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Introduction to Hierarchical Matrices with Applications
Steffen Börm, Lars Grasedyck and Wolfgang HackbuschAbstract
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 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 H-matrices, especially ofH2-matrices, and finally consider applications of the different methods to problems from integral equations, partial differential equations and control theory.
- Received:
- 20.02.02
- Published:
- 20.02.02
- MSC Codes:
- 65F05, 65F30, 65F50, 65N50
- Keywords:
- hierarchical matrices, formatted matrix operations, lyapunov equation, riccati equation, fast solvers