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Efficient Numerical Methods for Non-local Operators
EMS Tracts in Mathematics Vol. 14

Steffen Börm (Kiel University, Germany)

Efficient Numerical Methods for Non-local Operators

2-Matrix Compression, Algorithms and Analysis

Corrected 2nd printing, September 2013

ISBN print 978-3-03719-091-3, ISBN online 978-3-03719-591-8
DOI 10.4171/091
December 2010, 441 pages, hardcover, 17 x 24 cm.
58.00 Euro

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 × n matrix in standard representation requires n2 units of storage, a hierarchical matrix can approximate the matrix in a compact representation requiring only O(nk 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. ℋ2-matrices offer a refinement of hierarchical matrices: using a multilevel representation of submatrices, the efficiency can be significantly improved, particularly for large problems.

This books gives an introduction to the basic concepts and presents a general framework that can be used to analyze the complexity and accuracy of ℋ2-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 only required in isolated sections, e.g., for certain classes of model problems.

Keywords: Hierarchical matrix, data-sparse approximation, boundary element method, preconditioner

Further Information

Review in Zentralblatt MATH 1208.65037

Review in MR 2767920