An Adaptive Wavelet Method for Solving High-Dimensional Elliptic PDEs
TLDR
It will be demonstrated that the resulting approxIMations converge in energy norm with the same rate as the best approximations from the span of the best N tensor product wavelets, where moreover the constant factor that the authors may lose is independent of the space dimension n.Abstract:
Adaptive tensor product wavelet methods are applied for solving Poisson’s equation, as well as anisotropic generalizations, in high space dimensions. It will be demonstrated that the resulting approximations converge in energy norm with the same rate as the best approximations from the span of the best N tensor product wavelets, where moreover the constant factor that we may lose is independent of the space dimension n. The cost of producing these approximations will be proportional to their length with a constant factor that may grow with n, but only linearly.read more
Citations
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Journal ArticleDOI
Space-time adaptive wavelet methods for parabolic evolution problems
Christoph Schwab,Rob Stevenson +1 more
TL;DR: With respect to space-time tensor-product wavelet bases, parabolic initial boundary value problems are equivalently formulated as bi-infinite matrix problems and adaptive wavelet methods are shown to yield sequences of approximate solutions which converge at the optimal rate.
Journal ArticleDOI
Adaptive Petrov-Galerkin methods for first order transport equations !
TL;DR: Stable variational formulations are proposed for certain linear, unsymmetric operators with first order transport equations in bounded domains serving as the primary focus of this paper to adaptively resolve anisotropic solution features such as propagating singularities.
Journal ArticleDOI
Adaptive Near-Optimal Rank Tensor Approximation for High-Dimensional Operator Equations
Markus Bachmayr,Wolfgang Dahmen +1 more
TL;DR: A rigorous convergence analysis is conducted for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis, demonstrating that problems in very high dimensions can be treated with controlled solution accuracy.
Book ChapterDOI
Adaptive wavelet methods for solving operator equations: An overview
TL;DR: In this article, Cohen, Dahmen and DeVore introduced adaptive wavelet methods for solving operator equations and proved that their adaptive methods were not only proven to converge, but also with a rate better than that of their non-adaptive counterparts in cases where the latter methods converge with a reduced rate due a lacking regularity of the solution.
Journal ArticleDOI
Adaptive Wavelet Methods on Unbounded Domains
Sebastian Kestler,Karsten Urban +1 more
TL;DR: An adaptive wavelet method for operator equations on unbounded domains using wavelet bases on ℝn to equivalently express the operator equation in terms of a well-conditioned discrete problem on sequence spaces is introduced.
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