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TabeartEtAl2025
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<span id="TabeartEtAl2025">J. M. Tabeart, S. Gürol, J. W. Pearson, and A. T. Weaver, “Block Alpha-Circulant Preconditioners for All-at-Once Diffusion-Based Covariance Operators,” arXiv:2506.03947v2 [math.NA], 2025 [Online]. Available at: <a href="http://arxiv.org/abs/2506.03947v2" target="_blank">http://arxiv.org/abs/2506.03947v2</a></span>
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BibTeX entry <code>TabeartEtAl2025</code>
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<pre class="abstract">@unpublished{TabeartEtAl2025,
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author = {Tabeart, Jemima M. and Gürol, Selime and Pearson, John W. and Weaver, Anthony T.},
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howpublished = {arXiv:2506.03947v2 [math.NA]},
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title = {Block Alpha-Circulant Preconditioners for All-at-Once Diffusion-Based Covariance Operators},
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url = {http://arxiv.org/abs/2506.03947v2},
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year = {2025}
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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 parallellism 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 αour new approaches are robust and efficient in terms of outer iterations and matrix–vector products.
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