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Hierarchical Matrix Approximation for Kernel-Based Scattered Data Interpolation
Publikationstyp
Journal Article
Date Issued
2017-10-03
Sprache
English
Author(s)
Institut
TORE-URI
Volume
39
Issue
5
Start Page
A2287
End Page
A2316
Citation
SIAM Journal on Scientific Computing 39 (5): A2287-A2316 (2017)
Publisher DOI
Scattered data interpolation by radial kernel functions leads to linear equation systems with large, fully populated, ill-conditioned interpolation matrices. A successful iterative solution of such a system requires an efficient matrix-vector multiplication as well as an efficient preconditioner. While multipole approaches provide a fast matrix-vector multiplication, they avoid the explicit setup of the system matrix which hinders the construction of preconditioners, such as approximate inverses or factorizations which typically require the explicit system matrix for their construction. In this paper, we propose an approach that allows both an efficient matrix-vector multiplication as well as an explicit matrix representation which can then be used to construct a preconditioner. In particular, the interpolation matrix will be represented in hierarchical matrix format, and several approaches for the blockwise low-rank approximation are proposed and compared, of both analytical nature (separable expansions) and algebraic nature (adaptive cross approximation). The validity of using an approximate system matrix in the iterative solution of the interpolation equations is demonstrated through a range of numerical experiments.
DDC Class
510: Mathematik