Efficient arithmetic operations for rank-structured matrices based on hierarchical low-rank updates
Efficient arithmetic operations for rank-structured matrices based on hierarchical low-rank updates
复制标题
基于分层低秩更新的秩结构矩阵的高效算术运算
DOI：
10.1007/s00791-015-0233-3
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发表时间：
2015
影响因子：
--
通讯作者：
K. Reimer
中科院分区：
文献类型：
--
作者：
S. Börm;K. Reimer
在偏微分方程和积分方程的数值方法中，许多矩阵都是秩结构的，即，它们包含可以由低秩矩阵近似的子矩阵。一个相对一般的秩结构矩阵是-矩阵：它们可以达到最优的复杂度，但对于大量的实际应用来说仍然足够一般。我们考虑用矩阵进行代数运算的算法，即，用于近似矩阵乘积、逆或几乎线性复杂度的因子分解。新方法是基于局部低秩更新，可以在线性复杂度。这些更新可以与递归过程相结合来近似两个矩阵的乘积，并且这些乘积可以用于近似矩阵逆和LR或Cholesky分解。数值实验表明，新算法产生的预条件子需要存储单元，可在运算中求值，并需要运算来建立。
Many matrices appearing in numerical methods for partial differential equations and integral equations arerank-structured, i.e., they contain submatrices that can be approximated by matrices of low rank. A relatively general class of rank-structured matrices are-matrices: they can reach the optimal order of complexity, but are still general enough for a large number of practical applications. We consider algorithms for performing algebraic operations with-matrices, i.e., for approximating the matrix product, inverse or factorizations in almost linear complexity. The new approach is based on local low-rank updates that can be performed in linear complexity. These updates can be combined with a recursive procedure to approximate the product of two-matrices, and these products can be used to approximate the matrix inverse and the LR or Cholesky factorization. Numerical experiments indicate that the new algorithm leads to preconditioners that requireunits of storage, can be evaluated inoperations, and takeoperations to set up.
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影响因子：
3.7
作者：
Lintner, M
通讯作者：
Lintner, M
基于域分解 $${mathcal H}$$ -LU 预处理
影响因子：
2.1
作者：
L. Grasedyck;Ronald Kriemann;S. Borne
通讯作者：
S. Borne
影响因子：
2.1
作者：
H. Bungartz;R. Hoppe;C. Zenger
通讯作者：
C. Zenger
影响因子：
2.1
作者：
Boerm, Steffen
通讯作者：
Boerm, Steffen