On a Sparse Representation of an n-Dimensional Laplacian in Wavelet Coordinates

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• 作者：Dana Černá ; Václav Finěk
• 关键词：15A12 ; 41A15 ; 65F50 ; 65N12 ; 65T60 ; Wavelet ; Riesz bases ; cubic Hermite spline ; homogeneous Dirichlet boundary conditions ; condition numbers ; sparse representations
• 刊名：Results in Mathematics
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：69
• 期：1-2
• 页码：225-243
• 全文大小：721 KB
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• 作者单位：Dana Černá (1)
  Václav Finěk (1)
  
  1. Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Studentská 2, 461 17, Liberec, Czech Republic
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Birkh盲user Basel
• ISSN：1420-9012

文摘

Important parts of adaptive wavelet methods are well-conditioned wavelet stiffness matrices and an efficient approximate multiplication of quasi-sparse stiffness matrices with vectors in wavelet coordinates. Therefore it is useful to develop a well-conditioned wavelet basis with respect to which both the mass and stiffness matrices are sparse in the sense that the number of nonzero elements in each column is bounded by a constant. Consequently, the stiffness matrix corresponding to the n-dimensional Laplacian in the tensor product wavelet basis is also sparse. Then a matrix–vector multiplication can be performed exactly with linear complexity. In this paper, we construct a wavelet basis based on Hermite cubic splines with respect to which both the mass matrix and the stiffness matrix corresponding to a one-dimensional Poisson equation are sparse. Moreover, a proposed basis is well-conditioned on low decomposition levels. Small condition numbers for low decomposition levels and a sparse structure of stiffness matrices are kept for any well-conditioned second order partial differential equations with constant coefficients; furthermore, they are independent of the space dimension. Keywords Wavelet Riesz bases cubic Hermite spline homogeneous Dirichlet boundary conditions condition numbers sparse representations