DeepAI AI Chat
Log In Sign Up

Using Mixed Precision in Low-Synchronization Reorthogonalized Block Classical Gram-Schmidt

10/17/2022
∙
by   Eda Oktay, et al.
∙
0
∙

Using lower precision in algorithms can be beneficial in terms of reducing both computation and communication costs. Motivated by this, we aim to further the state-of-the-art in developing and analyzing mixed precision variants of iterative methods. In this work, we focus on the block variant of low-synchronization classical Gram-Schmidt with reorthogonalization, which we call BCGSI+LS. We demonstrate that the loss of orthogonality produced by this orthogonalization scheme can exceed O(u)Îș(𝒳), where u is the unit roundoff and Îș(𝒳) is the condition number of the matrix to be orthogonalized, and thus we can not in general expect this to result in a backward stable block GMRES implementation. We then develop a mixed precision variant of this algorithm, called BCGSI+LS-MP, which uses higher precision in certain parts of the computation. We demonstrate experimentally that for a number of challenging test problems, our mixed precision variant successfully maintains a loss of orthogonality below O(u)Îș(𝒳). This indicates that we can achieve a backward stable block GMRES algorithm that requires only one synchronization per iteration.

READ FULL TEXT

page 1

page 2

page 3

page 4

∙ 08/21/2022

Adaptively restarted block Krylov subspace methods with low-synchronization skeletons

With the recent realization of exascale performace by Oak Ridge National...
∙ 09/16/2018

Low synchronization GMRES algorithms

Communication-avoiding and pipelined variants of Krylov solvers are crit...
∙ 07/13/2021

Multistage Mixed Precision Iterative Refinement

Low precision arithmetic, in particular half precision (16-bit) floating...
∙ 03/16/2021

Mixed Precision s-step Lanczos and Conjugate Gradient Algorithms

Compared to the classical Lanczos algorithm, the s-step Lanczos variant ...
∙ 10/22/2020

An overview of block Gram-Schmidt methods and their stability properties

Block Gram-Schmidt algorithms comprise essential kernels in many scienti...
∙ 05/16/2022

Post-Modern GMRES

The GMRES algorithm of Saad and Schultz (1986) for nonsymmetric linear s...