高精度参数估计问题

对高精度参数的估计问题进行了研究.在观测数据无误差的情况下,将微分方程组转化为线性方程组,利用矩阵的奇异值分解给出了参数的最优解.在有观测数据误差的情况下,采用高斯-牛顿迭代法进行求解,给出了改进的高斯-牛顿法和阻尼最小二乘算法;通过灰色估计法给出了模型的初始解,通过微分方程数值解法计算模型迭代过程中误差和偏导数.最后,通过对迭代过程中的状态变量引入误差项,导出了基于总体最小二乘的高斯-牛顿迭代法,从系统的角度解决了观测时间有误差下的参数估计问题.     

Prediction of multiplicity of solutions of nonlinear boundary value problems: Novel application of homotopy analysis method

The purpose of the present paper is to introduce a method, probably for the first time, to predict the multiplicity of the solutions of nonlinear boundary value problems. This procedure can be easily applied on nonlinear ordinary differential equations with boundary conditions. This method, as will be seen, besides anticipating of multiplicity of the solutions of the nonlinear differential equations, calculates effectively the all branches of the solutions (on the condition that, there exist such solutions for the problem) analytically at the same time. In this manner, for practical use in science and engineering, this method might give new unfamiliar class of solutions which is of fundamental interest and furthermore, the proposed approach convinces to apply it on nonlinear equations by today’s powerful software programs so that it does not need tedious stages of evaluation and can be used without studying the whole theory. In fact, this technique has new point of view to well-known powerful analytical method for nonlinear differential equations namely homotopy analysis method (HAM). Everyone familiar to HAM knows that the convergence-controller parameter plays important role to guarantee the convergence of the solutions of nonlinear differential equations. It is shown that the convergence-controller parameter plays a fundamental role in the prediction of multiplicity of solutions and all branches of solutions are obtained simultaneously by one initial approximation guess, one auxiliary linear operator and one auxiliary function. The validity and reliability of the method is tested by its application to some nonlinear exactly solvable differential equations which is practical in science and engineering.     

Une méthode de décomposition en temps avec des schémas dʼintégration réversible pour la résolution de système dʼéquations différentielles ordinaires

We propose a time domain decomposition method that breaks the sequentiality of the integration scheme for systems of ODE. Under the condition of differentiability of the flow, we transform the initial value problem into a well-posed boundary values problem using the symmetrization of the interval of time integration and time-reversible integration scheme. For systems of linear ODE, we explicitly construct the block tridiagonal system satisfied by the solutions at the time sub-intervals extremities. We then propose an iterative algorithm of Schwarz type for updating the interfaces conditions which can extend the method to systems of nonlinear ODE.     

Exact and analytical solutions for a nonlinear sigma model

We consider wave solutions to nonlinear sigma models in n dimensions. First, we reduce the system of governing PDEs into a system of ODEs through a traveling wave assumption. Under a new transform, we then reduce this system into a single nonlinear ODE. Making use of the method of homotopy analysis, we are able to construct approximate analytical solutions to this nonlinear ODE. We apply two distinct auxiliary linear operators and show that one of these permits solutions with lower residual error than the other. This demonstrates the effectiveness of properly selecting the auxiliary linear operator when performing homotopy analysis of a nonlinear problem. From here, we then obtain residual error‐minimizing values of the convergence control parameter. We find that properly selecting the convergence control parameter makes a drastic difference in the magnitude of the residual error. Together, appropriate selection of the auxiliary linear operator and of the convergence control parameter is shown to allow approximate solutions that quickly converge to the true solution, which means that few terms are needed in the construction of such solution. This, in turn, greatly improves computational efficiency. Copyright © 2013 John Wiley & Sons, Ltd.     

On Gaussian decay estimates of solutions to some linear elliptic equations and their applications

In this paper we establish pointwise decay estimates of solutions to some linear elliptic equations by using the Nash–Moser iteration arguments and the ODE method. As applications we obtain sharp Gaussian decay estimates for solutions to nonlinear elliptic equations that are related with self-similar solutions to nonlinear heat equations and standing wave solutions to nonlinear Schrödinger equations with harmonic potential.     

Solution of Delay Differential Equation by Means of Homotopy Analysis Method

The algorithm of approximate analytical solution for delay differential equations (DDE) is obtained via homotopy analysis method (HAM) and modified homotopy analysis method (MHAM). Various examples of linear, nonlinear and system of initial value problems of DDE are solved and the results obtained show that these algorithms are accurate and efficient for the DDE. The convergence of this algorithm is also proved.     

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.     

An EM Algorithm for Estimating Equations

This article presents an algorithm for accommodating missing data in situations where a natural set of estimating equations exists for the complete data setting. The complete data estimating equations can correspond to the score functions from a standard, partial, or quasi-likelihood, or they can be generalized estimating equations (GEEs). In analogy to the EM, which is a special case, the method is called the ES algorithm, because it iterates between an E-Step wherein functions of the complete data are replaced by their expected values, and an S-Step where these expected values are substituted into the complete-data estimating equation, which is then solved. Convergence properties of the algorithm are established by appealing to general theory for iterative solutions to nonlinear equations. In particular, the ES algorithm (and indeed the EM) are shown to correspond to examples of nonlinear Gauss-Seidel algorithms. An added advantage of the approach is that it yields a computationally simple method for estimating the variance of the resulting parameter estimates.     

Stabilization of unstable stationary points in equations with delayed argument

We solve the problem of localization and stabilization of an unstable stationary point of a nonlinear system of ordinary differential equations (ODE) with a delayed argument for parameter values when the ODE system has chaotic dynamics. Translated from Nelineinaya Dinamika i Upravlenie, pp. 133–141, 1999.     

Weighted linear associative memory approach to nonlinear parameter estimation

The method of linear associative memory (LAM), a notion from the field of artificial neural nets, has been applied recently in nonlinear parameter estimation. In the LAM method, a model response, nonlinear with respect to the parameters, is approximated linearly by a matrix, which maps inversely from a response vector to a parameter vector. This matrix is determined from a set of initial training parameter vectors and their response vectors, and can be update recursively and adaptively with a pair of newly generated parameter response vectors. The LAM advantage is that it can yield a good estimation of the true parameters from a given observed response, even if the initial training parameter vectors are far from the true values.In this paper, we present a weighted linear associative memory (WLAM) for nonlinear parameter estimation. WLAM improves LAM by taking into account an observed response vector oriented weighting. The basic idea is to weight each pair of parameter response vectors in the cost function such that, if a response vector is closer to the observed one, then this pair plays a more important role in the cost function. This weighting algorithm improves significantly the accuracy of parameter estimation as compared to a LAM without weighting. In addition, we are able to construct the associative memory matrix recursively, while taking the weighting procedure into account, and simultaneously update the ridge parameter of the cost function further improving the efficiency of the WLAM estimation. These features enable WLAM to be a powerful tool for nonlinear parameter simulation.This work was supported by National Science Foundation, Grants BCS-93-15886 and INT-94-17206. We thank Mr. L. Yobas for fruitful discussions.     

Differential equations for ellipsoidal estimates for reachable sets of a nonlinear dynamical control system

The problem of estimating trajectory tubes of a nonlinear control dynamical system with uncertainty in initial data is considered. It is assumed that the dynamical system has a special structure, in which nonlinear terms are quadratic in phase coordinates and the values of the uncertain initial states and admissible controls are subject to ellipsoidal constraints. Differential equations are found that describe the dynamics of the ellipsoidal estimates of reachable sets of the nonlinear dynamical system under consideration. To estimate reachable sets of the nonlinear differential inclusion corresponding to the control system, we use results from the theory of ellipsoidal estimation and the theory of evolution equations for set-valued states of dynamical systems under uncertainty.     

Renormalization group and asymptotics of solutions of nonlinear parabolic equations

We present a general method for studying long-time asymptotics of nonlinear parabolic partial differential equations. The method does not rely on a priori estimates such as the maximum principle. It applies to systems of coupled equations, to boundary conditions at infinity creating a front, and to higher (possibly fractional) differential linear terms. We present in detail the analysis for nonlinear diffusion-type equations with initial data falling off at infinity and also for data interpolating between two different stationary solutions at infinity. In an accompanying paper, [5], the method is applied to systems of equations where some variables are “slaved,” such as the complex Ginzburg-Landau equation. © 1994 John Wiley & Sons, Inc.     

A Generalized Smoother for Linear Ordinary Differential Equations

Ordinary differential equations (ODEs) are equalities involving a function and its derivatives that define the evolution of the function over a prespecified domain. The applications of ODEs range from simulation and prediction to control and diagnosis in diverse fields such as engineering, physics, medicine, and finance. Parameter estimation is often required to calibrate these theoretical models to data. While there are many methods for estimating ODE parameters from partially observed data, they are invariably subject to several problems including high computational cost, complex estimation procedures, biased estimates, and large sampling variance. We propose a method that overcomes these issues and produces estimates of the ODE parameters that have less bias, a smaller sampling variance, and a 10-fold improvement in computational efficiency. The package GenPen containing the Matlab code to perform the methods described in this article is available online.     

A posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations

We present constructive a posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations (ODEs) on a bounded interval. Here, “constructive” indicates that we can obtain bounds of the operator norm in which all constants are explicitly given or are represented in a numerically computable form. In general, it is difficult to estimate these inverse operators a priori. We, therefore, propose a technique for obtaining a posteriori estimates by using Galerkin approximation of inverse operators. This type of estimation will play an important role in the numerical verification of solutions for initial value problems in nonlinear ODEs as well as for parabolic initial boundary value problems.     

Remarks on the optimal convolution kernel for CSOR waveform relaxation

The convolution SOR waveform relaxation method is a numerical method for solving large-scale systems of ordinary differential equations on parallel computers. It is similar in spirit to the SOR acceleration method for solving linear systems of algebraic equations, but replaces the multiplication with an overrelaxation parameter by a convolution with a time-dependent overrelaxation function. Its convergence depends strongly on the particular choice of this function. In this paper, an analytic expression is presented for the optimal continuous-time convolution kernel and its relation to the optimal kernel for the discrete-time iteration is derived. We investigate whether this analytic expression can be used in actual computations. Also, the validity of the formulae that are currently used to determine the optimal continuous-time and discrete-time kernels is extended towards a larger class of ODE systems.     

非线性随机分数阶微分方程Euler方法的弱收敛性

论文首先证明了非线性随机分数阶微分方程解的存在唯一性,然后构造了数值求解该方程的Euler方法,并证明了当方程满足一定约束条件时,该方法是弱收敛的.特别地,当分数阶α=0时,该方程退化为非线性随机微分方程,所获结论与现有文献中的相关结论是一致的;当α≠0,且初值条件为齐次时,所获结论可视为现有文献中线性随机分数阶微分方...     

Convergence of monotone iterations to initial value problems of functional differential equations

The method of quasilinearization is a well-known technique for obtaining approximate solutions of nonlinear differential equations. We use this technique to initial value problems of functional differential equations showing that corresponding linear iterations converge to the unique solution of our problem and this convergence is superlinear     

Numerical analysis of nonlinear model of excited carrier decay

This paper presents a mathematical model for photo-excited carrier decay in a semiconductor. Due to the carrier trapping states and recombination centers in the bandgap, the carrier decay process is defined by the system of nonlinear differential equations. The system of nonlinear ordinary differential equations is approximated by linearized backward Euler scheme. Some a priori estimates of the discrete solution are obtained and the convergence of the linearized backward Euler method is proved. The identifiability analysis of the parameters of deep centers is performed and the fitting of the model to experimental data is done by using the genetic optimization algorithm. Results of numerical experiments are presented.     

Chebyshev Approximation to Data by Positive Sums of Exponentials

The problem is considered of calculating Chebyshev approximationsto given data by sums of exponentials with positive coefficients,where the number of terms in the sum has to be obtained as partof the process. An exchange procedure based on linear programmingis developed for the estimation of the exponents, and this ismade efficient by the use of postoptimality theory and the applicationof the dual simplex algorithm. Rapid convergence to a best approximationcan then be obtained by the application of Newton's method tothe characterization conditions interpreted as a nonlinear systemof equations. The Newton step can be determined through thesolution of a quadratic programming problem, and advantage istaken of the structure so that the calculation can be simplifiedwithout inhibiting a second-order convergence rate. Numericalresults are presented for the application of an algorithm basedon these ideas to a number of data sets which have appearedin the literature.     

A linear code parameter search algorithm with applications to immunology

The immune system is modeled by way of a system of ordinary differential equations involving a large number of parameters, such as growth rates and initial conditions. Key to successful implementation of the model is the estimation of such parameters from available data. A parameter search algorithm based on linear codes is developed having as aim the identification of different regimes of behaviour of the model, the estimation of parameters in a high dimensional space, and the model calibration to data. This work was funded under the NIH grant GM67240.