Construction of optimally conditioned cubic spline wavelets on the interval
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Construction of optimally conditioned cubic spline wavelets on the interval ˇ ˇ Dana Cerná · Václav Finek
Received: 30 April 2009 / Accepted: 23 February 2010 / Published online: 8 June 2010 © Springer Science+Business Media, LLC 2010
Abstract The paper is concerned with a construction of new spline-wavelet bases on the interval. The resulting bases generate multiresolution analyses on the unit interval with the desired number of vanishing wavelet moments for primal and dual wavelets. Both primal and dual wavelets have compact support. Inner wavelets are translated and dilated versions of well-known wavelets designed by Cohen, Daubechies, and Feauveau. Our objective is to construct interval spline-wavelet bases with the condition number which is close to the condition number of the spline wavelet bases on the real line, especially in the case of the cubic spline wavelets. We show that the constructed set of functions is indeed a Riesz basis for the space L2 ([0, 1]) and for the Sobolev space H s ([0, 1]) for a certain range of s. Then we adapt the primal bases to the homogeneous Dirichlet boundary conditions of the first order and the dual bases to the complementary boundary conditions. Quantitative properties of the constructed bases are presented. Finally, we compare the efficiency of an adaptive wavelet scheme for several spline-wavelet bases and we show a superiority of our construction. Numerical examples are presented for the one-dimensional and two-dimensional Poisson equations where the solution has steep gradients. Keywords Biorthogonal wavelets · Interval · Spline · Condition number Mathematics Subject Classifications (2010) 65T60 · 65N99
Communicated by Rong-Qing Jia. ˇ ˇ D. Cerná (B) · V. Finek Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Studentská 2, 461 17 Liberec, Czech Republic e-mail: [email protected]
ˇ ˇ D. Cerná, V. Finek
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1 Introduction Wavelets are by now a widely accepted tool in signal and image processing as well as in numerical simulation. In the field of numerical analysis, methods based on wavelets are successfully used especially for preconditioning of large systems arising from discretization of elliptic partial differential equations, sparse representations of some types of operators and adaptive solving of operator equations. The quantitative performance of such methods strongly depends on a choice of a wavelet basis, in particular on its condition number. Wavelet bases on a bounded domain are usually constructed in the following way: Wavelets on the real line are adapted to the interval and then by a tensor product technique to the n-dimensional cube. Finally by splitting the domain into subdomains which are images of (0, 1)n under appropriate parametric mappings one can obtain wavelet bases on a fairly general domain. Thus, the properties of the employed wavelet basis on the interval are crucial for the properties of the resulting bases on general a domain. Biorthogonal spline-wavelet bases on the unit interval were constructed i
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