اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدImproved Accuracy and Parallelism for MRRR-based Eigensolvers -- A Mixed Precision Approach
101
0
0.0
(
0
)
تأليف
Matthias Petschow
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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.
قيم البحث
اقرأ أيضاً
We describe a strategy for rigorous arbitrary-precision evaluation of Legendre polynomials on the unit interval and its application in the generation of Gauss-Legendre quadrature rules. Our focus is on making the evaluation practical for a wide range
of realistic parameters, corresponding to the requirements of numerical integration to an accuracy of about 100 to 100 000 bits. Our algorithm combines the summation by rectangular splitting of several types of expansions in terms of hypergeometric series with a fixed-point implementation of Bonnets three-term recurrence relation. We then compute rigorous enclosures of the Gauss-Legendre nodes and weights using the interval Newton method. We provide rigorous error bounds for all steps of the algorithm. The approach is validated by an implementation in the Arb library, which achieves order-of-magnitude speedups over previous code for computing Gauss-Legendre rules with simultaneous high degree and precision.
Support for lower precision computation is becoming more common in accelerator hardware due to lower power usage, reduced data movement and increased computational performance. However, computational science and engineering (CSE) problems require dou
ble precision accuracy in several domains. This conflict between hardware trends and application needs has resulted in a need for mixed precision strategies at the linear algebra algorithms level if we want to exploit the hardware to its full potential while meeting the accuracy requirements. In this paper, we focus on preconditioned sparse iterative linear solvers, a key kernel in several CSE applications. We present a study of mixed precision strategies for accelerating this kernel on an NVIDIA V$100$ GPU with a Power 9 CPU. We seek the best methods for incorporating multiple precisions into the GMRES linear solver; these include iterative refinement and parallelizable preconditioners. Our work presents strategies to determine when mixed precision GMRES will be effective and to choose parameters for a mixed precision iterative refinement solver to achieve better performance. We use an implementation that is based on the Trilinos library and employs Kokkos Kernels for performance portability of linear algebra kernels. Performance results demonstrate the promise of mixed precision approaches and demonstrate even further improvements are possible by optimizing low-level kernels.
ODE Test Problems (OTP) is an object-oriented MATLAB package offering a broad range of initial value problems which can be used to test numerical methods such as time integration methods and data assimilation (DA) methods. It includes problems that a
re linear and nonlinear, homogeneous and nonhomogeneous, autonomous and nonautonomous, scalar and high-dimensional, stiff and nonstiff, and chaotic and nonchaotic. Many are real-world problems from fields such as chemistry, astrophysics, meteorology, and electrical engineering. OTP also supports partitioned ODEs for testing IMEX methods, multirate methods, and other multimethods. Functions for plotting solutions and creating movies are available for all problems, and exact solutions are provided when available. OTP is desgined for ease of use-meaning that working with and modifying problems is simple and intuitive.
Gaussian elimination with full pivoting generates a PLUQ matrix decomposition. Depending on the strategy used in the search for pivots, the permutation matrices can reveal some information about the row or the column rank profiles of the matrix. We p
ropose a new pivoting strategy that makes it possible to recover at the same time both row and column rank profiles of the input matrix and of any of its leading sub-matrices. We propose a rank-sensitive and quad-recursive algorithm that computes the latter PLUQ triangular decomposition of an m times n matrix of rank r in O(mnr^{omega-2}) field operations, with omega the exponent of matrix multiplication. Compared to the LEU decomposition by Malashonock, sharing a similar recursive structure, its time complexity is rank sensitive and has a lower leading constant. Over a word size finite field, this algorithm also improveLs the practical efficiency of previously known implementations.
Machine learning frameworks adopt iterative optimizers to train neural networks. Conventional eager execution separates the updating of trainable parameters from forward and backward computations. However, this approach introduces nontrivial training
time overhead due to the lack of data locality and computation parallelism. In this work, we propose to fuse the optimizer with forward or backward computation to better leverage locality and parallelism during training. By reordering the forward computation, gradient calculation, and parameter updating, our proposed method improves the efficiency of iterative optimizers. Experimental results demonstrate that we can achieve an up to 20% training time reduction on various configurations. Since our methods do not alter the optimizer algorithm, they can be used as a general plug-in technique to the training process.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
