PinT via Diagonalization Technique

报告人：吴树林教授

单位：东北师范大学

报告时间：2020.1.7下午2：40-5：40

报告地点：郑州大学数学与统计学院金融实验室

The diagonalization technique [1] provides new possibility for efficient parallel-in-time (PinT) computation of time-dependent PDEs. The original version of this technique is however only suitable for short time computation. In this talk, we first introduce a novel modification of this technique to handle both the parabolic and wave problems for long time computation [2]. The main idea for such a modification lies in the FFT-based factorization of an $\alpha$-circulant matrix, which is specified by the method of time discretization. Then, we show two applications of this technique. One is about reducing the cost of the coarse grid-correction of the parareal and MGRIT algorithms [3, 5, 6] and the other application is about how to solve the (optimal control of) wave propagation problems by the Krylov subspace solver [4, 7]. We will explain the details about the roundoff error arising from the diagonalization technique and how to reduce the communication cost for multi-cores parallel computer via the Cooley-Tukey radix-2 factorization of discrete FFT matrix.

References

[1] Y. Maday, E.M. Rønquist. Parallelization in time through tensor-product space-time solvers. C.R. Math., Vol. 346, pp. 113-118, 2008.

[2] M. J. Gander, S.-L. Wu*. Convergence analysis of a periodic-Like waveform relaxation method

for initial-value problems via the diagonalization technique. Numer. Math., Vol. 143, pp. 489-527,

2019.

[3] M. G. Gander, S.-L. Wu*. A diagonalization-based parareal algorithm for dissipative and wave

propagation problems. SIAM J. Numer. Anal., in revision.

[4] J. Liu, S.-L. Wu*. A block α-circulant preconditioner for all-at-once systems from wave equations.SIAM J. Matrix Anal. Appl., in revision.

[5] S.-L. Wu*. Toward parallel coarse grid correction for the parareal algorithm. SIAM J. Sci. Com put., Vol. 40, pp. A1446-A1472, 2018.

[6] S.-L. Wu*, T. Zhou. Acceleration of the two-level MGRIT algorithm via the diagonalization

technique. SIAM J. Sci. Comput., Vol. 41, pp. A3421-A3448, 2019.

[7] S.-L. Wu, T. Zhou*. Diagonalization-based parallel-in-time algorithms for parabolic PDE constrained optimization problems. ESAIM: COCV, in revision.

作者简介：

吴树林，1984年10月生于河南省固始县，2010年5月毕业于华中科技大学，获计算数学专业博士学位，现为东北师范大学教授、博士研究生导师。