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### SWUFE数学讲坛135：基于分块alpha循环预处理的MINRES算法求解发展型偏微分方程

Prior to the joining Hong Kong Baptist University as an Assistant Professor at Department of Mathematics in July of 2020, Sean Hon was a Research Fellow at National University of Singapore. He obtained his DPhil in mathematics from University of Oxford in 2018, supervised by Prof Andrew Wathen. In 2012 and 2014, he respectively finished his bachelor's degree (first class honour) and MPhil in mathematics from Hong Kong University of Science and Technology. Sean was awarded an Early Career Award by Hong Kong University Grants Committee and the University of Oxford Croucher Scholarship from the Croucher Foundation of Hong Kong in 2021 and 2014, respectively. (韩汝星博士现任香港浸会大学数学系助理教授，分别在20122014获得香港科技大学学士和硕士学位，2014年获得香港裘槎基金会奖学金，2018年获得牛津大学哲学博士学位（数学），2020年获得香港大学教育资助委员会青年科技奖)

In this work, we propose a novel parallel-in-time preconditioner for an all-at-once system arising from the numerical solution of evolutionary partial differential equations (PDEs). Namely, considering the wave equation as a model problem, our main result concerns a block $\alpha$-circulant matrix based preconditioner that can be fast diagonalized via fast Fourier transforms, whose effectiveness is theoretically supported for the modified block Toeplitz system arising from discretizing the concerned wave equation. Namely, after first transforming the original all-at-once linear system into a symmetric one, we develop the desired preconditioner based on the spectral information of the modified matrix. Then, we show that the eigenvalues of the preconditioned matrix are clustered around $1$ without any extreme outlier far away from the clusters. In other words, mesh-independent convergence is theoretically guaranteed when the minimal residual method is employed. Moreover, our proposed solver is further generalized to a full block triangular Toeplitz system which arises when a high order discretization scheme is used. Numerical experiments are given to support the effectiveness of our preconditioner, showing that the expected optimal convergence can be achieved. This is joint work with Xuelei Lin (Harbin Institute of Technology).