Error estimates for shooting methods in two-point boundary value problems for second-order equations

This paper is concerned with a procedure for estimating the global discretization error arising when a boundary value problem for a system of second order differential equations is solved by the simple shooting method, without transforming the original problem in an equivalent first order problem. Expressions of the global discretization error are derived for both linear and nonlinear boundary value problems, which reduce the error estimation for a boundary value problem to that for an initial value problem of same dimension. The procedure extends to second order equations a technique for global error estimation given elsewhere for first order equations. As a practical result the accuracy of the estimates for a second order problem is increased compared with the estimates for the equivalent first order problem.     

A Dirichlet problem for polyharmonic functions

In this article, the Dirichlet problem of polyharmonic functions is considered. As well the explicit expression of the unique solution to the simple Dirichlet problem for polyharmonic functions is obtained by using the decomposition of polyharmonic functions and turning the problem into an equivalent Riemann boundary value problem for polyanalytic functions, as the approach to find the kernel functions of the solution for the general Dirichlet problem is given. Project supported by NNSF of China.     

A Convergence Proof for Constrained Finite Element Solutions

A convergence proof is given for the finite-element solutionof the infinite dimensional quadratic programming problem ofminimizing a quadratic functional subject to linear constraints.The proof for the unconstrained problem is briefly reviewed,and then extended to the constrained case. Only the first partof the proof is given, in which necessary conditions for convergenceare derived for the specific problem and its finite-elementapproximation. The final step of proving that any problem doesobey these conditions will depend on the specific problem, butit is shown that if the finite element formulation is pointwiseconvergent and the unconstrained problem is convergent, thenso too will be the constrained problem.     

Continuous relaxation for discrete DC programming

Discrete DC programming with convex extensible functions is studied. A natural approach for this problem is a continuous relaxation that extends the problem to a continuous domain and applies the algorithm in continuous DC programming. By employing a special form of continuous relaxation, which is named “lin-vex extension,” the produced optimal solution of the extended continuous relaxation coincides with the solution of the original discrete problem. The proposed method is demonstrated for the degree-concentrated spanning tree problem, the unfair flow problem, and the correlated knapsack problem.     

An Exact Solution Method for the MTSP

A new mathematical model for the multi-travelling salesman problem (MTSP) is presented. The MTSP formulation is modified, and a branch-and-bound algorithm for solving this problem exactly is developed. The significance of this procedure is that it does not need to transform the problem into a single travelling salesman problem, which has been the case in the dominant algorithms for solving the above problem. Moreover, computational experience has shown that for a fixed number of cities to be visited, the time required to solve the problem decreases markedly as the number of salesmen increases.     

Appropriate problems for learning and for performing—an issue for teacher training

Selecting, modifying or creating appropriate problems for mathematics class has become an activity of increaing importance in the professional development of German mathematics teachers. But rather than asking in general: “What is a good problem?” there should be a stronger emphasis on considering the specific goal of a problem, e.g.: “What are the ingredients that make a problem appropriate for initiating a learning process” or “What are the characteristics that make a problem appropriate for its use in a central test?” We propose a guiding scheme for teachers that turns out to be especially helpful, since the newly introduced orientation on outcome standards a) leads to a critical predominance of test items and b) expects teachers to design adequate problems for specific learning processes (e.g. problem solving, reasoning and modelling activities).     

Inverse problems for quasi-variational inequalities

In this short note, our aim is to investigate the inverse problem of parameter identification in quasi-variational inequalities. We develop an abstract nonsmooth regularization approach that subsumes the total variation regularization and permits the identification of discontinuous parameters. We study the inverse problem in an optimization setting using the output-least squares formulation. We prove the existence of a global minimizer and give convergence results for the considered optimization problem. We also discretize the identification problem for quasi-variational inequalities and provide the convergence analysis for the discrete problem. We give an application to the gradient obstacle problem.     

Fixed Point Methods for the Complementarity Problem

This paper is concerned with iterative procedures for the monotone complementarity problem. Our iterative methods consist of finding fixed points of appropriate continuous maps. In the case of the linear complementarity problem, it is shown that the problem is solvable if and only if the sequence of iterates is bounded in which case summability methods are used to find a solution of the problem. This procedure is then used to find a solution of the nonlinear complementarity problem satisfying certain regularity conditions for which the problem has a nonempty bounded solution set.     

A new stabilized finite element method for optimal control for a Ladyzhenskaya model for unsteady flows

This article considers the time‐dependent optimal control problem of tracking the velocity for the viscous incompressible flows which is governed by a Ladyzhenskaya equations with distributed control. The existence of the optimal solution is shown and the first‐order optimality condition is established. The semidiscrete‐in‐time approximation of the optimal control problem is also given. The spatial discretization of the optimal control problem is accomplished by using a new stabilized finite element method which does not need a stabilization parameter or calculation of high order derivatives. Finally a gradient algorithm for the fully discrete optimal control problem is effectively proposed and implemented with some numerical examples. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 263–287, 2012     

Optimal control problems for wave equations

The problem on minimizing a quadratic functional on trajectories of the wave equation is considered. We assume that the density of external forces is a control function. A control problem for a partial differential equation is reduced to a control problem for a countable system of ordinary differential equations by use of the Fourier method. The controllability problem for this countable system is considered. Conditions of the noncontrollability for some wave equations were obtained.     

Heuristic Solution Methods for the Multilevel Generalized Assignment Problem

The multilevel generalized assignment problem is a problem of assigning agents to tasks where the agents can perform tasks at more than one efficiency level. A profit is associated with each assignment and the objective of the problem is profit maximization. Two heuristic solution methods are presented for the problem. The heuristics are developed from solution methods for the generalized assignment problem. One method uses a regret minimization approach whilst the other method uses a repair approach on a relaxation of the problem. The heuristics are able to solve moderately large instances of the problem rapidly and effectively. Procedures for deriving an upper bound on the solution of the problem are also described. On larger and harder instances of the problem one heuristic is particularly effective.     

大规模系统的全局优化

本文讨论了可分非凸大规模系统的全局优化控制问题 .提出了一种 3级递阶优化算法 .该算法首先把原问题转化为可分的多目标优化问题 ,然后凸化非劣前沿 ,再从非劣解集中挑出原问题的全局最优解 .建立了算法的理论基础 ,证明了算法的收敛性 .仿真结果表明算法是有效的 .     

Application of fast automatic differentiation for solving the inverse coefficient problem for the heat equation

The problem of determining the thermal conductivity coefficient that depends on temperature is studied. The consideration is based on the initial-boundary value problem for the one-dimensional unsteady heat equation. The mean-root-square deviation of the temperature distribution field and the heat flux from the experimental data on the left boundary of the domain is used as the objective functional. An analytical expression for the gradient of the objective functional is obtained. An algorithm for the numerical solution of the problem based on the modern fast automatic differentiation technique is proposed. Examples of solving the problem are discussed.     

Asymptotic Initial-Value Method for Singularly-Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations

A computational method is presented to solve a class of nonturning-point singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a small parameter multiplying the highest derivative, subject to Dirichlet-type boundary conditions. In this method, first we construct a zeroth order asymptotic expansion for the solution of the given boundary-value problem. Then, this problem is integrated to get an equivalent initial-value problem for first-order ordinary differential equations. This initial-value problem is solved by either a classical method or a fitted operator method after approximating some of the terms in the differential equations by using the zeroth order asymptotic expansion. This method is effective and easy to implement. An error estimate is derived for the numerical solution. Examples are given to illustrate the method.     

Computational results for extensions in quasilinearization techniques for optimal control

In a companion paper (Ref. 1), several extensions in quasilinearization were presented. These results are studied computationally in this paper for two problems: the brachistochrone problem and the reentry vehicle problem. For the brachistochrone free-time problem, it is shown that much more rapid convergence is obtained than that presented in previous literature (Ref. 2). It is also shown that, if the techniques of the companion paper are used, the normal region of convergence is extended significantly. Similar results are obtained for the reentry vehicle problem.The research presented in this paper was supported in part by the Air Force Office of Scientific Research, Grant No. AF-AFOSR-699-67.     

Multiobjective duality for convex ratios

In this paper we present a duality approach for a multiobjective fractional programming problem. The components of the vector objective function are particular ratios involving the square of a convex function and a positive concave function. Applying the Fenchel-Rockafellar duality theory for a scalar optimization problem associated to the multiobjective primal, a dual problem is derived. This scalar dual problem is formulated in terms of conjugate functions and its structure gives an idea about how to construct a multiobjective dual problem in a natural way. Weak and strong duality assertions are presented.     

Method for Constructing Piecewise Quadratic Value Functions in a Control Problem for a Switched System

The problem of constructing internal approximations to solvability sets and the control synthesis problem for a piecewise linear system with control parameters and disturbances (uncertainties) are solved. The solution is based on the comparison principle and piecewise quadratic value functions of a special form. Relations defining such functions and, in particular, “continuous binding conditions” for the functions and their first derivatives are obtained. The results are used to construct numerical methods for solving the control synthesis problem for the class of switched systems under study. An example of approximate solution of the control synthesis problem in a target control problem for a nonlinear mathematical model of a pendulum with a flywheel is considered.     

On the mode-change problem for random measures

The classical change-point problem in modern terms, i.e., the mode-change problem, is stated for sufficiently general set-indexed random processes, namely for random measures. A method is shown for solving this problem both in the general form and for the intensity of compound Poisson random measures. The results obtained are novel for the change-point problem, too.     

Solution procedures for three-dimensional eddy current problems

The problem under consideration is that of the scattering of time periodic electromagnetic fields by metallic obstacles. A common approximation here is that in which the metal is assumed to have infinite conductivity. The resulting problem, called the perfect conductor problem, involves solving Maxwell's equations in the region exterior to the obstacle with the tangential component of the electric field zero on the obstacle surface. In the interface problem different sets of Maxwell equations must be solved in the obstacle and outside while the tangential components of both electric and magnetic fields are continuous across the obstacle surface. Solution procedures for this problem are given. There is an exact integral equation procedure for the interface problem and an asymptotic procedure for large conductivity. Both are based on a new integral equation procedure for the perfect conductor problem. The asymptotic procedure gives an approximate solution by solving a sequence of problems analogous to the one for perfect conductors.