Efficient Numerical Methods for Non-local Operators: $mathcal{h}^2$-matrix Compression, Algorithms and Analysis (Ems Tracts in Mathematics) - Hardcover

Steffen Borm

Hardcover

ISBN 10: 3037190914 ISBN 13: 9783037190913

Publisher: European Mathematical Society, 2010

Synopsis

Hierarchical matrices present an efficient way of treating dense matrices that arise in the context of integral equations, elliptic partial differential equations, and control theory. While a dense $n\times n$ matrix in standard representation requires $n^2$ units of storage, a hierarchical matrix can approximate the matrix in a compact representation requiring only $O(n k \log n)$ units of storage, where $k$ is a parameter controlling the accuracy. Hierarchical matrices have been successfully applied to approximate matrices arising in the context of boundary integral methods, to construct preconditioners for partial differential equations, to evaluate matrix functions, and to solve matrix equations used in control theory. $\mathcal{H}^2$-matrices offer a refinement of hierarchical matrices: Using a multilevel representation of submatrices, the efficiency can be significantly improved, particularly for large problems. This book gives an introduction to the basic concepts and presents a general framework that can be used to analyze the complexity and accuracy of $\mathcal{H}^2$-matrix techniques. Starting from basic ideas of numerical linear algebra and numerical analysis, the theory is developed in a straightforward and systematic way, accessible to advanced students and researchers in numerical mathematics and scientific computing. Special techniques are required only in isolated sections, e.g., for certain classes of model problems. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Edition details

• Publisher: European Mathematical Society
• Publication date: 2010
• ISBN 10: 3037190914
• ISBN 13: 9783037190913
• Binding: Hardcover
• Language: English
• Number of pages: 441