Estimating the Accuracy of a Method of Auxiliary Boundary Conditions in Solving an Inverse Boundary Value Problem for a Nonlinear Equation

An inverse boundary value problem for a nonlinear parabolic equation is considered. Two-sided estimates for the norms of values of a nonlinear operator in terms of those of a corresponding linear operator are obtained.On this basis, two-sided estimates for the modulus of continuity of a nonlinear inverse problem in terms of that of a corresponding linear problem are obtained. A method of auxiliary boundary conditions is used to construct stable approximate solutions to the nonlinear inverse problem. An accurate (to an order) error estimate for the method of auxiliary boundary conditions is obtained on a uniform regularization class.     

Numerical treatment of singularly perturbed two point boundary value problems using initial-value method

In this paper, we describe an initial-value method for linear and nonlinear singularly perturbed boundary value problems in the interval [p,q]. For linear problems, the required approximate solution is obtained by solving the reduced problem and one initial-value problems directly deduced from the given problem. For nonlinear problems the original second-order nonlinear problem is linearized by using quasilinearization method. Then this linear problem is solved as previous method. The present method has been implemented on several linear and non-linear examples which approximate the exact solution. We also present the approximate and exact solutions graphically.     

粘弹性杆的某些问题

本文利用[3]所提出的方法讨论了粘弹性杆的几种线性和非线性问题.对于线性情形,[1]的结果在这里得到较为初等和简明的印证,同时还论述了有阻尼项的线性问题和有非线性强迫作用项Pu3的幂非线性问题,从而建立了一些新的结果.     

Fast stratified flow over several obstacles, including wings

Fast two-dimensional stratified flow over several obstaclesis considered, in particular over thin wings. The problem isreduced to a linear boundary-value problem by a nonlinear substitution.The linear problem is studied by potential theory. The solutionof the nonlinear problem is obtained.     

A trust region algorithm for solving bilevel programming problems

In this paper, we present a new trust region algorithm for a nonlinear bilevel programming problem by solving a series of its linear or quadratic approximation subproblems. For the nonlinear bilevel programming problem in which the lower level programming problem is a strongly convex programming problem with linear constraints, we show that each accumulation point of the iterative sequence produced by this algorithm is a stationary point of the bilevel programming problem.     

Column enumeration based decomposition techniques for a class of non-convex MINLP problems

We propose a decomposition algorithm for a special class of nonconvex mixed integer nonlinear programming problems which have an assignment constraint. If the assignment decisions are decoupled from the remaining constraints of the optimization problem, we propose to use a column enumeration approach. The master problem is a partitioning problem whose objective function coefficients are computed via subproblems. These problems can be linear, mixed integer linear, (non-)convex nonlinear, or mixed integer nonlinear. However, the important property of the subproblems is that we can compute their exact global optimum quickly. The proposed technique will be illustrated solving a cutting problem with optimum nonlinear programming subproblems.     

Some solvability problems for the Boltzmann equation in the framework of the Shakhov model

We consider the nonlinear Boltzmann equation in the framework of the Shakhov model for the classical problem of gas flow in a plane layer. The problem reduces to a system of nonlinear integral equations. The nonlinearity of the studied system can be partially simplified by passing to a new argument depending on the solution of the problem itself. We prove the existence theorem for a unique solution of the linear system and the existence theorem for a positive solution of the nonlinear Urysohn equation. We determine the temperature jumps on the lower and upper walls in the linear and nonlinear cases, and it turns out that the difference between them is rather small.     

A nonlinear boundary eigenvalue problem for TM-polarized electromagnetic waves in a nonlinear layer

We consider the propagation of TM-polarized electromagnetic waves in a nonlinear dielectric layer located between two linear media. The nonlinearity in the layer is described by the Kerr law. We reduce the problem to a nonlinear boundary eigenvalue problem for a system of ordinary differential equations. We obtain a dispersion relation and a first approximation for eigenvalues of the problem. We compare the results with those obtained for the case of a linear medium in the layer.     

Some sufficient conditions for the observability of nonlinear systems

Some sufficient conditions are presented for the observability of systems described by nonlinear, ordinary differential equations with linear observations. The point which the authors especially emphasize is the fact that the results obtained are global in some sense. In the beginning, the observability problem is reduced to the uniqueness problem of solutions of some nonlinear integral equations for general nonlinear systems with linear observations. Then, for some restricted systems, a simple sufficient condition is derived. The relation between global and local observability for nonlinear systems is also considered.The authors wish to thank their colleagues in Nagoya University.     

Simple solution methods for separable mixed linear and quadratic knapsack problem

Quadratic knapsack problem has a central role in integer and nonlinear optimization, which has been intensively studied due to its immediate applications in many fields and theoretical reasons. Although quadratic knapsack problem can be solved using traditional nonlinear optimization methods, specialized algorithms are much faster and more reliable than the nonlinear programming solvers. In this paper, we study a mixed linear and quadratic knapsack with a convex separable objective function subject to a single linear constraint and box constraints. We investigate the structural properties of the studied problem, and develop a simple method for solving the continuous version of the problem based on bi-section search, and then we present heuristics for solving the integer version of the problem. Numerical experiments are conducted to show the effectiveness of the proposed solution methods by comparing our methods with some state of the art linear and quadratic convex solvers.     

On a singular two dimensional nonlinear evolution equation with nonlocal conditions

This paper deals with a nonclassical initial boundary value problem for a two dimensional parabolic equation with Bessel operator. We prove the existence and uniqueness of the weak solution of the given nonlinear problem. We start by solving the associated linear problem. After writing this latter in its operator form, we establish an a priori bound from which we deduce the uniqueness of the strong solution. For the solvability of the associated linear problem, we prove that the range of the operator generated by the considered problem is dense. On the basis of the obtained results of the linear problem, we apply an iterative process to establish the existence and uniqueness of the nonlinear problem.     

线性比式和规划问题的输出空间分支定界算法

本文旨在针对线性比式和规划这一NP-Hard非线性规划问题提出新的全局优化算法.首先,通过引入p个辅助变量把原问题等价的转化为一个非线性规划问题,这个非线性规划问题的目标函数是乘积和的形式并给原问题增加了p个新的非线性约束,再通过构造凸凹包络的技巧对等价问题的目标函数和约束条件进行相应的线性放缩,构成等价问题的一个下界线性松弛规划问题,从而提出了一个求解原问题的分支定界算法,并证明了算法的收敛性.最后,通过数值结果比较表明所提出的算法是可行有效的.     

A full multigrid method for nonlinear eigenvalue problems

We introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of the nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and nonlinear eigenvalue problems on the coarsest finite element space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as the linear problem solver. We prove that the computational work of this new scheme is truly optimal, the same as solving the linear corresponding boundary value problem. In this case, this type of iteration scheme certainly improves the overfull efficiency of solving nonlinear eigenvalue problems. Some numerical experiments are presented to validate the efficiency of the new method.     

Ranking in quadratic integer programming problems

The present paper develops an algorithm for ranking the integer feasible solutions of a quadratic integer programming (QIP) problem. A linear integer programming (LIP) problem is constructed which provides bounds on the values of the objective function of the quadratic problem. The integer feasible solutions of this related integer linear programming problem are systematically scanned to rank the integer feasible solutions of the quadratic problem in non-decreasing order of the objective function values. The ranking in the QIP problem is useful in solving a nonlinear integer programming problem in which some other complicated nonlinear restrictions are imposed which cannot be included in the simple linear constraints of QIP, the objective function being still quadratic.     

Fast nonlinear stratified flow over an obstacle

Fast 2-D stratified flow over a hard obstacle is considered. The problem is reduced to a linear boundary value problem by a nonlinear substitution. The linear problem is studied by potential theory. The solution of the nonlinear problem is justified by some estimates.     

一类非线性抛物方程Cauchy问题解的存在性条件

研究了带源项的非线性抛物方程Cauchy问题解存在的必要条件以及解所应具有的性质,通过把文献中的线性算子推广到形式较一般的带源项的非线性抛物算子,利用其中处理线性问题的方法来处理非线性问题．     

A new approach for asymptotic stability system of the nonlinear ordinary differential equations

In this paper, we use measure theory for considering asymptotically stable of an autonomous system [1] of first order nonlinear ordinary differential equations(ODE’s). First, we define a nonlinear infinite-horizon optimal control problem related to the ODE. Then, by a suitable change of variable, we transform the problem to a finite-horizon nonlinear optimal control problem. Then, the problem is modified into one consisting of the minimization of a linear functional over a set of positive Radon measures. The optimal measure is approximated by a finite combination of atomic measures and the problem converted to a finite-dimensional linear programming problem. The solution to this linear programming problem is used to find a piecewise-constant control, and by using the approximated control signals, we obtain the approximate trajectories and the error functional related to it. Finally the approximated trajectories and error functional is used to for considering asymptotically stable of the original problem.     

Design and optimal tuning of nonlinear PI compensators

In this paper, linear time-invariant single-input single-output (SISO) systems that are stabilizable by linear proportional and integral (PI) compensators are considered. For such systems, a five-parameter nonlinear PI compensator is proposed. The parameters of the proposed compensator are tuned by solving an optimization problem. The optimization problem always has a solution.Additionally, a general nonlinear PI compensator is proposed and is approximated by easy-to-compute compensators, for instance, a six-parameter nonlinear PI compensator. The parameters of the approximate compensators are tuned to satisfy an optimality condition. The superiority of the proposed nonlinear PI compensators over linear PI compensators is discussed and is demonstrated for two feedback systems.     

Two-grid finite volume element method for linear and nonlinear elliptic problems

Two-grid finite volume element discretization techniques, based on two linear conforming finite element spaces on one coarse and one fine grid, are presented for the two-dimensional second-order non-selfadjoint and indefinite linear elliptic problems and the two-dimensional second-order nonlinear elliptic problems. With the proposed techniques, solving the non-selfadjoint and indefinite elliptic problem on the fine space is reduced into solving a symmetric and positive definite elliptic problem on the fine space and solving the non-selfadjoint and indefinite elliptic problem on a much smaller space; solving a nonlinear elliptic problem on the fine space is reduced into solving a linear problem on the fine space and solving the nonlinear elliptic problem on a much smaller space. Convergence estimates are derived to justify the efficiency of the proposed two-grid algorithms. A set of numerical examples are presented to confirm the estimates. The work is supported by the National Natural Science Foundation of China (Grant No: 10601045).