# Efficient Numerical Methods for Non-local Operators: \(\mathcal{H}^{2}\)-Matrix Compression, Algorithms and Analysis

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*Steffen Börm*

A publication of the European Mathematical Society

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.

#### Readership

Graduate students and research mathematicians interested in applications.