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Introduction to hierarchical matrices with applications. (English) Zbl 1035.65042
Summary: We give a short introduction to methods for the data-sparse approximation of matrices resulting from the discretisation of non-local operators occurring in boundary integral methods, as the inverses of partial differential operators or as solutions of control problems.
The result of the approximation will be so-called hierarchical matrices (or short $$\mathcal H$$-matrices). These matrices form a subset of the set of all matrices and have a data-sparse representation. The essential operations for these matrices (matrix-vector and matrix-matrix multiplication, addition and inversion) can be performed in, up to logarithmic factors, optimal complexity.
We give a review of specialised variants of $$\mathcal H$$-matrices, especially of $$\mathcal H^2$$-matrices, and finally consider applications of the different methods to problems from integral equations, partial differential equations and control theory.

##### MSC:
 65F30 Other matrix algorithms (MSC2010) 15A24 Matrix equations and identities 65F50 Computational methods for sparse matrices 65N38 Boundary element methods for boundary value problems involving PDEs 65K10 Numerical optimization and variational techniques
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