Efficient Numerical Methods for Non-local Operators

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.
$H^2$-matrices
offer a refinement of hierarchical matrices: using a
multilevel representation of submatrices, the efficiency can be
significantly improved, particularly for large problems.

This books gives an introduction to the basic concepts and presents a
general framework that can be used to analyze the complexity and
accuracy of $H^2$-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 only required
in isolated sections, e.g., for certain classes of model problems.