偏微分方程的区间小波自适应精细积分法

利用插值小波理论构造了拟Shannon区间小波,并结合外推法给出了一种求解非线性常微分方程组的时间步长自适应精细积分法,在此基础上构造了求解非线性偏微分方程的区间小波自适应精细积分法（AIWPIM）．数值结果表明,该方法在计算精度上优于将小波和四阶Runge-Kutta法组合得到的偏微分方程的数值求解方法,而计算量则相差不大．该文方法通过Burgers方程给出,但适用于一般情形．     

Integration of barotropic vorticity equation over spherical geodesic grid using multilevel adaptive wavelet collocation method

In this paper, we present the multilevel adaptive wavelet collocation method for solving non-divergent barotropic vorticity equation over spherical geodesic grid. This method is based on multi-dimensional second generation wavelet over a spherical geodesic grid. The method is more useful in capturing, identifying, and analyzing local structure [1] than any other traditional methods (i.e. finite difference, spectral method), because those methods are either full or partial miss important phenomena such as trends, breakdown points, discontinuities in higher derivatives of the solution. Wavelet decomposition is used for interpolation and adaptive grid refinement on different levels.     

On a Sparse Representation of an n-Dimensional Laplacian in Wavelet Coordinates

Important parts of adaptive wavelet methods are well-conditioned wavelet stiffness matrices and an efficient approximate multiplication of quasi-sparse stiffness matrices with vectors in wavelet coordinates. Therefore it is useful to develop a well-conditioned wavelet basis with respect to which both the mass and stiffness matrices are sparse in the sense that the number of nonzero elements in each column is bounded by a constant. Consequently, the stiffness matrix corresponding to the n-dimensional Laplacian in the tensor product wavelet basis is also sparse. Then a matrix–vector multiplication can be performed exactly with linear complexity. In this paper, we construct a wavelet basis based on Hermite cubic splines with respect to which both the mass matrix and the stiffness matrix corresponding to a one-dimensional Poisson equation are sparse. Moreover, a proposed basis is well-conditioned on low decomposition levels. Small condition numbers for low decomposition levels and a sparse structure of stiffness matrices are kept for any well-conditioned second order partial differential equations with constant coefficients; furthermore, they are independent of the space dimension.     

Adaptive Wavelet Schwarz Methods for the Navier-Stokes Equation

In this article, we are concerned with domain decomposition methods for the stationary incompressible Navier-Stokes equation. We construct an adaptive additive Schwarz method based on discretization by means of a divergence-free wavelet frame. We prove that the method is convergent and asymptotically optimal with respect to the degrees of freedom involved.     

Symplectic and multi-symplectic wavelet collocation methods for two-dimensional Schrödinger equations

In this paper, we develop symplectic and multi-symplectic wavelet collocation methods to solve the two-dimensional nonlinear Schrödinger equation in wave propagation problems and the two-dimensional time-dependent linear Schrödinger equation in quantum physics. The Hamiltonian and the multi-symplectic formulations of each equation are considered. For both formulations, wavelet collocation method based on the autocorrelation function of Daubechies scaling functions is applied for spatial discretization and symplectic method is used for time integration. The conservation of energy and total norm is investigated. Combined with splitting scheme, splitting symplectic and multi-symplectic wavelet collocation methods are also constructed. Numerical experiments show the effectiveness of the proposed methods.     

Adaptive Wavelet BEM for Boundary Integral Equations: Theory and Numerical Experiments

We are concerned with the numerical treatment of boundary integral equations by the adaptive wavelet boundary element method. In particular, we consider the second kind Fredholm integral equation for the double layer potential operator on patchwise smooth manifolds contained in ?³. The corresponding operator equations are treated by adaptive implementations that are in complete accordance with the underlying theory. The numerical experiments demonstrate that adaptive methods really pay off in this setting. The observed convergence rates fit together very well with the theoretical predictions based on the Besov regularity of the exact solution.     

Adaptive Wavelet Methods for Elliptic Operator Equations with Nonlinear Terms

We present an adaptive wavelet method for the numerical solution of elliptic operator equations with nonlinear terms. This method is developed based on tree approximations for the solution of the equations and adaptive fast reconstruction of nonlinear functionals of wavelet expansions. We introduce a constructive greedy scheme for the construction of such tree approximations. Adaptive strategies of both continuous and discrete versions are proposed. We prove that these adaptive methods generate approximate solutions with optimal order in both of convergence and computational complexity when the solutions have certain degree of Besov regularity.     

Adaptive Wavelet Solution to the Stokes Problem

This paper deals with the design and analysis of adaptive wavelet method for the Stokes problem. First, the limitation of Richardson iteration is explained and the multiplied matrix M0 in the paper of Bramble and Pasciak is proved to be the simplest possible in an appropiate sense. Similar to the divergence operator, an exact application of its dual is shown; Second, based on these above observations, an adaptive wavelet algorithm for the Stokes problem is designed. Error analysis and computational complexity are given; Finally, since our algorithm is mainly to deal with an elliptic and positive definite operator equation, the last section is devoted to the Galerkin solution of an elliptic and positive definite equation. It turns out that the upper bound for error estimation may be improved.     

粘弹性介质中波场重建方法

传统的波动方程波场重建基于完全弹性介质，不能获得满意的地震资料分辨率，本基于能描述大地吸收弹性介质中地震波传播的斯托克斯波动方程．提出了一种新的多尺度粘弹性波动方程波场重建的方法，根据地震波传播核函数的物理特性，研究了一种新的物理小波，提出了小波多尺度波场重建方法。达到对吸收信息的补偿，提高地震资料的分辨率。     

求解非线性Black-Scholes模型的自适应小波精细积分法

针对非线性Black-Scholes方程,基于quasi-Shannon小波函数给出了一种求解非线性偏微分方程的自适应多尺度小波精细积分法.该方法首先利用插值小波理论构造了用于逼近连续函数的多尺度小波插值算子,利用该算子可以将非线性Black-Scholes方程自适应离散为非线性常微分方程组;然后将用于求解常微分方程组的精细积分法和小波变换的动态过程相结合,并利用非线性处理技术(如同伦分析技术)可有效求解非线性Black-Scholes方程.数值结果表明了该方法在数值精度和计算效率方面的优越性.     

Reduced basis methods with adaptive snapshot computations

We use asymptotically optimal adaptive numerical methods (here specifically a wavelet scheme) for snapshot computations within the offline phase of the Reduced Basis Method (RBM). The resulting discretizations for each snapshot (i.e., parameter-dependent) do not permit the standard RB ‘truth space’, but allow for error estimation of the RB approximation with respect to the exact solution of the considered parameterized partial differential equation. The residual-based a posteriori error estimators are computed by an adaptive dual wavelet expansion, which allows us to compute a surrogate of the dual norm of the residual. The resulting adaptive RBM is analyzed. We show the convergence of the resulting adaptive greedy method. Numerical experiments for stationary and instationary problems underline the potential of this approach.     

A wavelet-based nested iteration–inexact conjugate gradient algorithm for adaptively solving elliptic PDEs

In Cohen et al. (Math Comput 70:27–75, 2001), a new paradigm for the adaptive solution of linear elliptic partial differential equations (PDEs) was proposed, based on wavelet discretizations. Starting from a well-conditioned representation of the linear operator equation in infinite wavelet coordinates, one performs perturbed gradient iterations involving approximate matrix–vector multiplications of finite portions of the operator. In a bootstrap-type fashion, increasingly smaller tolerances guarantee convergence of the adaptive method. In addition, coarsening performed on the iterates allow one to prove asymptotically optimal complexity results when compared to the wavelet best N-term approximation. In the present paper, we study adaptive wavelet schemes for symmetric operators employing inexact conjugate gradient routines. Inspired by fast schemes on uniform grids, we incorporate coarsening and the adaptive application of the elliptic operator into a nested iteration algorithm. Our numerical results demonstrate that the runtime of the algorithm is linear in the number of unknowns and substantial savings in memory can be achieved in two and three space dimensions.     

Treatment of Dirichlet-type boundary conditions in the spline-based wavelet Galerkin method employing multiple point constraints

The wavelet methods have been extensively adopted and integrated in various numerical methods to solve partial differential equations. The wavelet functions, however, do not satisfy the Kronecker delta function properties, special treatment methods for imposing the Dirichlet-type boundary conditions are thus required. It motivates us to present in this paper a novel treatment technique for the essential boundary conditions (BCs) in the spline-based wavelet Galerkin method (WGM), taking the advantages of the multiple point constraints (MPCs) and adaptivity. The linear B-spline scaling function and multilevel wavelet functions are employed as basis functions. The effectiveness of the present method is addressed, and in particular the applicability of the MPCs is also investigated. In the proposed technique, MPC equations based on the tying relations of the wavelet basis functions along the essential BCs are developed. The stiffness matrix is degenerated based on the MPC equations to impose the BCs. The numerical implementation is simple, and no additional degrees of freedom are needed in the system of linear equations. The accuracy of the present formulation in treating the BCs in the WGM is high, which is illustrated through a number of representative numerical examples including an adaptive analysis.     

A fast adaptive spectral graph wavelet method for the viscous Burgers' equation on a star-shaped connected graph

Water wave propagation in an open channel network can be described by the viscous Burgers' equation on the corresponding connected graph, possibly with small viscosity. In this paper, we propose a fast adaptive spectral graph wavelet method for the numerical solution of the viscous Burgers' equation on a star-shaped connected graph. The vital feature of spectral graph wavelets is that they can be constructed on any complex network using the graph Laplacian. The essence of the method is that the same operator can be used for the construction of the spectral graph wavelet and the approximation of the differential operator involved in the Burgers' equation. In this paper, two test problems are considered with homogeneous Dirichlet boundary condition. The numerical results show that the method accurately captures the evolution of the localized patterns at all the scales, and the adaptive node arrangement is accordingly obtained. The convergence of the given method is verified, and efficiency is shown using CPU time.     

一类自适应的奇异小波有限元法

本文对一类具有孤立奇点的奇异椭圆型微分方程,通过小波变换自动检测出奇点和奇异度,基于此,导出了一类自适应的奇异小波有限元法,在网格均匀剖分下可以达到饱和阶,     

Wavelet adaptive method for second order elliptic problems: boundary conditions and domain decomposition

Summary. Wavelet methods allow to combine high order accuracy, multilevel preconditioning techniques and adaptive approximation, in order to solve efficiently elliptic operator equations. One of the main difficulty in this context is the efficient treatment of non-homogeneous boundary conditions. In this paper, we propose a strategy that allows to append such conditions in the setting of space refinement (i.e. adaptive) discretizations of second order problems. Our method is based on the use of compatible multiscale decompositions for both the domain and its boundary, and on the possibility of characterizing various function spaces from the numerical properties of these decompositions. In particular, this allows the construction of a lifting operator which is stable for a certain range of smoothness classes, and preserves the compression of the solution in the wavelet basis. An explicit construction of the wavelet bases and the lifting is proposed on fairly general domains, based on conforming domain decomposition techniques. Received November 2, 1998 / Published online April 20, 2000     

Adaptive frame methods for elliptic operator equations

This paper is concerned with the development of adaptive numerical methods for elliptic operator equations. We are especially interested in discretization schemes based on frames. The central objective is to derive an adaptive frame algorithm which is guaranteed to converge for a wide range of cases. As a core ingredient we use the concept of Gelfand frames which induces equivalences between smoothness norms and weighted sequence norms of frame coefficients. It turns out that this Gelfand characteristic of frames is closely related to their localization properties. We also give constructive examples of Gelfand wavelet frames on bounded domains. Finally, an application to the efficient adaptive computation of canonical dual frames is presented.     

On the Convergence Analysis of the Inexact Linearly Implicit Euler Scheme for a Class of Stochastic Partial Differential Equations

This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe’s method. We use the implicit Euler scheme for the time discretization. Consequently, in each step, an elliptic equation with random right-hand side has to be solved. In practice, this cannot be performed exactly, so that efficient numerical methods are needed. Well-established adaptive wavelet or finite-element schemes, which are guaranteed to converge with optimal order, suggest themselves. We investigate how the errors corresponding to the adaptive spatial discretization propagate in time, and we show how in each time step the tolerances have to be chosen such that the resulting perturbed discretization scheme realizes the same order of convergence as the one with exact evaluations of the elliptic subproblems.     

Besov regularity for second order elliptic boundary value problems with variable coefficients

This paper is concerned with some theoretical foundations for adaptive numerical methods for elliptic boundary value problems. The approximation order that can be achieved by such an adaptive method is determined by certain Besov regularity of the weak solution. We study Besov regularity for second order elliptic problems in bounded domains in ℝ ^d . The investigations are based on intermediate Schauder estimates and on some potential theoretic framework. Moreover, we use characterizations of Besov spaces by wavelet expansions. This work has been supported by the Deutsche Forschungsgemeinschaft (Da 360/1-1)     

An adaptive multiresolution method on dyadic grids: Application to transport equations

We propose a modified adaptive multiresolution scheme for solving d$d$-dimensional hyperbolic conservation laws which is based on cell-average discretization in dyadic grids. Adaptivity is obtained by interrupting the refinement at the locations where appropriate scale (wavelet) coefficients are sufficiently small. One important aspect of such a multiresolution representation is that we can use the same binary tree data structure for domains of any dimension. The tree structure allows us to succinctly represent the data and efficiently navigate through it. Dyadic grids also provide a more gradual refinement as compared with the traditional quad-trees (2D) or oct-trees (3D) that are commonly used for multiresolution analysis. We show some examples of adaptive binary tree representations, with significant savings in data storage when compared to quad-tree based schemes. As a test problem, we also consider this modified adaptive multiresolution method, using a dynamic binary tree data structure, applied to a transport equation in 2D domain, based on a second-order finite volume discretization.