用Levenberg-Marquardt类的投影收缩方法解运输问题

For solving linear variational inequalities (LVI), the projection and contraction method of Levenberg-Marquardt type needs less iterations than an elementary projection and contraction method. However, the method of Levenberg-Marquardt type has to calculate the inverse of a matrix and hence it is unsuitable for large problems. In this paper, using the special structure of the constraint matrix, we present a PC method of Levenberg-Marquardt type for LVI arising from transportation problem without calculating any inverse matrices.Several computational experiments are presentded to indicate that the methods is good for solving the transportation problem.     

A Minimum Residual Based Gradient Iterative Method for a Class of Matrix Equations

In this paper, we present a minimum residual based gradient iterative method for solving a class of matrix equations including Sylvester matrix equations and general coupled matrix equations. The iterative method uses a negative gradient as steepest direction and seeks for an optimal step size to minimize the residual norm of next iterate. It is shown that the iterative sequence converges unconditionally to the exact solution for any initial guess and that the norm of the residual matrix and err...     

ON THE FINITE VOLUME ELEMENT VERSION OF RITZ-VOLTERRA PROJECTION AND APPLICATIONS TO RELATED EQUATIONS

In this paper, we present a general error analysis framework for the finite volume element (FVE) approximation to the Ritz-Volterra projection, the Sobolev equations and parabolic integro-differential equations. The main idea in our paper is to consider the FVE methods as perturbations of standard finite element methods which enables us to derive the optimal L2 and H1 norm error estimates, and the L∞ and W∞1 norm error estimates by means of the time dependent Green functions. Our disc ussions also include elliptic and parabolic problems as the special cases.     

无记忆非拟 Newton 算法的导出和全局收敛性

In this paper, a new class of memoryless non-quasi-Newton method for solving unconstrained optimization problems is proposed, and the global convergence of this method with inexact line search is proved. Furthermore, we propose a hybrid method that mixes both the memoryless non-quasi-Newton method and the memoryless Perry-Shanno quasi-Newton method. The global convergence of this hybrid memoryless method is proved under mild assumptions. The initial results show that these new methods are efficient for the given test problems. Especially the memoryless non-quasi-Newton method requires little storage and computation, so it is able to efficiently solve large scale optimization problems.     

A PARALLEL COMPUTATION SCHEME FOR IMPLICIT RUNGE-KUTTA METHODS AND THE ITERATIVELY B-CONVERGENCE OF ITS NEWTON ITERATIVE PROCESS

In this paper, based on the implicit Runge-Kutta(IRK) methods, we derive a class of parallel scheme that can be implemented on the parallel computers with Ns(N is a positive even number) processors efficiently, and discuss the iteratively B-convergence of the Newton iterative process for solving the algebraic equations of the scheme, secondly we present a strategy providing initial values parallelly for the iterative process. Finally, some numerical results show that our parallel scheme is higher efficient as N is not so large.     

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.     

Allen-Cahn方程的虚拟元方法（英文）

In this article, the virtual element method of the Allen-Cahn equation on a polygon grid is discussed in the fully discrete formulation. With the help of the energy projection operator, we give the corresponding error estimates in the L2norm and H1norm.     

Properties and Iterative Methods for the Lasso and Its Variants

The lasso of Tibshirani （1996） is a least-squares problem regularized by the l1 norm. Due to the sparseness promoting property of the l1 norm, the lasso has been received much attention in recent years. In this paper some basic properties of the lasso and two variants of it are exploited. Moreover, the proximal method and its variants such as the relaxed proximal algorithm and a dual method for solving the lasso by iterative algorithms are presented.     

A CLASS OF TRUST REGION METHODS FOR LINEAR INEQUALITY CONSTRAINED OPTIMIZATION AND ITS THEORY ANALYSIS：Ⅰ. ALGORITHM AND GLOBAL CONVERGENCE

A class of trust region methods tor solving linear inequality constrained problems is propo6ed in this paper. It is shown that the algorithm is of global convergence. The algorithm uses a version of the two-slded projection and the strategy of the unconstrained trust region methods. It keeps the good convergence properties of the unconstrained case and has the merits of the projection method. In some sense, our algorithm can be regarded as an extension and improvement of the projected type algorithm.     

A class of globally convergent conjugate gradient methods

Conjugate gradient methods are very important ones for solving nonlinear optimization problems, especially for large scale problems. However, unlike quasi-Newton methods, conjugate gradient methods were usually analyzed individually. In this paper, we propose a class of conjugate gradient methods, which can be regarded as some kind of convex combination of the Fletcher-Reeves method and the method proposed by Dai et al. To analyze this class of methods, we introduce some unified tools that concern a general method with the scalar βk having the form of φk/φk-1. Consequently, the class of conjugate gradient methods can uniformly be analyzed.     

SOME LEAST SQUARES ESTIMATES OF THE AUTOREGRESSIVE MODELS

In this paper, we present some iterative methods for solving lth order autoregressive models, prove global convergence for l=1 case, and the numerical results of new algorithms seem to be more efficient than the ones of Cochrane-Orcutt iterative method.     

GENERALIZED CONJUGATE A-ORTHOGONAL RESIDUAL SQUARED METHOD FOR COMPLEX NON-HERMITIAN LINEAR SYSTEMS

Recently numerous numerical experiments on realistic calculation have shown that the conjugate A-orthogonal residual squared （CORS） method is often competitive with other popular methods. However, the CORS method, like the CGS method, shows irreg- ular convergence, especially appears large intermediate residual norm, which may lead to worse approximate solutions and slower convergence rate. In this paper, we present a new product-type method for solving complex non-Hermitian linear systems based on the bicon- jugate A-orthogonal residual （BiCOR） method, where one of the polynomials is a BiCOR polynomial, and the other is a BiCOR polynomial with the same degree corresponding to different initial residual. Numerical examples are given to illustrate the effectiveness of the proposed method.     

A PRIORI AND A POSTERIORI ERROR ESTIMATES OF A WEAKLY OVER-PENALIZED INTERIOR PENALTY METHOD FOR NON-SELF-ADJOINT AND INDEFINITE PROBLEMS

In this paper, we study a weakly over-penalized interior penalty method for non-self- adjoint and indefinite problems. An optimal a priori error estimate in the energy norm is derived. In addition, we introduce a residual-based a posteriori error estimator, which is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to validate our theoretical analysis.     

A TRULY GLOBALLY CONVERGENT FEASIBLE NEWTON-TYPE METHOD FOR MIXED COMPLEMENTARITY PROBLEMS

Typical solution methods for solving mixed complementarity problems either generatefeasible iterates but have to solve relatively complicated subproblems such as quadraticprograms or linear complementarity problems,or(those methods)have relatively simplesubproblems such as system of linear equations but possibly generate infeasible iterates.In this paper,we propose a new Newton-type method for solving monotone mixed com-plementarity problems,which ensures to generate feasible iterates,and only has to solve asystem of well-conditioned linear equations with reduced dimension per iteration.Withoutany regularity assumption,we prove that the whole sequence of iterates converges to a so-lution of the problem(truly globally convergent).Furthermore,under suitable conditions,the local superlinear rate of convergence is also established.     

Three-stage Stiffly Accurate Runge-Kutta Methods for Stiff Stochastic Differential Equations

In this paper we discuss diagonally implicit and semi-implicit methods based on the three-stage stiffly accurate Runge-Kutta methods for solving Stratonovich stochastic differential equations（SDEs）.Two methods,a three-stage stiffly accurate semi-implicit（SASI3） method and a three-stage stiffly accurate diagonally implicit （SADI3） method,are constructed in this paper.In particular,the truncated random variable is used in the implicit method.The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of stiff SDEs.     

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.     

A Note on Preconditioned Conjugate Gradients for Solving Singular Systems

The polynomial accelerations are very efficient methods for solving largesparse linear systems.In[2]we discussed the general polynomial accelerationmethods based on basic iteration methods for solving singular systems,especi-ally,the Chebyshev semi-iteration and the preconditioned conjugate gradientacceleration.In[4]E.F.Kaasschieter discussed the preconditioned conjugategradients for solving singular systems of which the coefficient matrices aresingular Stieljes matrices by using incomplete Cholesky decomposition as     

STRONG CONVERGENCE OF AN INERTIAL EXTRAGRADIENT METHOD WITH AN ADAPTIVE NONDECREASING STEP SIZE FOR SOLVING VARIATIONAL INEQUALITIES

In this work,we investigate a classical pseudomonotone and Lipschitz continuous variational inequality in the setting of Hilbert space,and present a projection-type approximation method for solving this problem.Our method requires only to compute one projection onto the feasible set per iteration and without any linesearch procedure or additional projections as well as does not need to the prior knowledge of the Lipschitz constant and the sequentially weakly continuity of the variational inequal...     

An Efficient Variant of the Restarted Shifted GMRES Method for Solving Shifted Linear Systems

We investigate the restart of the Restarted Shifted GMRES method for solving shifted linear systems.Recently the variant of the GMRES(m) method with the unfixed update has been proposed to improve the convergence of the GMRES(m) method for solving linear systems,and shown to have an efficient convergence property.In this paper,by applying the unfixed update to the Restarted Shifted GMRES method,we propose a variant of the Restarted Shifted GMRES method.We show a potentiality for efficient convergence within the variant by some numerical results.     

A REDUCED FE FORMULATION BASED ON POD METHOD FOR HYPERBOLIC EQUATIONS

A proper orthogonal decomposition(POD) method was successfully used in the reduced-order modeling of complex systems.In this paper,we extend the applications of POD method,namely,apply POD method to a classical finite element(FE) formulation for second-order hyperbolic equations with real practical applied background,establish a reduced FE formulation with lower dimensions and high enough accuracy,and provide the error estimates between the reduced FE solutions and the classical FE solutions and the implementation of algorithm for solving reduced FE formulation so as to provide scientific theoretic basis for service applications.Some numerical examples illustrate the fact that the results of numerical computation are consistent with theoretical conclusions.Moreover,it is shown that the reduced FE formulation based on POD method is feasible and efficient for solving FE formulation for second-order hyperbolic equations.