Applications of the Generalized Fourier Transform in Numerical Linear Algebra |
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Authors: | Email author" target="_blank">Krister??hlanderEmail author Hans?Munthe-Kaas |
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Institution: | (1) Department of Information Technology, Uppsala University, Box 337, SE-751 05 Uppsala, Sweden;(2) Department of Mathematics, University of Bergen, Joh. Brunsgate 12, N-5008 Bergen, Norway |
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Abstract: | Equivariant matrices, commuting with a group of permutation matrices, are considered. Such matrices typically arise from PDEs
and other computational problems where the computational domain exhibits discrete geometrical symmetries. In these cases,
group representation theory provides a powerful tool for block diagonalizing the matrix via the Generalized Fourier Transform
(GFT). This technique yields substantial computational savings in problems such as solving linear systems, computing eigenvalues
and computing analytic matrix functions such as the matrix exponential.
The paper is presenting a comprehensive self contained introduction to this field. Building upon the familiar special (finite
commutative) case of circulant matrices being diagonalized with the Discrete Fourier Transform, we generalize the classical
convolution theorem and diagonalization results to the noncommutative case of block diagonalizing equivariant matrices.
Applications of the GFT in problems with domain symmetries have been developed by several authors in a series of papers. In
this paper we elaborate upon the results in these papers by emphasizing the connection between equivariant matrices, block
group algebras and noncommutative convolutions. Furthermore, we describe the algebraic structure of projections related to
non-free group actions. This approach highlights the role of the underlying mathematical structures, and provides insight
useful both for software construction and numerical analysis. The theory is illustrated with a selection of numerical examples.
AMS subject classification (2000) 43A30, 65T99, 20B25 |
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Keywords: | non commutative Fourier analysis equivariant operators block diagonalization |
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