Adaptive wavelet methods for solving operator equations: An overview

Abstract

In [Math. Comp, 70 (2001), 27-75] and [Found. Comput. Math., 2(3) (2002), 203-245], Cohen, Dahmen and DeVore introduced adaptive wavelet methods for solving operator equations. These papers meant a break-through in the field, because 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. Until then, adaptive methods were usually assumed to converge via a saturation assumption. An exception was given by the work of Dörfler in [SIAM J. Numer. Anal., 33 (1996), 1106-1124, where an adaptive finite element method was proven to converge, with no rate though.This work contains a complete analysis of the methods from the aforementioned two papers of Cohen, Dahmen and DeVore. Furthermore, we give an overview over the subsequent developments in the field of adaptive wavelet methods. This includes a precise analysis of the near-sparsity of an operator in wavelet coordinates needed to obtain optimal computational complexity; the avoidance of coarsening; quantitative improvements of the algorithms; their generalization to frames; and their application with tensor product wavelet bases which give dimension independent rates.

Reduced basis methods with adaptive snapshot computations

“…The number of terms in these expansions as well as their localization have a strong influence on the decision if an adaptive snapshot computation is indeed required or if, e.g., an adaptively generated common truth as in [35] might be sufficient. Details concerning the decay of wavelet coefficients can be found in [8,30,31].…”

Contraction and Optimality Properties of an Adaptive Legendre–Galerkin Method: The Multi-Dimensional Case

“…The latter is always measured in an algebraic approximation class, i.e., the best N -term approximation error decays at least as a power of N −1 . We refer to the surveys [27] by Nochetto, Siebert and Veeser for AFEM and [30] by Stevenson for adaptive wavelets.…”

A note on quarkonial systems and multilevel partition of unity methods

“…The MPUM can be considered as a merger of spectral and multiscale approximation schemes, and offers great potential and flexibility for developing adaptive schemes necessary for large-scale modeling and computation. However, it is fair to say that the theoretical understanding of PUM and MPUM methods is not as complete as that of multiscale finite element and wavelet methods [5], [6], [12], [18], on the one hand, and of spectral methods [3], on the other. In particular, the parallel developments on quarkonial decompositions in function space theory have not been taken notice of.…”

“…This property is important for the construction of fast and efficient solvers for such operator equations. Results in this direction are discussed in, e.g., [18], mostly for wavelet systems and closely related multilevel frame systems for which there is no or little redundancy in the admissible representations (1.5). More redundant representation systems, such as quarkonial systems, have not yet been touched to any generality.…”