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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2404.19163 |
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| _version_ | 1866914777255641088 |
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| author | Habib, Adela Finkelstein, Joshua Niklasson, Anders M. N. |
| author_facet | Habib, Adela Finkelstein, Joshua Niklasson, Anders M. N. |
| contents | In recent years, a new kind of accelerated hardware has gained popularity in the Artificial Intelligence (AI) and Machine Learning (ML) communities which enables extremely high-performance tensor contractions in reduced precision for deep neural network calculations. In this article, we exploit Nvidia Tensor cores, a prototypical example of such AI/ML hardware, to develop a mixed precision approach for computing a dense matrix factorization of the inverse overlap matrix in electronic structure theory, $S^{-1}$. This factorization of $S^{-1}$, written as $ZZ^T=S^{-1}$, is used to transform the general matrix eigenvalue problem into a standard matrix eigenvalue problem. Here we present a mixed precision iterative refinement algorithm where $Z$ is given recursively using matrix-matrix multiplications and can be computed with high performance on Tensor cores. To understand the performance and accuracy of Tensor cores, comparisons are made to GPU-only implementations in single and double precision. Additionally, we propose a non-parametric stopping criteria which is robust in the face of lower precision floating point operations. The algorithm is particularly useful when we have a good initial guess to $Z$, for example, from previous time steps in quantum-mechanical molecular dynamics simulations or from a previous iteration in a geometry optimization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_19163 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Efficient Mixed-Precision Matrix Factorization of the Inverse Overlap Matrix in Electronic Structure Calculations with AI-Hardware and GPUs Habib, Adela Finkelstein, Joshua Niklasson, Anders M. N. Computational Physics Materials Science Mathematical Physics In recent years, a new kind of accelerated hardware has gained popularity in the Artificial Intelligence (AI) and Machine Learning (ML) communities which enables extremely high-performance tensor contractions in reduced precision for deep neural network calculations. In this article, we exploit Nvidia Tensor cores, a prototypical example of such AI/ML hardware, to develop a mixed precision approach for computing a dense matrix factorization of the inverse overlap matrix in electronic structure theory, $S^{-1}$. This factorization of $S^{-1}$, written as $ZZ^T=S^{-1}$, is used to transform the general matrix eigenvalue problem into a standard matrix eigenvalue problem. Here we present a mixed precision iterative refinement algorithm where $Z$ is given recursively using matrix-matrix multiplications and can be computed with high performance on Tensor cores. To understand the performance and accuracy of Tensor cores, comparisons are made to GPU-only implementations in single and double precision. Additionally, we propose a non-parametric stopping criteria which is robust in the face of lower precision floating point operations. The algorithm is particularly useful when we have a good initial guess to $Z$, for example, from previous time steps in quantum-mechanical molecular dynamics simulations or from a previous iteration in a geometry optimization. |
| title | Efficient Mixed-Precision Matrix Factorization of the Inverse Overlap Matrix in Electronic Structure Calculations with AI-Hardware and GPUs |
| topic | Computational Physics Materials Science Mathematical Physics |
| url | https://arxiv.org/abs/2404.19163 |