Using divergence-free and curl-free wavelets for the simulation of turbulent flows

We present a numerical method based on divergence-free and curl-free wavelets to solve the incompressible Navier-Stokes equations. We introduce a new scheme which uses anisotropic divergence-free wavelets for the decomposition of the velocity, and which only needs Fast Wavelet Transform algorithms. Numerical results finally show the feasibility of the method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Anisotropic curl-free wavelet bases on the unit cube

This paper deals with the construction of anisotropic curl-free wavelets on the cube [0, 1]³, which satisfies the specific boundary conditions. First, one constructs curl-free wavelets on the unit cube based on one dimensional wavelets on the interval [0, 1] with some boundary conditions. Then, the stability of the corresponding wavelets in curl-free space and the characterization of Sobolev spaces are studied. Finally, one gives a Helmholtz decomposition and the representation of curl and div operators in wavelet coordinates.     

Craya decomposition using compactly supported biorthogonal wavelets

We present a new local Craya–Herring decomposition of three-dimensional vector fields using compactly supported biorthogonal wavelets. Therewith vector-valued function spaces are split into two orthogonal components, i.e., curl-free and divergence-free spaces. The latter is further decomposed into toroidal and poloidal parts to decorrelate horizontal from vertical contributions which are of particular interest in geophysical turbulence. Applications are shown for isotropic, rotating and stratified turbulent flows. A comparison between isotropic and anisotropic orthogonal Craya–Herring wavelets, built in Fourier space and thus not compactly supported, is also given.     

Improved stability estimates and a characterization of the native space for matrix-valued RBFs

In this paper we derive several new results involving matrix-valued radial basis functions (RBFs). We begin by introducing a class of matrix-valued RBFs which can be used to construct interpolants that are curl-free. Next, we offer a characterization of the native space for divergence-free and curl-free kernels based on the Fourier transform. Finally, we investigate the stability of the interpolation matrix for both the divergence-free and curl-free cases, and when the kernel has finite smoothness we obtain sharp estimates. An erratum to this article can be found at     

Optimal design via variational principles: the three-dimensional case

The analysis of some three-dimensional optimal design problems leads us to study variational principles under curl-free and divergence-free constraints simultaneously. We explicitly exploit the relationship between curl-free and div-free restrictions in order to take advantage of the accumulated experience in the classical curl case by the introduction of potentials. Our discussion takes place in the three-dimensional situation. This is a first contribution in the sense that we only deal with the most basic issues.     

On interpolatory divergence-free wavelets

We construct interpolating divergence-free multiwavelets based on cubic Hermite splines. We give characterizations of the relevant function spaces and indicate their use for analyzing experimental data of incompressible flow fields. We also show that the standard interpolatory wavelets, based on the Deslauriers-Dubuc interpolatory scheme or on interpolatory splines, cannot be used to construct compactly supported divergence-free interpolatory wavelets.

     

Divergence-free wavelet solution to the Stokes problem

In this paper, we use divergence-free wavelets to give an adaptive solution to the velocity field of the Stokes problem. We first use divergence-free wavelets to discretize the divergence-free weak formulation of the Stokes problem and obtain a discrete positive definite linear system of equations whose coefficient matrix is quasi-sparse; Secondly, an adaptive scheme is used to solve the discrete linear system of equations and the error estimation and complexity analysis are given.     

A discrete weighted Helmholtz decomposition and its application

We propose a discrete weighted Helmholtz decomposition for edge element functions. The decomposition is orthogonal in a weighted $L^2$ inner product and stable uniformly with respect to the jumps in the discontinuous weight function. As an application, the new Helmholtz decomposition is applied to demonstrate the quasi-optimality of a preconditioned edge element system for solving a saddle-point Maxwell system in non-homogeneous media by a non-overlapping domain decomposition preconditioner, i.e., the condition number grows only as the logarithm of the dimension of the local subproblem associated with an individual subdomain, and more importantly, it is independent of the jumps of the physical coefficients across the interfaces between any two subdomains of different media. Numerical experiments are presented to validate the effectiveness of the non-overlapping domain decomposition preconditioner.     

Dimension Functions of Rationally Dilated GMRAs and Wavelets

In this paper we study properties of generalized multiresolution analyses (GMRAs) and wavelets associated with rational dilations. We characterize the class of GMRAs associated with rationally dilated wavelets extending the result of Baggett, Medina, and Merrill. As a consequence, we introduce and derive the properties of the dimension function of rationally dilated wavelets. In particular, we show that any mildly regular wavelet must necessarily come from an MRA (possibly of higher multiplicity) extending Auscher’s result from the setting of integer dilations to that of rational dilations. We also characterize all 3 interval wavelet sets for all positive dilation factors. Finally, we give an example of a rationally dilated wavelet dimension function for which the conventional algorithm for constructing integer dilated wavelet sets fails.     

分数阶尺度函数的分解与重构算法

引入分数阶多分辨分析与分数阶尺度函数的概念.运用时频分析方法与分数阶小波变换,研究了分数阶正交小波的构造方法,得到分数阶正交小波存在的充要条件.给出分数阶尺度函数与小波的分解与重构算法,算法比经典的尺度函数与小波的分解与重构算法更具有一般性.     

Designing Multiresolution Analysis-type Wavelets and Their Fast Algorithms

Often, the Dyadic Wavelet Transform is performed and implemented with the Daubechies wavelets, the Battle-Lemarie wavelets, or the splines wavelets, whereas in continuous-time wavelet decomposition a much larger variety of mother wavelets is used. Maintaining the dyadic time-frequency sampling and the recursive pyramidal computational structure, we present various methods for constructing wavelets ψ_wanted, with some desired shape and properties and which are associated with semi-orthogonal multiresolution analyses. We explain in detail how to design any desired wavelet, starting from any given multiresolution analysis. We also explicitly derive the formulae of the filter bank structure that implements the designed wavelet. We illustrate these wavelet design techniques with examples that we have programmed with Matlab routines.     

Reconstructed Discontinuous Approximation to Stokes Equation in a Sequential Least Squares Formulation

We propose a new least squares finite element method to solve the Stokes problem with two sequential steps. The approximation spaces are constructed by the patch reconstruction with one unknown per element. For the first step, we reconstruct an approximation space consisting of piecewise curl-free polynomials with zero trace. By this space, we minimize a least squares functional to obtain the numerical approximations to the gradient of the velocity and the pressure. In the second step, we minimize another least squares functional to give the solution to the velocity in the reconstructed piecewise divergence-free space. We derive error estimates for all unknowns under both $L^2$ norms and energy norms. Numerical results in two dimensions and three dimensions verify the convergence rates and demonstrate the great flexibility of our method.     

Divergence-Free Wavelets on the Hypercube: General Boundary Conditions

On the n-dimensional hypercube, for given \(k\in {\mathbb {N}}\), wavelet Riesz bases are constructed for the subspace of divergence-free vector fields of the Sobolev space \(H^k((0,1)^n)^n\) with general homogeneous Dirichlet boundary conditions, including slip or no-slip boundary conditions. Both primal and suitable dual wavelets can be constructed to be locally supported. The construction of the isotropic wavelet bases is restricted to the square, but that of the anisotropic wavelet bases applies for any space dimension n.     

Ascending and descending regions of a discrete Morse function

We present an algorithm which produces a decomposition of a regular cellular complex with a discrete Morse function analogous to the Morse–Smale decomposition of a smooth manifold with respect to a smooth Morse function. The advantage of our algorithm compared to similar existing results is that it works, at least theoretically, in any dimension. Practically, there are dimensional restrictions due to the size of cellular complexes of higher dimensions, though. We prove that the algorithm is correct in the sense that it always produces a decomposition into descending and ascending regions of the critical cells in a finite number of steps, and that, after a finite number of subdivisions, all the regions are topological disks. The efficiency of the algorithm is discussed and its performance on several examples is demonstrated.     

Nonconservative Estimating Functions and Approximate Quasi-Likelihoods

The estimating function approach unifies two dominant methodologies in statistical inferences: Gauss's least square and Fisher's maximum likelihood. However, a parallel likelihood inference is lacking because estimating functions are in general not integrable, or nonconservative. In this paper, nonconservative estimating functions are studied from vector analysis perspective. We derive a generalized version of the Helmholtz decomposition theorem for estimating functions of any dimension. Based on this theorem we propose locally quadratic potentials as approximate quasi-likelihoods. Quasi-likelihood ratio tests are studied. The ideas are illustrated by two examples: (a) logistic regression with measurement error model and (b) probability estimation conditional on marginal frequencies.     

On divergence-free wavelets

This paper is concerned with the construction of compactly supported divergence-free vector wavelets. Our construction is based on a large class of refinable functions which generate multivariate multiresolution analyses which includes, in particular, the non tensor product case.For this purpose, we develop a certain relationship between partial derivatives of refinable functions and wavelets with modifications of the coefficients in their refinement equation. In addition, we demonstrate that the wavelets we construct form a Riesz-basis for the space of divergence-free vector fields.Work supported by the Deutsche Forschungsgemeinschaft in the Graduiertenkolleg Analyse und Konstruktion in der Mathematik at the RWTH Aachen.     

PERIODIC CARDINAL INTERPOLATORY WAVELETS

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An alternative to the Kelvin decomposition for plane anisotropic elasticity

In this paper, we propose an alternative tensorial decomposition to the Kelvin's one (introduced by Kelvin in 1856) for plane anisotropic elasticity using the polar formalism (introduced by Verchery in 1979). In the first part of the paper, a parallel between the two approaches is proposed. Thanks to it, some new results are found; namely, the projectors introduced have a direct interpretation in terms of material symmetry and are intrinsic for any type of symmetry considered, that is, they do not depend on any elastic modulus for any type of symmetry, unlike in the Kelvin decomposition. The introduction of what we call, in the paper, the polar projectors, stresses and strains gives a new insight into the polar formalism. The results proposed in this paper will hopefully be useful in some cases, for example, in the modeling of anisotropic damage evolution in solids. Copyright © 2013 John Wiley & Sons, Ltd.     

Decomposition of refinable spaces and applications to operator equations

This paper presents a brief survey of several recent applications of multilevel techniques, in particular, in connection with the solution of periodic pseudodifferential equations. It is pointed out that these applications naturally lead to certain decompositions of refinable spaces which are induced by a class of linear projectors. Then recent results on the construction of such nonorthogonal wavelets are reviewed and extended to the particular needs of the present context.     

Preconditioning by Projectors in the Solution of Contact Problems: A Parallel Implementation

A non-overlapping domain decomposition algorithm of the Neumann–Neumann type for solving contact problems of elasticity is presented. Using the duality theory of convex programming, the discretized problem turns into a quadratic one with equality and bound constraints. The dual problem is modified by orthogonal projectors to the natural coarse space. The resulting problem is solved by an augmented Lagrangian algorithm. The projectors ensure an optimal convergence rate for the solution of the auxiliary linear problems by the preconditioned conjugate gradient method. Relevant aspects on the numerical linear algebra of these problems are presented, together with an efficient parallel implementation of the method.