A mixed precision LOBPCG algorithm

The locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm is a popular approach for computing a few smallest eigenvalues and the corresponding eigenvectors of a large Hermitian positive definite matrix A . In this work, we propose a mixed precision variant of LOBPCG that uses a...

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Bibliographic Details
Published in:Numerical algorithms Vol. 94; no. 4; pp. 1653 - 1671
Main Authors: Kressner, Daniel, Ma, Yuxin, Shao, Meiyue
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2023
Springer Nature B.V
Subjects:
Algebra
Algorithms
Approximation
Cholesky factorization
Computer Science
Convergence
Eigenvalues
Eigenvectors
Error analysis
Impact analysis
Mathematical analysis
Matrices (mathematics)
Numeric Computing
Numerical Analysis
Numerical prediction
Original Paper
Theory of Computation
Symmetric eigenvalue problem
65F50
65F15
LOBPCG algorithm
65F08
Mixed precision algorithm
ISSN:1017-1398, 1572-9265
Online Access:Get full text
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Summary:The locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm is a popular approach for computing a few smallest eigenvalues and the corresponding eigenvectors of a large Hermitian positive definite matrix A . In this work, we propose a mixed precision variant of LOBPCG that uses a (sparse) Cholesky factorization of A computed in lower precision as the preconditioner. To further enhance performance, a mixed precision orthogonalization strategy is proposed. To analyze the impact of reducing precision in the preconditioner on performance, we carry out a rounding error and convergence analysis of PINVIT, a simplified variant of LOBPCG. Our theoretical results predict and our numerical experiments confirm that the impact on convergence remains marginal. In practice, our mixed precision LOBPCG algorithm typically reduces the computation time by a factor of 1.4 – 2.0 on both CPUs and GPUs.
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ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-023-01550-9