QR factorization of ill-conditioned tall-and-skinny matrices on distributed-memory systems
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913343386681344 |
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| author | Mijić, Nenad Kaushik, Abhiram Davidović, Davor |
| author_facet | Mijić, Nenad Kaushik, Abhiram Davidović, Davor |
| contents | In this paper we present a novel algorithm developed for computing the QR factorisation of extremely ill-conditioned tall-and-skinny matrices on distributed memory systems. The algorithm is based on the communication-avoiding CholeskyQR2 algorithm and its block Gram-Schmidt variant. The latter improves the numerical stability of the CholeskyQR2 algorithm and significantly reduces the loss of orthogonality even for matrices with condition numbers up to $10^{15}$. Currently, there is no distributed GPU version of this algorithm available in the literature which prevents the application of this method to very large matrices. In our work we provide a distributed implementation of this algorithm and also introduce a modified version that improves the performance, especially in the case of extremely ill-conditioned matrices. The main innovation of our approach lies in the interleaving of the CholeskyQR steps with the Gram-Schmidt orthogonalisation, which ensures that update steps are performed with fully orthogonalised panels. The obtained orthogonality and numerical stability of our modified algorithm is equivalent to CholeskyQR2 with Gram-Schmidt and other state-of-the-art methods. Weak scaling tests performed with our test matrices show significant performance improvements. In particular, our algorithm outperforms state-of-the-art Householder-based QR factorisation algorithms available in ScaLAPACK by a factor of $6$ on CPU-only systems and up to $80\times$ on GPU-based systems with distributed memory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04237 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | QR factorization of ill-conditioned tall-and-skinny matrices on distributed-memory systems Mijić, Nenad Kaushik, Abhiram Davidović, Davor Distributed, Parallel, and Cluster Computing Data Structures and Algorithms Performance In this paper we present a novel algorithm developed for computing the QR factorisation of extremely ill-conditioned tall-and-skinny matrices on distributed memory systems. The algorithm is based on the communication-avoiding CholeskyQR2 algorithm and its block Gram-Schmidt variant. The latter improves the numerical stability of the CholeskyQR2 algorithm and significantly reduces the loss of orthogonality even for matrices with condition numbers up to $10^{15}$. Currently, there is no distributed GPU version of this algorithm available in the literature which prevents the application of this method to very large matrices. In our work we provide a distributed implementation of this algorithm and also introduce a modified version that improves the performance, especially in the case of extremely ill-conditioned matrices. The main innovation of our approach lies in the interleaving of the CholeskyQR steps with the Gram-Schmidt orthogonalisation, which ensures that update steps are performed with fully orthogonalised panels. The obtained orthogonality and numerical stability of our modified algorithm is equivalent to CholeskyQR2 with Gram-Schmidt and other state-of-the-art methods. Weak scaling tests performed with our test matrices show significant performance improvements. In particular, our algorithm outperforms state-of-the-art Householder-based QR factorisation algorithms available in ScaLAPACK by a factor of $6$ on CPU-only systems and up to $80\times$ on GPU-based systems with distributed memory. |
| title | QR factorization of ill-conditioned tall-and-skinny matrices on distributed-memory systems |
| topic | Distributed, Parallel, and Cluster Computing Data Structures and Algorithms Performance |
| url | https://arxiv.org/abs/2405.04237 |