Improved Accuracy and Parallelism for MRRR-based Eigensolvers -- A Mixed Precision Approach

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) is among the fastest methods. Although fast, the solvers based on MRRR do not deliver the same accuracy as competing methods like Divide & Conquer or the QR algorithm. In this paper, we demonstrate that the use of mixed precisions leads to improved accuracy of MRRR-based eigensolvers with limited or no performance penalty. As a result, we obtain eigensolvers that are not only equally or more accurate than the best available methods, but also -in most circumstances- faster and more scalable than the competition

Lanczos eigensolution method for high-performance computers

The theory, computational analysis, and applications are presented of a Lanczos algorithm on high performance computers. The computationally intensive steps of the algorithm are identified as: the matrix factorization, the forward/backward equation solution, and the matrix vector multiples. These computational steps are optimized to exploit the vector and parallel capabilities of high performance computers. The savings in computational time from applying optimization techniques such as: variable band and sparse data storage and access, loop unrolling, use of local memory, and compiler directives are presented. Two large scale structural analysis applications are described: the buckling of a composite blade stiffened panel with a cutout, and the vibration analysis of a high speed civil transport. The sequential computational time for the panel problem executed on a CONVEX computer of 181.6 seconds was decreased to 14.1 seconds with the optimized vector algorithm. The best computational time of 23 seconds for the transport problem with 17,000 degs of freedom was on the the Cray-YMP using an average of 3.63 processors

A bibliography on parallel and vector numerical algorithms

This is a bibliography of numerical methods. It also includes a number of other references on machine architecture, programming language, and other topics of interest to scientific computing. Certain conference proceedings and anthologies which have been published in book form are listed also

GHOST: Building blocks for high performance sparse linear algebra on heterogeneous systems

While many of the architectural details of future exascale-class high performance computer systems are still a matter of intense research, there appears to be a general consensus that they will be strongly heterogeneous, featuring "standard" as well as "accelerated" resources. Today, such resources are available as multicore processors, graphics processing units (GPUs), and other accelerators such as the Intel Xeon Phi. Any software infrastructure that claims usefulness for such environments must be able to meet their inherent challenges: massive multi-level parallelism, topology, asynchronicity, and abstraction. The "General, Hybrid, and Optimized Sparse Toolkit" (GHOST) is a collection of building blocks that targets algorithms dealing with sparse matrix representations on current and future large-scale systems. It implements the "MPI+X" paradigm, has a pure C interface, and provides hybrid-parallel numerical kernels, intelligent resource management, and truly heterogeneous parallelism for multicore CPUs, Nvidia GPUs, and the Intel Xeon Phi. We describe the details of its design with respect to the challenges posed by modern heterogeneous supercomputers and recent algorithmic developments. Implementation details which are indispensable for achieving high efficiency are pointed out and their necessity is justified by performance measurements or predictions based on performance models. The library code and several applications are available as open source. We also provide instructions on how to make use of GHOST in existing software packages, together with a case study which demonstrates the applicability and performance of GHOST as a component within a larger software stack.Comment: 32 pages, 11 figure

Solving large sparse eigenvalue problems on supercomputers

An important problem in scientific computing consists in finding a few eigenvalues and corresponding eigenvectors of a very large and sparse matrix. The most popular methods to solve these problems are based on projection techniques on appropriate subspaces. The main attraction of these methods is that they only require the use of the matrix in the form of matrix by vector multiplications. The implementations on supercomputers of two such methods for symmetric matrices, namely Lanczos' method and Davidson's method are compared. Since one of the most important operations in these two methods is the multiplication of vectors by the sparse matrix, methods of performing this operation efficiently are discussed. The advantages and the disadvantages of each method are compared and implementation aspects are discussed. Numerical experiments on a one processor CRAY 2 and CRAY X-MP are reported. Possible parallel implementations are also discussed

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

Temporal unpredictability detection of real-time video sequence

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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