Lebesgue piecewise affine approximation of nonlinear systems

This paper addresses a piecewise affine (PWA) approximation problem, i.e., a problem of finding a PWA system model which approximates a given nonlinear system. First, we propose a new class of PWA systems, called the Lebesgue PWA approximation systems, as a model to approximate nonlinear systems. Next, we derive an error bound of the PWA approximation model, and provide a technique for constructing the approximation model with specified accuracy. Finally, the proposed method is applied to a gene regulatory network with nonlinear dynamics, which shows that the method is a useful approximation tool.     

CVaR-constrained stochastic programming reformulation for stochastic nonlinear complementarity problems

We reformulate a stochastic nonlinear complementarity problem as a stochastic programming problem which minimizes an expected residual defined by a restricted NCP function with nonnegative constraints and CVaR constraints which guarantee the stochastic nonlinear function being nonnegative with a high probability. By applying smoothing technique and penalty method, we propose a penalized smoothing sample average approximation algorithm to solve the CVaR-constrained stochastic programming. We show that the optimal solution of the penalized smoothing sample average approximation problem converges to the solution of the corresponding nonsmooth CVaR-constrained stochastic programming problem almost surely. Finally, we report some preliminary numerical test results.     

Approximation for Navier-Stokes equations around a rotating obstacle

This work presents an approximation method for Navier-Stokes equations around a rotating obstacle. The detail of this method is that the exterior domain is truncated into a bounded domain and a new exterior domain by introducing a large ball. The approximation problem is composed of the nonlinear problem in the bounded domain and the linear problem in the new exterior domain. We derive the approximation error between the solutions of Navier-Stokes equations and the approximation problem.     

非线性l1问题的光滑近似算法

为非线性l1问题的求解构造了光滑逼近函数.首先将非线性l1问题转化为等价的不可微优化问题;其次通过两步提出光滑逼近函数的一般性构造方法;最后进行了数值仿真.文中介绍了光滑逼近函数的有关性质,指出相关文献已有的光滑函数方法是本文的特例,并证明了方法的收敛性及有效性.     

The numerical solution of a periodic parabolic problem subject to a nonlinear boundary condition

The numerical solution by finite differences of a periodic parabolic problem subject to a nonlinear boundary condition is considered. It is shown that Newton's method can be used to solve the nonlinear equations provided a suitable initial approximation is known, and a method for constructing this first approximation is given.     

Exact and approximation algorithms for a soft rectangle packing problem

We consider the problem of finding an arrangement of rectangles with given areas that minimizes the total length of all inner and outer border lines. We present a polynomial time approximation algorithm and derive an upper bound estimation on its approximation ratio. Furthermore, we give a formulation of the problem as mixed-integer nonlinear program and show that it can be approximatively reformulated as linear mixed-integer program. On a test set of problem instances, we compare our approximation algorithm with another one from the literature. Using a standard numerical mixed-integer linear solver, we show that adding the solutions from the approximation algorithm as advanced starter helps to reduce the overall solution time for proven global optimality, or gives better primal and dual bounds if a certain time-limit is reached before.     

A NEW SMOOTHING EQUATIONS APPROACH TO THE NONLINEAR COMPLEMENTARITY PROBLEMS

The nonlinear complementarity problem can be reformulated as a nonsmooth equation. In this paper we propose a new smoothing Newton algorithm for the solution of the nonlinear complementarity problem by constructing a new smoothing approximation function. Global and local superlinear convergence results of the algorithm are obtained under suitable conditions. Numerical experiments confirm the good theoretical properties of the algorithm.     

非线性扰动Klein-Gordon方程初值问题的渐近理论

在二维空间中研究一类非线性扰动Klein-Gordon方程初值问题解的渐近理论. 首先利用压缩映象原理,结合一些先验估计式及Bessel函数的收敛性,根据Klein-Gordon方程初值问题的等价积分方程,在二次连续可微空间中得到了初值问题解的适定性;其次,利用扰动方法构造了初值问题的形式近似解,并得到了该形式近似解的渐近合理性;最后给出了所得渐近理论的一个应用,用渐近近似定理分析了一个具体的非线性Klein-Gordon方程初值问题解的渐近近似程度．     

Nonlinear Methods of Approximation

Abstract. Our main interest in this paper is nonlinear approximation. The basic idea behind nonlinear approximation is that the elements used in the approximation do not come from a fixed linear space but are allowed to depend on the function being approximated. While the scope of this paper is mostly theoretical, we should note that this form of approximation appears in many numerical applications such as adaptive PDE solvers, compression of images and signals, statistical classification, and so on. The standard problem in this regard is the problem of m -term approximation where one fixes a basis and looks to approximate a target function by a linear combination of m terms of the basis. When the basis is a wavelet basis or a basis of other waveforms, then this type of approximation is the starting point for compression algorithms. We are interested in the quantitative aspects of this type of approximation. Namely, we want to understand the properties (usually smoothness) of the function which govern its rate of approximation in some given norm (or metric). We are also interested in stable algorithms for finding good or near best approximations using m terms. Some of our earlier work has introduced and analyzed such algorithms. More recently, there has emerged another more complicated form of nonlinear approximation which we call highly nonlinear approximation. It takes many forms but has the basic ingredient that a basis is replaced by a larger system of functions that is usually redundant. Some types of approximation that fall into this general category are mathematical frames, adaptive pursuit (or greedy algorithms), and adaptive basis selection. Redundancy on the one hand offers much promise for greater efficiency in terms of approximation rate, but on the other hand gives rise to highly nontrivial theoretical and practical problems. With this motivation, our recent work and the current activity focuses on nonlinear approximation both in the classical form of m -term approximation (where several important problems remain unsolved) and in the form of highly nonlinear approximation where a theory is only now emerging.     

Analytical substrate flux approximation for the Monod boundary value problem

We present an analytical approximation for the diffusive flux of a substrate into a reactive layer, in which the substrate is degraded according to Monod kinetics. This problem is described by a nonlinear two-point boundary value problem. The approximation is derived based on a Homotopy Analysis Method idea and verified computationally, by comparison against a numerical solution of the problem. The analytical approximation is easy to evaluate and depends only on model parameters.     

Fixed-size Least Squares Support Vector Machines: A Large Scale Application in Electrical Load Forecasting

Based on the Nyström approximation and the primal-dual formulation of the least squares support vector machines, it becomes possible to apply a nonlinear model to a large scale regression problem. This is done by using a sparse approximation of the nonlinear mapping induced by the kernel matrix, with an active selection of support vectors based on quadratic Renyi entropy criteria. The methodology is applied to the case of load forecasting as an example of a real-life large scale problem in industry. The forecasting performance, over ten different load series, shows satisfactory results when the sparse representation is built with less than 3% of the available sample.     

Local strong uniqueness for nonlinear approximation

The local constant of strong uniqueness for nonlinear approximation with respect to a norm is the local constant of strong uniqueness for approximation in the associated problem of linear approximation by the tangent space.     

Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations

We study the initial-value problem for a general class of nonlinear nonlocal coupled wave equations. The problem involves convolution operators with kernel functions whose Fourier transforms are nonnegative. Some well-known examples of nonlinear wave equations, such as coupled Boussinesq-type equations arising in elasticity and in quasi-continuum approximation of dense lattices, follow from the present model for suitable choices of the kernel functions. We establish local existence and sufficient conditions for finite-time blow-up and as well as global existence of solutions of the problem.     

Existence and uniqueness of solutions to singular Cahn–Hilliard equations with nonlinear viscosity terms and dynamic boundary conditions

We prove global existence and uniqueness of solutions to a Cahn–Hilliard system with nonlinear viscosity terms and nonlinear dynamic boundary conditions. The problem is highly nonlinear, characterized by four nonlinearities and two separate diffusive terms, all acting in the interior of the domain or on its boundary. Through a suitable approximation of the problem based on abstract theory of doubly nonlinear evolution equations, existence and uniqueness of solutions are proved using compactness and monotonicity arguments. The asymptotic behaviour of the solutions as the diffusion operator on the boundary vanishes is also shown.     

Approximation of nonlinear functional differential equation control systems

We develop a general approximation framework for use in optimal control problems governed by nonlinear functional differential equations. Our approach entails only the use of linear semigroup approximation results, while the nonlinearities are treated as perturbations of a linear system. Numerical results are presented for several simple nonlinear optimal control problem examples.This research was supported in part by the US Air Force under Contract No. AF-AFOSR-76-3092 and in part by the National Science Foundation under Grant No. NSF-GP-28931x3.     

Nonlinear and bilinear models for chemical reactor control

The dynamic behavior of a continuously stirred tank reactor (CSTR) with an exothermic reversible reaction is studied. The balance equations of the reaction lead to a set of highly nonlinear differential equations. For system analysis and control synthesis the dynamic equation are rewritten as state space model. From this nonlinear model a bilinear model is derived. Then, two optimization problems are solved: The time optimal problem for the nonlinear model and the quadratic problem for the bilinear model. In case of the finite time bilinear-quadratic problem a modified Riccati approximation algorithm for a stabilizing feedback controller is presented.     

修正的完全近似法的改进形式及其应用

本文对文［１］、［２］提出的修正的完全近似法作了改进：将自变量变换中的待定非线性泛函用特定非线性函数来代替，使运算简化；给出一种新的变换形式，解决了修正的完全近似法往往导出现长期项的问题。     

An age-dependent population dynamical model with a small parameter

In this paper, we consider an age-dependent population dynamical model with age interaction, where the nonlinear term depends on a small positive parameter. We prove that under weak assumptions, the solutions of the nonlinear problem give a good approximation to the solution of the linear problem.     

Analytical solution of a near-equidiffusional flame problem

The near-equidiffusional flame problem for determining the effectof acoustic disturbances having a time scale comparable withthe diffusion time is reduced to a set of nonlinear Volterraequations for a pair of functions related to the burning rateand the flame temperature perturbation. Further analysis yieldsa small-time approximation of general validity and an explicitsolution for arbitrary small-amplitude disturbances. A numericalmethod for solving the Volterra-equation problem is described.Numerical examples are exhibited to compare the behaviour ofthe linear approximation to the solution of the full model.Slowly decaying oscillatory behaviour for flames with Lewisnumber near the right stability boundary is predicted by thelinear approximation and observed in numerical examples forthe nonlinear model.     

Shape optimization of materially non-linear bodies in contact

Optimal shape design problem for a deformable body in contact with a rigid foundation is studied. The body is made from material obeying a nonlinear Hooke's law. We study the existence of an optimal shape as well as its approximation with the finite element method. Practical realization with nonlinear programming is discussed. A numerical example is included.