H²-matrix preconditioners for integral and elliptic partial differential equations

积分和椭圆偏微分方程的 H² 矩阵预条件子

基本信息

批准号：
229645647
负责人：
Professor Dr. Steffen Börm
金额：
--
依托单位：
Mathematisches Seminar
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2012
资助国家：
德国
起止时间：
2011-12-31 至 2014-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/229645647?language=en
关键词：
matrix preconditioners integral elliptic partial

项目摘要

The recently developed H-matrix method can be used to construct efficient solvers for elliptic partial differential equations and integral equations. This approach is particularly robust when considering differential equations with discontinuous or anisotropic coefficients, e.g., appearing in simulations of composite materials. An H-matrix usually requires O(n k log n) units of storage, where n is the matrix dimension and k depends on the condition number of the linear system and the desired accuracy of the approximation.H²-matrices combine the H-matrix technique with multilevel structures in order to reduce the storage requirements to O(n k) and handle significantly larger linear systems. Numerical experiments show that H²-matrices indeed require less storage than their H-matrix counterparts, but the setup time of currently existing algorithms compares unfavorably to those for H-matrices.This research project focuses on developing a new, more efficient method for the construction of H²-matrix solvers. The approach is based on a new algorithm that performs local low-rank updates for H²-matrices in linear, i.e., optimal, complexity. This flexible new algorithm can be used to construct preconditioners for differential and integral equations more efficiently and it is also expected to pave the way for the development of efficient techniques for other arithmetic operations in the set of H²-matrices.

最近开发的H-Matrix方法可用于为椭圆形偏微分方程和积分方程构建有效的求解器。当考虑不连续或各向异性系数的微分方程时，例如出现在复合材料的模拟中时，这种方法尤其强大。 H-Matrix通常需要O（n K log n）存储单元，其中n是矩阵维度，K取决于线性系统的条件数和近似值的所需准确性。H²-Matrices将H-矩阵技术与多层结构结合在一起，以减少对O（N K）的存储需求，以减少O（N K）和较大较大的系统。数值实验表明，H²Matrices确实需要比H-Matrices对应物更少的存储空间，但是当前现有算法的设置时间与H-矩阵的设置时间不利。此研究项目着重于开发一种新的，更有效的H²-Matrix求解器的方法。该方法基于一种新算法，该算法在线性（即最佳，复杂性）中执行h²-matrix的局部低级更新。这种灵活的新算法可用于更有效地为差异方程式构建预处理，也有望为在H²-Matrices集合中为其他算术操作开发有效技术的有效技术铺平道路。