Börm S. Efficient Numerical Methods for Non-local Operators: H2 Matrix Compression, Algorithms and Analysis

European Mathematical Society, 2010. — 442 p. — (EMS Tracts in Mathematics 14). — ISBN: 3037190914.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 Hn\times n$ matrix in standard representation requires Hn2 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 Hk 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.Model problem
Hierarchical matrices
Application to integral operators
Orthogonal cluster bases and matrix projections
Compression
A priori matrix arithmetic
A posteriori matrix arithmetic
Application to elliptic partial differential operators
Applications