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美国普渡大学 夏建林教授:Fast Randomized and Matrix-Free Direct Solvers

([西财新闻] 发布于 :2018-05-17 )

光华讲坛——社会名流与企业家论坛第4955期

 

主题Fast Randomized and Matrix-Free Direct Solvers

主讲人美国普渡大学 夏建林教授

主持人经济数学学院 顾先明

时间2018年5月18日(星期五)上午10:30

地点西南财经大学柳林校区通博楼B412

主办单位:经济数学学院 科研处

 

主讲人简介:

夏建林Jianlin Xia,普渡大学Purdue University数学系教授,博士毕业于美国加州大学伯克利分校。主要研究兴趣包括数值线性代数、科学计算等,自2017年起担任国际SCI期刊Applied Numerical Mathematics的编委,特别是近年来在结构类矩阵(如HSS-和MSSS-类矩阵)的高性能算法研究中取得很多重要成果,曾获得美国国家科学基金会颁发的职业生涯奖(计算数学方向,2013-2018)。截至目前,已在SIAM Journal of Scientific ComputingSIAM Journal of Matrix Analysis and ApplicationsMathematics of Computation等国际一流学术期刊上发表论文55篇。

内容提要:

In this talk, we will discuss how randomized techniques can be used in structured matrix computations, especially in matrix approximation and in turn in solving large dense and sparse linear systems. It is known that randomized sampling can help compute approximate SVDs via matrix-vector products. Such randomized ideas have been applied to some structured matrices for the fast compression of off-diagonal blocks. This leads to randomized and even matrix-free direct solvers for large dense linear systems.

 Furthermore, the techniques can be extended to sparse direct solvers, where randomization helps compress dense fill-in in the factorization into skinny matrix-vector products. This has a significant advantage over dense or structured fill-in used before, since the processing and propagation of the skinny products are much simpler. For some sparse discretized problems (often elliptic), the randomized sparse direct solvers can reach nearly O(n) complexity.

 We also show how to control the approximation accuracy in randomized structured solution, and further prove the superior backward stability of these randomized methods. Applications to spectral methods and Jacobi transformations will also be mentioned. Part of the work is joint with Yuanzhe Xi.


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