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Results in Mathematics

, Volume 69, Issue 1–2, pp 225–243 | Cite as

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

  • Dana Černá
  • Václav FiněkEmail author
Article

Abstract

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 

Mathematics Subject Classification

15A12 41A15 65F50 65N12 65T60 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Mathematics and Didactics of MathematicsTechnical University of LiberecLiberecCzech Republic

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