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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hierarchical matrix ： ウィキペディア英語版 Hierarchical matrix In numerical mathematics, hierarchical matrices (H-matrices) 〔W. Hackbusch, ''A sparse matrix arithmetic based on H-matrices. Part I: Introduction to H-matrices'', Computing (1999), 62:89–108〕 〔M. Bebendorf, ''Hierarchical matrices: A means to efficiently solve elliptic boundary value problems'', Springer (2008)〕 〔W. Hackbusch, ''Hierarchische Matrizen. Algorithmen und Analysis'', Springer (2009)〕 are used as data-sparse approximations of non-sparse matrices. While a sparse matrix of dimension $n$ can be represented efficiently in $O(n)$ units of storage by storing only its non-zero entries, a non-sparse matrix would require $O(n^2)$ units of storage, and using this type of matrices for large problems would therefore be prohibitively expensive in terms of storage and computing time. Hierarchical matrices provide an approximation requiring only $O(n\; k\backslash ,\backslash log(n))$ units of storage, where $k$ is a parameter controlling the accuracy of the approximation. In typical applications, e.g., when discretizing integral equations 〔W. Hackbusch and B. N. Khoromskij, ''A sparse H-Matrix Arithmetic. Part II: Application to Multi-Dimensional Problems'', Computing (2000), 64:21–47〕 〔M. Bebendorf, ''Approximation of boundary element matrices'', Num. Math. (2000), 86:565--589〕 〔M. Bebendorf and S. Rjasanow, ''Adaptive low-rank approximation of collocation matrices'', Computing (2003), 70:1–24〕 〔S. Börm and L. Grasedyck, ''Hybrid cross approximation of integral operators'', Num. Math. (2005), 101:221–249〕 or solving elliptic partial differential equations 〔M. Bebendorf and W. Hackbusch, ''Existence of H-matrix approximants to the inverse FE-matrix of elliptic operators with $L^\backslash infty$-coefficients'', Num. Math. (2003), 95:1–28〕 〔S. Börm, ''Approximation of solution operators of elliptic partial differential equations by H- and H2-matrices'', Num. Math. (2010), 115:165–193〕 〔M. Faustmann, J. M. Melenk and D. Praetorius, ''H-matrix approximability of the inverses of FEM matrices'', to appear in Num. Math., preprint available at (arXiv.org )〕 a rank proportional to $\backslash log(1/\backslash epsilon)^\backslash gamma$ with a small constant $\backslash gamma$ is sufficient to ensure an accuracy of $\backslash epsilon$. Compared to many other data-sparse representations of non-sparse matrices, hierarchical matrices offer a major advantage: the results of matrix arithmetic operations like matrix multiplication, factorization or inversion can be approximated in $O(n\; k^\backslash alpha\backslash ,\backslash log(n)^\backslash beta)$ operations, where $\backslash alpha,\backslash beta\backslash in\backslash .$〔L. Grasedyck and W. Hackbusch, ''Construction and Arithmetics of H-Matrices'', Computing (2003), 70:295–334〕 == Basic idea == Hierarchical matrices rely on local low-rank approximations: let $I,J$ be index sets, and let $G\backslash in^$ denote the matrix we have to approximate. In many applications (see above), we can find subsets $t\backslash subseteq\; I,s\backslash subseteq\; J$ such that $G|\_$ can be approximated by a rank-$k$ matrix. This approximation can be represented in factorized form $G|\_\backslash approx\; A\; B^$ * with factors $A\backslash in^,B\backslash in^$. While the standard representation of the matrix $G|\_$ requires $O((\backslash \#t)(\backslash \#s))$ units of storage, the factorized representation requires only $O(k(\backslash \#t+\backslash \#s))$ units. If $k$ is not too large, the storage requirements are reduced significantly. In order to approximate the entire matrix $G$, it is split into a family of submatrices. Large submatrices are stored in factorized representation, while small submatrices are stored in standard representation in order to improve the efficiency. Low-rank matrices are closely related to degenerate expansions used in panel clustering and the fast multipole method to approximate integral operators. In this sense, hierarchical matrices can be considered the algebraic counterparts of these techniques. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hierarchical matrix」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.