一类奇性积分方程的多尺度配置快速算法

This paper develops fast multiscale collocation methods for a class of Fredholm integral equations of the second kind with singular kernels. A truncation strategy for the coefficient matrix of the corresponding discrete system is proposed, which forms a basis for fast algorithms. The convergence, stability and computational complexity of these algorithms are analyzed.     

MULTI-PARAMETER TIKHONOV REGULARIZATION FOR LINEAR ILL-POSED OPERATOR EQUATIONS

We consider solving linear ill-posed operator equations. Based on a multi-scale decomposition for the solution space, we propose a multi-parameter regularization for solving the equations. We establish weak and strong convergence theorems for the multi-parameter regularization solution. In particular, based on the eigenfunction decomposition, we develop a posteriori choice strategy for multi-parameters which gives a regularization solution with the optimal error bound. Several practical choices of multi-parameters are proposed. We also present numerical experiments to demonstrate the outperformance of the multiparameter regularization over the single parameter regularization.     

快速配置法中数值积分的误差控制策略

We propose two error control techniques for numerical integrations in fast multiscale collocation methods for solving Fredholm integral equations of the second kind with weakly singular kernels. Both techniques utilize quadratures for singular integrals using graded points. One has a polynomial order of accuracy if the integrand has a polynomial order of smoothness except at the singular point and the other has exponential order of accuracy if the integrand has an infinite order of smoothness except at the singular point. We estimate the order of convergence and computational complexity of the corresponding approximate solutions of the equation. We prove that the second technique preserves the order of convergence and computational complexity of the original collocation method. Numerical experiments are presented to illustrate the theoretical estimates.     

EFFICIENT NUMERICAL ALGORITHMS FOR THREE-DIMENSIONAL FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

This paper detailedly discusses the locally one-dimensional numerical methods for ef- ficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional diffusion equation. The second order finite difference scheme is used to discretize the space fractional derivative and the Crank-Nicolson procedure to the time derivative. We theoretically prove and numerically verify that the presented numerical methods are unconditionally stable and second order convergent in both space and time directions. In particular, for the Riesz fractional dif- fusion equation, the idea of reducing the splitting error is used to further improve the algorithm, and the unconditional stability and convergency are also strictly proved and numerically verified for the improved scheme.     

Fourth-Order Splitting Methods for Time-Dependant Differential Equations

This study was suggested by previous work on the simulation of evolution equations with scale-dependent processes,e.g.,wave-propagation or heat-transfer,that are modeled by wave equations or heat equations.Here,we study both parabolic and hyperbolic equations.We focus on ADI (alternating direction implicit) methods and LOD (locally one-dimensional) methods,which are standard splitting methods of lower order,e.g.second-order.Our aim is to develop higher-order ADI methods,which are performed by Richardson extrapolation,Crank-Nicolson methods and higher-order LOD methods,based on locally higher-order methods.We discuss the new theoretical results of the stability and consistency of the ADI methods.The main idea is to apply a higher- order time discretization and combine it with the ADI methods.We also discuss the dis- cretization and splitting methods for first-order and second-order evolution equations. The stability analysis is given for the ADI method for first-order time derivatives and for the LOD (locally one-dimensional) methods for second-order time derivatives.The higher-order methods are unconditionally stable.Some numerical experiments verify our results.     

Generalized Jacobi and Gauss-Seidel Methods for Solving Linear System of Equations

The Jacobi and Gauss-Seidel algorithms are among the stationary iterative methods for solving linear system of equations. They are now mostly used as precondition-ers for the popular iterative solvers. In this paper a generalization of these methods are proposed and their convergence properties are studied. Some numerical experiments are given to show the efficiency of the new methods.     

MIXED INTERIOR PENALTY DISCONTINUOUS GALERKIN METHODS FOR ONE-DIMENSIONAL FULLY NONLINEAR SECOND ORDER ELLIPTIC AND PARABOLIC EQUATIONS

This paper is concerned with developing accurate and efficient numerical methods for one-dimensional fully nonlinear second order elliptic and parabolic partial differential equations （PDEs）. In the paper we present a general framework for constructing high order interior penalty discontinuous Galerkin （IP-DG） methods for approximating viscosity solutions of these fully nonlinear PDEs. In order to capture discontinuities of the second order derivative uxx of the solution u, three independent functions p1,p2 and p3 are introduced to represent numerical derivatives using various one-sided limits. The proposed DG frame- work, which is based on a nonstandard mixed formulation of the underlying PDE, embeds a nonlinear problem into a mostly linear system of equations where the nonlinearity has been modified to include multiple values of the second order derivative uxz. The proposed framework extends a companion finite difference framework developed by the authors in [9] and allows for the approximation of fully nonlinear PDEs using high order polynomials and non-uniform meshes. In addition to the nonstandard mixed formulation setting, another main idea is to replace the fully nonlinear differential operator by a numerical operator which is consistent with the differential operator and satisfies certain monotonicity （called g-monotonicity） properties. To ensure such a g-monotonicity, the crux of the construction is to introduce the numerical moment, which plays a critical role in the proposed DG frame- work. The g-monotonicity gives the DG methods the ability to select the mathematically ＂correct＂ solution （i.e., the viscosity solution） among all possible solutions. Moreover, the g-monotonicity allows for the possible development of more efficient nonlinear solvers as the special nonlinearity of the algebraic systems can be explored to decouple the equations. This paper also presents and analyzes numerical results for several numerical test problems which are used to guage the accuracy and efficiency of the proposed DG methods.     

FINITE ELEMENT METHODS FOR A BI-WAVE EQUATION MODELING D-WAVE SUPERCONDUCTORS

In this paper we develop two conforming finite element methods for a fourth order bi-wave equation arising as a simplified Ginzburg-Landau-type model for d-wave superconductors in absence of applied magnetic field. Unlike the biharmonic operator A2, the bi-wave operator □^2 is not an elliptic operator, so the energy space for the bi-wave equation is much larger than the energy space for the biharmonic equation. This then makes it possible to construct low order conforming finite elements for the bi-wave equation. However, the existence and construction of such finite elements strongly depends on the mesh. In the paper, we first characterize mesh conditions which allow and not allow construction of low order conforming finite elements for approximating the bi-wave equation. We then construct a cubic and a quartic conforming finite element. It is proved that both elements have the desired approximation properties, and give optimal order error estimates in the energy norm, suboptimal （and optimal in some cases） order error estimates in the H1 and L^2 norm. Finally, numerical experiments are presented to guage the efficiency of the proposed finite element methods and to validate the theoretical error bounds.     

ON NEWTON-HSS METHODS FOR SYSTEMS OF NONLINEAR EQUATIONS WITH POSITIVE-DEFINITE JACOBIAN MATRICES

The Hermitian and skew-Hermitian splitting （HSS） method is an unconditionally convergent iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By making use of the HSS iteration as the inner solver for the Newton method, we establish a class of Newton-HSS methods for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. For this class of inexact Newton methods, two types of local convergence theorems are proved under proper conditions, and numerical results are given to examine their feasibility and effectiveness. In addition, the advantages of the Newton-HSS methods over the Newton-USOR, the Newton-GMRES and the Newton-GCG methods are shown through solving systems of nonlinear equations arising from the finite difference discretization of a two-dimensional convection-diffusion equation perturbed by a nonlinear term. The numerical implemen- tations also show that as preconditioners for the Newton-GMRES and the Newton-GCG methods the HSS iteration outperforms the USOR iteration in both computing time and iteration step.     

GENERALIZED MATRIX MULTISPLITTING RELAXATION METHODS AND THEIR CONVERGENCE

In this paper, we set up a general framework of parallel matrix mullisplitting relaxation methods for solving large scale system of linear equations. We investigate the convergence properties of this framework and give several sufficient conditions ensuring it to converge as well as diverge. At last, we conclude a necessary and sufficient condition for the convergence of this framework when the coefficient matrix is an L-matrix.     

The Petrov–Galerkin method for second kind integral equations II: multiwavelet schemes

This paper continues the theme of the recent work [Z. Chen and Y. Xu, The Petrov–Galerkin and iterated Petrov–Galerkin methods for second kind integral equations, SIAM J. Numer. Anal., to appear] and further develops the Petrov–Galerkin method for Fredholm integral equations of the second kind. Specifically, we study wavelet Petrov–Galerkin schemes based on discontinuous orthogonal multiwavelets and prove that the condition number of the coefficient matrix for the linear system obtained from the wavelet Petrov–Galerkin scheme is bounded. In addition, we propose a truncation strategy which forms a basis for fast wavelet algorithms and analyze the order of convergence and computational complexity of these algorithms. This revised version was published online in June 2006 with corrections to the Cover Date.     

Multilevel augmentation methods for solving the sine-Gordon equation

In this paper we develop the multilevel augmentation method for solving nonlinear operator equations of the second kind and apply it to solving the one-dimensional sine-Gordon equation. We first give a general setting of the multilevel augmentation method for solving the second kind nonlinear operator equations and prove that the multilevel augmentation method preserves the optimal convergence order of the projection method while reducing computational cost significantly. Then we describe the semi-discrete scheme and the fully-discrete scheme based on multiscale methods for solving the sine-Gordon equation, and apply the multilevel augmentation method to solving the discrete equation. A complete analysis for convergence order is proposed. Finally numerical experiments are presented to confirm the theoretical results and illustrate the efficiency of the method.     

Discrete Wavelet Petrov–Galerkin Methods

In this paper, we develop a discrete wavelet Petrov–Galerkin method for integral equations of the second kind with weakly singular kernels suitable for solving boundary integral equations. A compression strategy for the design of a fast algorithm is suggested. Estimates for the rate of convergence and computational complexity of the method are provided.     

An extrapolation method for a class of boundary integral equations

Boundary value problems of the third kind are converted into boundary integral equations of the second kind with periodic logarithmic kernels by using Green's formulas. For solving the induced boundary integral equations, a Nyström scheme and its extrapolation method are derived for periodic Fredholm integral equations of the second kind with logarithmic singularity. Asymptotic expansions for the approximate solutions obtained by the Nyström scheme are developed to analyze the extrapolation method. Some computational aspects of the methods are considered, and two numerical examples are given to illustrate the acceleration of convergence.

     

一类弱非线性方程组的Picard-MHSS迭代方法

修正的Hermite/反Hermite分裂(MHSS)迭代方法是一类求解大型稀疏复对称线性代数方程组的无条件收敛的迭代算法.基于非线性代数方程组的特殊结构和性质,我们选取Picard迭代为外迭代方法,MHSS迭代作为内迭代方法,构造了求解大型稀疏弱非线性代数方程组的Picard-MHSS和非线性MHSS-like方法.这两类方法的优点是不需要在每次迭代时均精确计算和存储Jacobi矩阵,仅需要在迭代过程中求解两个常系数实对称正定子线性方程组.除此之外,在一定条件下,给出了两类方法的局部收敛性定理.数值结果证明了这两类方法是可行、有效和稳健的.     

Circulant Wavelet Preconditioners for Solving Elliptic Differential Equations and Boundary Integral Equations

In this paper is discussed solving an elliptic equation and a boundary integral equation of the second kind by representation of compactly supported wavelets. By using wavelet bases and the Galerkin method for these equations, we obtain a stiff sparse matrix that can be ill-conditioned. Therefore, we have to introduce an operator which maps every sparse matrix to a circulant sparse matrix. This class of circulant matrices is a class of preconditioners in a Banach space. Based on having some properties in the spectral theory for this class of matrices, we conclude that the circulant matrices are a good class of preconditioners for solving these equations. We called them circulant wavelet preconditioners (CWP). Therefore, a class of algorithms is introduced for rapid numerical application.     

A Jacobi-collocation method for solving second kind Fredholm integral equations with weakly singular kernels

In this work, we propose a Jacobi-collocation method to solve the second kind linear Fredholm integral equations with weakly singular kernels. Particularly, we consider the case when the underlying solutions are sufficiently smooth. In this case, the proposed method leads to a fully discrete linear system. We show that the fully discrete integral operator is stable in both infinite and weighted square norms. Furthermore, we establish that the approximate solution arrives at an optimal convergence order under the two norms. Finally, we give some numerical examples, which confirm the theoretical prediction of the exponential rate of convergence.     

Multigrid methods for boundary integral equations

Summary Multigrid methods are applied for solving algebraic systems of equations that occur to the numerical treatment of boundary integral equations of the first and second kind. These methods, originally formulated for partial differential equations of elliptic type, combine relaxation schemes and coarse grid corrections. The choice of the relaxation scheme is found to be essential to attain a fast convergent iterative process. Theoretical investigations show that the presented relaxation scheme provides a multigrid algorithm of which the rate of convergence increases with the dimension of the finest grid. This is illustrated for the calculation of potential flow around an aerofoil.     

求解一类具有Hibert核的奇异积分方程的小波方法

1　引　　言近年来,用小波方法数值求解积分方程越来越引起人们的注意.文献[1]提出的算法可将一类积分算子所对应的矩阵稀疏化,为小波方法快速求解积分方程开辟了一条新的道路这方面的研究不仅可以深入发展小波理论和应用算法,深入发展小波方法的功效,而且对边界元方法有重要的指导意义.然而研究稳健快速的数值方法,一直是这方面研究的难点问题.本文考虑带Hilbert核的奇异积分方程q(y)=12π∫2π0f(x)ctg12(x-y)dx,y∈[0,2π],(1.1)的小波数值解法;其中f(x)∈H2π,q(y)∈H2π是以2π为周期的Holder类函数;q(y)已知,f(x)待求解;(1.1)式右…     

Solving nonlinear integral equations of Fredholm type with high order iterative methods

The application of high order iterative methods for solving nonlinear integral equations is not usual in mathematics. But, in this paper, we show that high order iterative methods can be used to solve a special case of nonlinear integral equations of Fredholm type and second kind. In particular, those that have the property of the second derivative of the corresponding operator have associated with them a vector of diagonal matrices once a process of discretization has been done.