Adaptive wavelet methods for elliptic operator equations: Convergence rates

By Albert Cohen, Wolfgang Dahmen, Ronald A. DeVore. Mathematics of Computation, 2001

502 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0e9d694ce0b846cd9077eca68b0bd6895e990ec9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Computer Vision and Pattern Recognition

In Mathematical Physics

Citations in the corpus

Adaptive wavelet methods for solving operator equations: An overview (2009, methodology): “Secondly, and more importantly, instead of taking as in [CDD01] an a priori distribution of the accuracies of the approximations of B over the parts, which distribution was chosen to yield an error below the prescribed tolerance in a worst case scenario, to enhance its quantitative performance, the ”
Approximation of high-dimensional parametric PDEs * (2015, background): “Let us point out that a similar objective can be attained when considering the spatial discretization of a single PDE, by either adaptive wavelet methods [21, 22, 38] or by adaptive finite element methods [70, 7, 86].”
Adaptive low rank wavelet methods and applications to two-electron Schrödinger equations (2012, methodology): “An alternative would be the mentioned class of adaptive wavelet schemes based on approximately solving a sequence of Galerkin discretizations, as proposed originally in [30].”
Concepts of Data-Sparse Tensor-Product Approximation in Many-Particle Modelling (2008, background): “[8, 17, 18, 19, 76], since they are smooth in the far field region.”
Multilevel preconditioning and low‐rank tensor iteration for space–time simultaneous discretizations of parabolic PDEs (2015, background): “If the matrix can be suitably approximated by matrices with only finitely many nonzero entries—this is the case for many relevant parabolic equations—then the adaptive wavelet methods of [9, 10] yield optimal rates of convergence.”
Convergence of Adaptive Finite Element Methods (2002, background): “Mesh optimality has been recently studied in [5] via wavelet algorithms which resort to the entire wavelet expansions of u and f .”
An adaptive multilevel wavelet collocation method for elliptic problems (2005, background): “Furthermore, it was shown that adaptive wavelet schemes are asymptotically optimal for elliptic equations, in the sense that the rate of convergence to the exact solution with respect to the number of degrees of freedom is the same as the rate of convergence of the best N-term approximation, which i”
CHARMS: a simple framework for adaptive simulation (2002, background): “Such refinement algorithms and their associated unrefinement operators have the advantage that they come with well established theory [Cohen et al. 2001] and result in nested meshes, and by implication nested approximation spaces.”
Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs (2011, methodology): “We should point out that a similar program was developed when solving a single PDE by either adaptive wavelet methods [7,8,14] or by adaptive finite element methods [4,12,21,26].”
Adaptive Finite Element Methods with convergence rates (2004, methodology): “Recently, an analysis of rates of convergence for wavelet based adaptive methods was given in [6],[7].”
Numerical solution for a class of fractional convection–diffusion equations using the flatlet oblique multiwavelets (2014, methodology): “basis can be successfully used for representing differential operator modeling to solve partial differential equations (Alpert, 1993; Alpert et al., 1993; Cohen et al., 2001).”
Multiscale and Wavelet Methods for Operator Equations (2003, background): “2, see [26].”
Weighted Fourier Inequalities: New Proofs and Generalizations (2003, methodology): “Besides the inherent mathematical motivation of going beyond the Plancherel theorem in this way, we are motivated to understand and apply general inequalities (1.2) in a manner analogous to recent developments and applications of comparably general notions such as Wiener amalgam spaces and Besov spa”
Best N Term Approximation Spaces for Tensor Product Wavelet Bases (2006, methodology): “For instance, the algorithm given in [ 6 ] yields an approximation rate in the H 1 norm of 1 for solutions to smooth uniformly elliptic PDEs on two-dimensional polygonal domains using isotropically supported piecewise bilinear ansatz functions; the rate 1 is”