Block Alpha-Circulant Preconditioners for All-at-Once Diffusion-Based Covariance Operators
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| Format: | Preprint |
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2025
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| _version_ | 1866913099867488256 |
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| author | Tabeart, Jemima M. Gürol, Selime Pearson, John W. Weaver, Anthony T. |
| author_facet | Tabeart, Jemima M. Gürol, Selime Pearson, John W. Weaver, Anthony T. |
| contents | Covariance matrices are central to data assimilation and inverse methods derived from statistical estimation theory. Previous work has considered the application of an all-at-once diffusion-based representation of a covariance matrix operator in order to exploit inherent parallelism in the underlying problem. In this paper, we provide practical methods to apply block $α$-circulant preconditioners to the all-at-once system for the case where the main diffusion operation matrix cannot be readily diagonalized using a discrete Fourier transform. Our new framework applies the block $α$-circulant preconditioner approximately by solving an inner block diagonal problem via a choice of inner iterative approaches. Our first method applies Chebyshev semi-iteration to a symmetric positive definite matrix, shifted by a complex scaling of the identity. We extend theoretical results for Chebyshev semi-iteration in the symmetric positive definite setting, to obtain computable bounds on the asymptotic convergence factor for each of the complex sub-problems. The second approach transforms the complex sub-problem into a (generalized) saddle point system with real coefficients. Numerical experiments reveal that in the case of unlimited computational resources, both methods can match the iteration counts of the `best-case' block $α$-circulant preconditioner. We also provide a practical adaptation to the nested Chebyshev approach, which improves performance in the case of a limited computational budget. Using an appropriate choice of $α$ our new approaches are robust and efficient in terms of outer iterations and matrix--vector products. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_03947 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Block Alpha-Circulant Preconditioners for All-at-Once Diffusion-Based Covariance Operators Tabeart, Jemima M. Gürol, Selime Pearson, John W. Weaver, Anthony T. Numerical Analysis Covariance matrices are central to data assimilation and inverse methods derived from statistical estimation theory. Previous work has considered the application of an all-at-once diffusion-based representation of a covariance matrix operator in order to exploit inherent parallelism in the underlying problem. In this paper, we provide practical methods to apply block $α$-circulant preconditioners to the all-at-once system for the case where the main diffusion operation matrix cannot be readily diagonalized using a discrete Fourier transform. Our new framework applies the block $α$-circulant preconditioner approximately by solving an inner block diagonal problem via a choice of inner iterative approaches. Our first method applies Chebyshev semi-iteration to a symmetric positive definite matrix, shifted by a complex scaling of the identity. We extend theoretical results for Chebyshev semi-iteration in the symmetric positive definite setting, to obtain computable bounds on the asymptotic convergence factor for each of the complex sub-problems. The second approach transforms the complex sub-problem into a (generalized) saddle point system with real coefficients. Numerical experiments reveal that in the case of unlimited computational resources, both methods can match the iteration counts of the `best-case' block $α$-circulant preconditioner. We also provide a practical adaptation to the nested Chebyshev approach, which improves performance in the case of a limited computational budget. Using an appropriate choice of $α$ our new approaches are robust and efficient in terms of outer iterations and matrix--vector products. |
| title | Block Alpha-Circulant Preconditioners for All-at-Once Diffusion-Based Covariance Operators |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2506.03947 |