Non-uniform multiresolution analysis with supercompact multiwavelets

Abstract

a b s t r a c tThis work is devoted to a generalization of the framework presented in Beam and Warming (2000) [6], where a multiresolution analysis scheme with supercompact multiwavelets was presented. The approach considers uniform partitions of a nested grid hierarchy in the framework of Harten's multi-scale representations. In this paper we study the non-uniform case. The non-uniform analysis is well adapted to more realistic contexts and makes it possible to improve the approximation.

A Wavelet-Free Approach for Multiresolution-Based Grid Adaptation for Conservation Laws

“…The basic idea is to find a completion of , i.e., for a given This idea originates from [18], where a very general setting is considered. It has been carried out in the particular setting of DG spaces for the one-dimensional uniform dyadic hierarchy in [13] and for the one-dimensional non-uniform dyadic case in [4]. In both works an explicit formula for ,1 is derived.…”

“…To find a completion for the general non-uniform multi-dimensional case is not trivial and it is open whether an explicit formula similar to the one-dimensional case can be derived. Furthermore, the computation of the completions from [4,13] for the one-dimensional case is still costly since it requires the inversion of a matrix.…”

“…Corresponding to the grid on a fixed maximum refinement level L, we consider an MRA of the semi-discrete DG solution L defined by (3). The weak formulations defining the evolution of this solution can be written in a compact form as where the inner product is applied component-wise and L ∶ (S L ) m × S L → m is defined in (4). Note that L (⋅, ⋅) is linear in the second argument.…”

Solving Nonlinear Pdes Using the Higher Order Haar Wavelet Method on Nonuniform and Adaptive Grids

“…They were first used to solve integral and differential equations by Lepik [55]. The nonuniform Haar wavelets have since been used in multiresolution analysis [3], boundary value problems [32], fractional order problems [64,78] as well as two dimensional problems [68]. An overview of uniform and non-uniform wavelet based methods can be found in [48].…”

Multiresolution analysis and supercompact multiwavelets for surfaces

“…[4,5]). The non uniform case have been recently studied in [1]. For both cases the nested grid hierarchy is defined on the unit interval.…”