A mixed precision LOBPCG algorithm
Fuente:
arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866929334190604288 |
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| author | Kressner, Daniel Ma, Yuxin Shao, Meiyue |
| author_facet | Kressner, Daniel Ma, Yuxin Shao, Meiyue |
| contents | 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 reduced 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_12528 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A mixed precision LOBPCG algorithm Kressner, Daniel Ma, Yuxin Shao, Meiyue Numerical Analysis 65F15, 65F50 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 reduced 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. |
| title | A mixed precision LOBPCG algorithm |
| topic | Numerical Analysis 65F15, 65F50 |
| url | https://arxiv.org/abs/2302.12528 |