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

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.

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Correspondence to Václav Finěk.

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The authors have been supported by the SGS project “Numerical Methods II”.

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Černá, D., Finěk, V. On a Sparse Representation of an n-Dimensional Laplacian in Wavelet Coordinates. Results. Math. 69, 225–243 (2016). https://doi.org/10.1007/s00025-015-0488-5

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Mathematics Subject Classification

  • 15A12
  • 41A15
  • 65F50
  • 65N12
  • 65T60

Keywords

  • Wavelet
  • Riesz bases
  • cubic Hermite spline
  • homogeneous Dirichlet boundary conditions
  • condition numbers
  • sparse representations