大型線性方程組解法

20190715第十五期

作者:許進超

本文預覽

      • 作者簡介
        許進超是美國賓州州立大學數學系的威拉曼數學講座教授(Verne M. Willaman Professor of Mathematics),同時也是該校的計算數學與應用中心主任。他是數值偏微分方程快速算法的專家,在開發、設計和分析解決大型方程組快速算法有許多開創性的研究。
      • 重點摘要
        ◊ 如何找線性代數方程組的解?這在數學與應用中是一個非常基本的問題,作者在本文討論求解的數值方法。
        ◊ 從最古老的高斯消去法到現代的多層網格法,作者在文中演示如何以數學技術研發大規模線性方程組的有效方法。
        ◊ 作者也有對於電腦硬體的開發與數學演算法的開發之間作出比較與探討。
      • 本文出處
        本文是許進超教授在 ICCM 思廉講座的講稿。本刊感謝許教授同意刊登。
      • 參考資料
        [1] Douglas N Arnold, Richard S Falk, and Ragnar Winther. Finite element exterior calculus, homological techniques, and applications. Acta numerica, 15:1–155, 2006.
        [2] James H Bramble, Joseph E Pasciak, and Jinchao Xu. Parallel multilevel preconditioners. Mathematics of Computation, 55(191):1–22, 1990.
        [3] Michael Griebel. Multilevel algorithms considered as iterative methods on semidefinite systems. SIAM Journal on Scientific Computing, 15(3):547–565, 1994.
        [4] Ralf Hiptmair. Finite elements in computational electromagnetism. Acta Numerica, 11:237–339, 2002.
        [5] Ralf Hiptmair and Jinchao Xu. Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. SIAM Journal on Numerical Analysis, 45(6):2483–2509, 2007.
        [6] Kaibo Hu, Yicong Ma, and Jinchao Xu. Stable finite element methods preserving \nabla\cdot\varvec {B}= 0·b= 0 exactly for mhd models. Numerische Mathematik, 135(2):371–396, 2017.
        [7] Kaibo Hu and Jinchao Xu. Structure-preserving finite element methods for stationary mhd models. arXiv preprint arXiv:1503.06160, 2015.
        [8] Xiaozhe Hu, Jinchao Xu, and Chen-Song Zhang. Application of auxiliary space preconditioning in field-scale reservoir simulations. Science in China Series A: Mathematics, 56(12):2737–2751, 2013.
        [9] Yicong Ma, Kaibo Hu, Xiaozhe Hu, and Jinchao Xu. Robust preconditioners for incompressible mhd models. Journal of Computational Physics, 316:721–746, 2016.
        [10] SV Nepomnyaschikh. Decomposition and fictitious domains methods for elliptic boundary value problems. In Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations, pages 62–72. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1992.
        [11] Changhe Qiao, Shuhong Wu, Jinchao Xu, and Chen-Song Zhang. Analytical decoupling techniques for fully implicit reservoir simulation. Journal of Computational Physics, 336:664–681, 2017.
        [12] Jinchao Xu. Theory of multilevel methods, volume 8924558. Cornell University Ithaca, NY, 1989.
        [13] Jinchao Xu. Iterative methods by space decomposition and subspace correction. SIAM review, 34(4):581–613, 1992.
        [14] Jinchao Xu. The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing, 56(3):215–235, 1996.
        [15] Jinchao Xu. Multilevel Interative Methods. Preprint, 2018.
        [16] Jinchao Xu and Kai Yang. Well-posedness and robust preconditioners for discretized fluid-structure interaction systems. Computer Methods in Applied Mechanics and Engineering, 292:69–91, 2015.
        [17] D. Young. Iterative methods for solving partial difference equations of elliptic type. PhD thesis, Department of Mathematics, Harvard University, 1950.

     

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