An achievement rate approach to linear programming problems with an interval objective function

In this paper, we focus on a treatment of a linear programming problem with an interval objective function. From the viewpoint of the achievement rate, a new solution concept, the maximin achievement rate solution, is proposed. Nice properties of this solution are shown: a maximin achievement rate solution is necessarily optimal when a necessarily optimal solution exists, and if not, then it is still a possibly optimal solution. An algorithm for a maximin achievement rate solution is proposed based on a relaxation procedure together with a simplex method. A numerical example is given to demonstrate the proposed solution algorithm.     

On fourth-order elliptic boundary value problems with nonmonotone nonlinear function

This paper is concerned with the fourth-order elliptic boundary value problems with nonmonotone nonlinear function. The existence and uniqueness of a solution is proven by the method of upper and lower solutions. A monotone iteration is developed so that the iteration sequence converges monotonically to a maximal solution or a minimal solution, depending on whether the initial iteration is an upper solution or a lower solution.     

Analytical solution for laterally loaded long piles based on Fourier–Laplace integral

Piles are frequently used to support lateral loads. Elastic solutions based on the Winkler foundation model are widely used to design laterally loaded piles at working load. This paper reports a simplified analytical solution for laterally loaded long piles in a soil with stiffness linearly increasing with depth. Based on a Fourier–Laplace integral, a power series solution for small depth and a Wentzel–Kramers–Brillouin (WKB) asymptotic solution for large depth are derived. By using this analytical solution, the deflection and bending moment profiles of a laterally loaded pile can be obtained through simple calculation. The proposed power series solution is exact for infinitely long piles. Numerical examples show that this solution agrees well with other existing methods on predicting the deflection and bending moment of laterally loaded piles. The WKB asymptotic solution developed in this study has never been introduced before. The simplified analytical solution obtained in this study provides a better approach for engineers to analyze the responses and design of laterally loaded long piles.     

Minimax regret solution to multiobjective linear programming problems with interval objective functions coefficients

The current paper focuses on a multiobjective linear programming problem with interval objective functions coefficients. Taking into account the minimax regret criterion, an attempt is being made to propose a new solution i.e. minimax regret solution. With respect to its properties, a minimax regret solution is necessarily ideal when a necessarily ideal solution exists; otherwise it is still considered a possibly weak efficient solution. In order to obtain a minimax regret solution, an algorithm based on a relaxation procedure is suggested. A numerical example demonstrates the validity and strengths of the proposed algorithm. Finally, two special cases are investigated: the minimax regret solution for fixed objective functions coefficients as well as the minimax regret solution with a reference point. Some of the characteristic features of both cases are highlighted thereafter.     

A simple characterization of solution sets of convex programs

By means of elementary arguments we first show that the gradient of the objective function of a convex program is constant on the solution set of the problem. Furthermore the solution set lies in an affine subspace orthogonal to this constant gradient, and is in fact in the intersection of this affine subspace with the feasible region. As a consequence we give a simple polyhedral characterization of the solution set of a convex quadratic program and that of a monotone linear complementarity problem. For these two problems we can also characterize a priori the boundedness of their solution sets without knowing any solution point. Finally we give an extension to non-smooth convex optimization by showing that the intersection of the subdifferentials of the objective function on the solution set is non-empty and equals the constant subdifferential of the objective function on the relative interior of the optimal solution set. In addition, the solution set lies in the intersection with the feasible region of an affine subspace orthogonal to some subgradient of the objective function at a relative interior point of the optimal solution set.     

The Cahn–Hilliard equation with time-dependent constraint

We propose an abstract variational inequality formulation of the Cahn–Hilliard equation with a time-dependent constraint. We introduce notions of strong and weak solutions, and prove that a strong solution, if it exists, is a weak solution, and that the existence of a unique weak solution holds under an appropriate time-dependence condition on the constraint. We also show that the weak solution is a strong solution under appropriate assumptions on the data. Our abstract results can be applied to various concrete problems.     

Two-Grid Finite Element Method with Crank-Nicolson Fully Discrete Scheme for the Time-Dependent Schrödinger Equation

In this paper, we study the Crank-Nicolson Galerkin finite element method and construct a two-grid algorithm for the general two-dimensional time-dependent Schrödinger equation. Firstly, we analyze the superconvergence error estimate of the finite element solution in $H^1$ norm by use of the elliptic projection operator. Secondly, we propose a fully discrete two-grid finite element algorithm with Crank-Nicolson scheme in time. With this method, the solution of the Schrödinger equation on a fine grid is reduced to the solution of original problem on a much coarser grid together with the solution of two Poisson equations on the fine grid. Finally, we also derive error estimates of the two-grid finite element solution with the exact solution in $H^1$ norm. It is shown that the solution of two-grid algorithm can achieve asymptotically optimal accuracy as long as mesh sizes satisfy $H = \mathcal{O}(h^{\frac{1}{2}})$.     

Discrete ill-posed least-squares problems with a solution norm constraint

Straightforward solution of discrete ill-posed least-squares problems with error-contaminated data does not, in general, give meaningful results, because propagated error destroys the computed solution. Error propagation can be reduced by imposing constraints on the computed solution. A commonly used constraint is the discrepancy principle, which bounds the norm of the computed solution when applied in conjunction with Tikhonov regularization. Another approach, which recently has received considerable attention, is to explicitly impose a constraint on the norm of the computed solution. For instance, the computed solution may be required to have the same Euclidean norm as the unknown solution of the error-free least-squares problem. We compare these approaches and discuss numerical methods for their implementation, among them a new implementation of the Arnoldi–Tikhonov method. Also solution methods which use both the discrepancy principle and a solution norm constraint are considered.     

Characterizations of optimal solution sets of convex infinite programs

In this paper, several Lagrange multiplier characterizations of the solution set of a convex infinite programming problem are given. Characterizations of solution sets of cone-constrained convex programs are derived as well. The procedure is then adopted to a semi-convex problem with convex constraints. For this problem, we present firstly a necessary and sufficient condition for optimality and secondly a characterization of its optimal solution set, based on a Lagrange multiplier associated with a given solution and on directional derivatives of the objective function.      

Bernstein series solution of a class of linear integro-differential equations with weakly singular kernel

In this study, a new collocation method based on the Bernstein polynomials is introduced for the approximate solution of a class of linear Volterra integro-differential equations with weakly singular kernel. If the exact solution is polynomial, then the exact solution can be obtained. If the exact solution is not a polynomial, then an accurate solution can be obtained with a combination of choice in the number of nodes and the number of digits in the solver. In addition, the method is presented with error and stability analysis.     

Convergence of a FEM and two-grid algorithms for elliptic problems on disjoint domains

In this paper, we analyze a FEM and two-grid FEM decoupling algorithms for elliptic problems on disjoint domains. First, we study the rate of convergence of the FEM and, in particular, we obtain a superconvergence result. Then with proposed algorithms, the solution of the multi-component domain problem (simple example — two disjoint rectangles) on a fine grid is reduced to the solution of the original problem on a much coarser grid together with solution of several problems (each on a single-component domain) on fine meshes. The advantage is the computational cost although the resulting solution still achieves asymptotically optimal accuracy. Numerical experiments demonstrate the efficiency of the algorithms.     

热传导方程小波解的点态收敛

本文主要考虑热传导方程uxx=ut,0≤x<1,t≥0;u(1,t)=g(t),其中边界条件g(t)为已知函数．此定解问题为一不适定问题,也就是说当边界条件有微小扰动时,将会引起解大的扰动．本文将利用多分辨率分析构造一小波解,且证明此解是适定的,并给出所定义小波解与定解问题的真正解在点态意义下的误差估计．     

SOME RECENT PROGRESS ON STOCHASTIC HEAT EQUATIONS

This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations(for example, stochastic heat equations)driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical(square integrable) solution(mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution; Feynman-Kac formula for the moments of the solution; and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.     

不通过Poisson公式解调和方程在球上的一类Dirichlet问题

得到了一个解调和方程在球上的一类Dirichlet问题的简单方法,即不通过Poisson公式而实际上只解一个Euler方程,从而较容易地求出其解.     

A two-grid discretization method for decoupling systems of partial differential equations

In this paper, we propose a two-grid finite element method for solving coupled partial differential equations, e.g., the Schrödinger-type equation. With this method, the solution of the coupled equations on a fine grid is reduced to the solution of coupled equations on a much coarser grid together with the solution of decoupled equations on the fine grid. It is shown, both theoretically and numerically, that the resulting solution still achieves asymptotically optimal accuracy.

     

Regularization of the Solution of the Cauchy Problem: The Quasi-Reversibility Method

Some regularization algorithm is proposed related to the problem of continuation of the wave field from the planar boundary into the half-plane. We consider a hyperbolic equation whose main part coincideswith the wave operator, whereas the lowest term contains a coefficient depending on the two spatial variables. The regularization algorithm is based on the quasi-reversibility method proposed by Lattes and Lions. We consider the solution of an auxiliary regularizing equation with a small parameter; the existence, the uniqueness, and the stability of the solution in the Cauchy data are proved. The convergence is substantiated of this solution to the exact solution as the small parameter vanishes. A solution of an auxiliary problem is constructed with the Cauchy data having some error. It is proved that, for a suitable choice of a small parameter, the approximate solution converges to the exact solution.     

(2+1)维AKNS方程的行波解研究(英文)

Based on the travelling wave method, a(2 + 1)-dimensional AKNS equation is considered. Elliptic solution and soliton solution are presented and it is shown that the soliton solution can be reduced from the elliptic solution. It also proves that the result is consistent with the soliton solution of simplify Hirota bilinear method by Wazwaz and illustrate the solution are right travelling wave solution.     

Cooperative games with coalition structures

Many game-theoretic solution notions have been defined or can be defined not only with reference to the all-player coalition, but also with reference to an arbitrary coalition structure. In this paper, theorems are established that connect a given solution notion, defined for a coalition structure ? with the same solution notion applied to appropriately defined games on each of the coalitions in ?. This is done for the kernel, nucleolus, bargaining set, value, core, and thevon Neumann-Morgenstern solution. It turns out that there is a single function that plays the central role in five out of the six solution notions in question, though each of these five notions is entirely different. This is an unusual instance of a game theoretic phenomenon that does not depend on a particular solution notion but holds across a wide class of such notions.     

Convergence of a FEM and two-grid algorithms for elliptic problems on disjoint domains

In this paper, we analyze a FEM and two-grid FEM decoupling algorithms for elliptic problems on disjoint domains. First, we study the rate of convergence of the FEM and, in particular, we obtain a superconvergence result. Then with proposed algorithms, the solution of the multi-component domain problem (simple example — two disjoint rectangles) on a fine grid is reduced to the solution of the original problem on a much coarser grid together with solution of several problems (each on a single-component domain) on fine meshes. The advantage is the computational cost although the resulting solution still achieves asymptotically optimal accuracy. Numerical experiments demonstrate the efficiency of the algorithms.     

Optimality conditions for proper efficient solutions of vector set-valued optimization

Based on the concept of an epiderivative for a set-valued map introduced in J. Nanchang Univ. 25 (2001) 122-130, in this paper, we present a few necessary and sufficient conditions for a Henig efficient solution, a globally proper efficient solution, a positive properly efficient solution, an f-efficient solution and a strongly efficient solution, respectively, to a vector set-valued optimization problem with constraints.