methods. To compute k eigenpairs of a real n-by-n tridiagonal T, MRRR only requires O(kn) arithmetic operations; in contrast, all the other practical methods require O(k^2 n) or O(n^3) operations in the worst case. This thesis centers around the performance and accuracy of MRRR.Comment: PhD thesi

MPI-CUDA parallel linear solvers for block-tridiagonal matrices in the context of SLEPc's eigensolvers

[EN] We consider the computation of a few eigenpairs of a generalized eigenvalue problem Ax = lambda Bx with block-tridiagonal matrices, not necessarily symmetric, in the context of Krylov methods. In this kind of computation, it is often necessary to solve a linear system of equations in each iteration of the eigensolver, for instance when B is not the identity matrix or when computing interior eigenvalues with the shift-and-invert spectral transformation. In this work, we aim to compare different direct linear solvers that can exploit the block-tridiagonal structure. Block cyclic reduction and the Spike algorithm are considered. A parallel implementation based on MPI is developed in the context of the SLEPc library. The use of GPU devices to accelerate local computations shows to be competitive for large block sizes.This work was supported by Agencia Estatal de Investigacion (AEI) under grant TIN2016-75985-P, which includes European Commission ERDF funds. Alejandro Lamas Davina was supported by the Spanish Ministry of Education, Culture and Sport through a grant with reference FPU13-06655.Lamas Daviña, A.; Roman, JE. (2018). MPI-CUDA parallel linear solvers for block-tridiagonal matrices in the context of SLEPc's eigensolvers. Parallel Computing. 74:118-135. https://doi.org/10.1016/j.parco.2017.11.006S1181357

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