pint.bib updates

_bibliography/pint.bib

@@ -7839,6 +7839,15 @@ @article{SunEtAl2025
 	year = {2025},
 }
 
+@unpublished{TabeartEtAl2025,
+	abstract = {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 $\alpha$-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 $\alpha$-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 $\alpha$-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 $\alpha$ our new approaches are robust and efficient in terms of outer iterations and matrix--vector products.},
+	author = {Jemima M. Tabeart and Selime Gürol and John W. Pearson and Anthony T. Weaver},
+	howpublished = {arXiv:2506.03947v2 [math.NA]},
+	title = {Block Alpha-Circulant Preconditioners for All-at-Once Diffusion-Based Covariance Operators},
+	url = {http://arxiv.org/abs/2506.03947v2},
+	year = {2025},
+}
+
 @article{TangEtAl2025,
 	author = {Tang, Changyang and Wu, Shu-Lin and Zhou, Tao and Zhou, Yuancheng},
 	doi = {10.1007/s10915-025-02899-w},