Journal ArticleDOI
Locally Supported, Piecewise Polynomial Biorthogonal Wavelets on Nonuniform Meshes
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TLDR
In this paper, biorthogonal wavelets are constructed on nonuniform ��meshes and both primal and dual wavelets generate Riesz bases for the Sobolev spaces for (|s| < 3/2) and the primal side span standard Lagrange finite element spaces, respectively.Abstract:
In this paper, biorthogonal wavelets are constructed on nonuniform
meshes. Both primal and dual wavelets are
locally supported, continuous piecewise polynomials. The wavelets generate
Riesz bases for the Sobolev spaces (H
s
) for (|s| < 3/2). The
wavelets at the primal side span standard Lagrange finite element spaces.read more
Citations
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Journal ArticleDOI
Adaptive Solution of Operator Equations Using Wavelet Frames
TL;DR: This paper writes the domain or manifold on which the operator equation is posed as an overlapping union of subdomains, each of them being the image under a smooth parametrization of the hypercube, and proves that this adaptive method has optimal computational complexity.
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A sparse grid space-time discretization scheme for parabolic problems
Michael Griebel,Daniel Oeltz +1 more
TL;DR: The space-time sparse grid approach can be employed together with adaptive refinement in space and time and then leads to similar approximation rates as the non-adaptive method for smooth functions.
Book
Spline Functions: Computational Methods
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On the compressibility of operators in wavelet coordinates
TL;DR: In [Found. Math., 2 (2002), pp. 203--245], Cohen, Dahmen, and DeVore proposed an adaptive wavelet algorithm for solving operator equations, assuming that the operator defines a boundedly in the wavelet.
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Adaptive boundary element methods with convergence rates
TL;DR: The main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods, and an inverse-type inequality involving boundary integral operators on locally refined finite element spaces.
References
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Journal ArticleDOI
Biorthogonal bases of compactly supported wavelets
TL;DR: In this paper, it was shown that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual Riesz bases of compactly supported wavelets.
The lifting scheme: A construction of second generation wavelets
TL;DR: The lifting scheme is presented, a simple construction of second generation wavelets; these are wavelets that are not necessarily translates and dilates of one fixed function, and can be adapted to intervals, domains, surfaces, weights, and irregular samples.
Journal ArticleDOI
The lifting scheme: a construction of second generation wavelets
TL;DR: The lifting wavelet as discussed by the authors is a simple construction of second generation wavelets that can be adapted to intervals, domains, surfaces, weights, and irregular samples, and it leads to a faster, in-place calculation of the wavelet transform.
Journal ArticleDOI
Wavelet and multiscale methods for operator equations
TL;DR: Although there remain many unresolved problems, multigrid or multilevel schemes in the classical framework of finite difference and finite element discretizations exhibit by now a comparatively clear profile.
Book ChapterDOI
Wavelet methods in numerical analysis
TL;DR: The chapter describes the decomposition and reconstruction algorithms that can be used to compute the coefficients of a function in two elementary wavelet bases and it investigates the way these schemes can be generalized in a natural way to multivariate functions.