On structure-oriented hybrid two-stage iteration methods for the large and sparse blocked system of linear equations

In this paper, we first present a class of structure-oriented hybrid two-stage iteration methods for solving the large and sparse blocked system of linear equations, as well as the saddle point problem as a special case. And the new methods converge to the solution under suitable restrictions, for instance, when the coefficient matrix is positive stable matrix generally. Numerical experiments for a model generalized saddle point problem are given, and the results show that our new methods are feasible and efficient, and converge faster than the Classical Uzawa Method.     

Universal Iterative Methods for Computing Generalized Inverses

In this paper we construct a few iterative processes for computing {1,2} inverses of a linear bounded operator, based on the hyper-power iterative method or the Neumann-type expansion. Under suitable conditions these methods converge to the {1,2,3} or {1,2,4} inverses. Also, we specify conditions when the iterative processes converge to the Moore-Penrose inverse, the weighted Moore-Penrose inverse or to the group inverse. A few error estimates are derived. The advantages of the introduced methods over Tanabe's method [16] for computing reflexive generalized inverses are also investigated.     

Dual interior point methods for linear semidefinite programming problems

Dual interior point methods for solving linear semidefinite programming problems are proposed. These methods are an extension of dual barrier-projection methods for linear programs. It is shown that the proposed methods converge locally at a linear rate provided that the solutions to the primal and dual problems are nondegenerate.     

A unified study of continuous and discontinuous Galerkin methods

A unified study is presented in this paper for the design and analysis of different finite element methods(FEMs), including conforming and nonconforming FEMs, mixed FEMs, hybrid FEMs, discontinuous Galerkin(DG) methods, hybrid discontinuous Galerkin(HDG) methods and weak Galerkin(WG) methods.Both HDG and WG are shown to admit inf-sup conditions that hold uniformly with respect to both mesh and penalization parameters. In addition, by taking the limit of the stabilization parameters, a WG method is shown to converge to a mixed method whereas an HDG method is shown to converge to a primal method. Furthermore,a special class of DG methods, known as the mixed DG methods, is presented to fill a gap revealed in the unified framework.     

Certain first-order iterative methods for mixed variational inequalities in a Hilbert space

For a certain class of mixed variational inequalities in a Hilbert space, first-order iterative methods of the proximal type are constructed, and sufficient conditions for them to converge strongly to a solution of the original problem are obtained.     

The convergence of shooting methods

Shooting methods for nonlinear boundary value problems are examined. It is shown that the methods converge whenever the problem is well posed in the sense that the solution to be computed is isolated.     

A numerical method for detecting singular minimizers

Summary A numerical method for computing minimizers in one-dimensional problems of the calculus of variations is described. Such minimizers may have unbounded derivatives, even when the integrand is smooth and regular. In such cases, because of the Lavrentiev phenomenon, standard finite element methods may fail to converge to a minimizer. The scheme proposed is shown to converge to an absolute minimizer and is tested on an example. The effect of quadrature is analyzed. The implications for higher-dimensional problems, and in particular for fracture in nonlinear elasticity, are discussed.     

Block-iterative methods for consistent and inconsistent linear equations

Summary We shall in this paper consider the problem of computing a generalized solution of a given linear system of equations. The matrix will be partitioned by blocks of rows or blocks of columns. The generalized inverses of the blocks are then used as data to Jacobi- and SOR-types of iterative schemes. It is shown that the methods based on partitioning by rows converge towards the minimum norm solution of a consistent linear system. The column methods converge towards a least squares solution of a given system. For the case with two blocks explicit expressions for the optimal values of the iteration parameters are obtained. Finally an application is given to the linear system that arises from reconstruction of a two-dimensional object by its one-dimensional projections.     

Stability of backward Euler multirate methods and convergence of waveform relaxation

For a large class of traditional backward Euler multirate methods we show that stability is preserved when the methods are applied to certain stable (but not necessarily monotonic) non-linear systems. Methods which utilize waveform relaxation sweeps are shown to be stable and converge for certain monotonic systems.     

Nonlinear extended variational inequalities without differentiability: Applications and solution methods

We consider a class of nonlinear problems which is intermediate between equilibrium and variational inequality ones. Several classes of applications of such problems are described. Iterative methods are proposed for finding a solution. The methods are utilized without differentiability properties of the mappings and converge to a solution under weakened monotonicity type assumptions.     

Some computational methods for systems of nonlinear equations and systems of polynomial equations

This paper gives a brief survey and assessment of computational methods for finding solutions to systems of nonlinear equations and systems of polynomial equations. Starting from methods which converge locally and which find one solution, we progress to methods which are globally convergent and find an a priori determinable number of solutions. We will concentrate on simplicial algorithms and homotopy methods. Enhancements of published methods are included and further developments are discussed.     

Banach空间中非扩张映象的黏性逼近方法

借助Banach空间中非扩张映象的黏性逼近方法,得出了非扩张映象迭代序列收敛于其不动点的充分必要条件．本文结果推广和改进了一些人的最新结果．     

Extensions of block-projections methods with relaxation parameters to inconsistent and rank-deficient least-squares problems

There exist many classes of block-projections algorithms for approximating solutions of linear least-squares problems. Generally, these methods generate sequences convergent to the minimal norm least-squares solution only for consistent problems. In the inconsistent case, which usually appears in practice because of some approximations or measurements, these sequences do no longer converge to a least-squares solution or they converge to the minimal norm solution of a “perturbed” problem. In the present paper, we overcome this difficulty by constructing extensions for almost all the above classes of block-projections methods. We prove that the sequences generated with these extensions always converge to a least-squares solution and, with a suitable initial approximation, to the minimal norm solution of the problem. Numerical experiments, described in the last section of the paper, confirm the theoretical results obtained.     

EXPECTATION STABILITY OF SECOND-ORDER WEAK NUMERICAL METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS

In recent years, many numerical methods for solving stochastic differential equations have been developed. Some of these methods converge in the weak sense and some others converge in the mean square sense. One of the important features of numerical methods is their stability behavior. In this paper, we focus our attention on stability in expectation (e. stability) of numerical methods of second-order accuracy in the weak sense. The region of e. stability for these methods will be discussed. The possibility of enlarging regions of e. stability will be described. Some numerical examples will be discussed to support the theoretical study.     

一般混合似变分不等式的隐式迭代算法

对一般混合似变分不等式的若干隐式迭代算法进行了研究;利用一般混合似变分不等式与不动点问题和预解方程的等价关系,采用分裂技巧和自适应迭代技巧结合,提出了一个求解一般混合似变分不等式的新的隐式迭代算法;并证明了该算法在算子T是g-单调连续的条件下收敛．     

ERROR ANALYSIS OF NONLINEAR GALERKIN METHODS FOR KURAMOTO-SIVASHINSKY EQUATIONS

Nonlinear Galerkin methods are new schemes for integrating dissipative systems:In the present paper, we obtain the estimates to the rate of convergence of such methods for Kuramoto-Sivashinsky equations. In particular, by an illustrative example, we show that nonlinear Galerkin methods converge faster than the usual Galerkin method.     

An efficient and accurate lattice for pricing derivatives under a jump-diffusion process

Derivatives are popular financial instruments whose values depend on other more fundamental financial assets (called the underlying assets). As they play essential roles in financial markets, evaluating them efficiently and accurately is critical. Most derivatives have no simple valuation formulas; as a result, they must be priced by numerical methods such as lattice methods. In a lattice, the prices of the derivatives converge to theoretical values when the number of time steps increases. Unfortunately, the nonlinearity error introduced by the nonlinearity of the option value function may cause the pricing results to converge slowly or even oscillate significantly. The lognormal diffusion process, which has been widely used to model the underlying asset’s price dynamics, does not capture the empirical findings satisfactorily. Therefore, many alternative processes have been proposed, and a very popular one is the jump-diffusion process. This paper proposes an accurate and efficient lattice for the jump-diffusion process. Our lattice is accurate because its structure can suit the derivatives’ specifications so that the pricing results converge smoothly. To our knowledge, no other lattices for the jump-diffusion process have successfully solved the oscillation problem. In addition, the time complexity of our lattice is lower than those of existing lattice methods by at least half an order. Numerous numerical calculations confirm the superior performance of our lattice to existing methods in terms of accuracy, speed, and generality.     

Approximate Methods for Generalized Inverses of Operators in Banach Spaces

In this paper, we present the necessary and sufficient condition of convergence of several iterative methods for computing the generalized inverses of operators in Banach spaces. It is proved that the iterative methods converge to the generalized inverse of an Operator in Banach spaces if and only if these conditions are satisfied.     

Two fundamental convergence theorems for nonlinear conjugate gradient methods and their applications

1. IntroductionWe consider the global convergence of conjugate gradient methods for the unconstrainednonlinear optimization problemadn f(x),where f: Re - RI is continuously dtherelltiable and its gradiellt is denoted by g. Weconsider only the cajse where the methods are implemented without regular restarts. Theiterative formula is given byXk 1 = Xk Akdk, (1'1).and the seaxch direction da is defined bywhere gb is a scalar, ^k is a stenlength, and gb denotes g(xk).The best-known formulas fo…     

Revised DEMATEL: Resolving the Infeasibility of DEMATEL

Decision Making Trial and Evaluation Laboratory (DEMATEL) has been applied in many situations, such as marketing strategies, control systems, safety problems, developing the competencies of global managers and group decision making. It has been incorporated into other methods such as Analytical Network Process (ANP), Multiple Criteria Decision Making (MCDM), fuzzy set theory, etc., to vitalize these traditional methods and explore new applications for the hybrid methods. DEMATEL models the influences of components of a system with an initial direct relation matrix. Influences of components can ripple transitively to other components, which is modeled by raising the initial direct relation matrix to powers. The total influence is computed by summing up matrices of all powers based on the assumption that the matrix raising to the power of infinity would converge to zero. The current paper shows that raising the initial relation matrix to the power of infinity may not converge to zero and hence total influence may not converge. The current paper also shows that our revised DEMATEL guarantees that the initial direct-relation matrix to infinite power will converge to zero and the total influence can be obtained accordingly. The newly developed approach is illustrated with numerical examples.