随机微分方程的适定性及微分方程参数的贝叶斯估计方法

发布时间：2018-03-04 07:31

本文选题：随机薛定谔方程　切入点：适定性　出处：《东北师范大学》2016年博士论文　论文类型：学位论文

【摘要】：本论文主要研究了以下两个方面的内容。一是讨论了随机薛定谔方程解的适定性,包括解的爆破性质和整体解的存在性和唯一性；二是用贝叶斯惩罚B样条方法给出了几类常微分方程模型中参数(常值参数以及时变参数)的估计.关于这些问题的研究背景和动机我们在第一章中给予介绍。微分方程的数学理论研究在物理学,医学,生物学,金融学等应用科学中发挥着重要作用。薛定谔方程是一类特殊的微分方程,其在原子、分子、固体物理、核物理、化学等领域中被广泛应用.然而,在现实生活中,很多事情都是不确定的,是受随机因素干扰的,本文在第一部分考虑了在噪声影响下的薛定谔方程即随机薛定谔方程的解的动力学性质。具体来说,在第二章,我们讨论了在可加噪声和二次位势双重作用下,随机薛定谔方程解的爆破性质,我们得到了不管位势是排斥型还是吸引型,任意有限能量的初值均可能产生爆破解,并且爆破时间可以任意小.这与确定型薛定谔方程不同,对确定性方程来说,排斥型位势具有阻止解爆破的效应。因此,这部分结果表明,噪声对薛定谔方程解的动力学行为的影响比位势的影响要强。与爆破性质对应的,我们在第三章讨论了在Stratonovich型乘积噪声影响下的薛定谔-泊松方程组整体解的适定性。与确定型薛定谔-泊松方程组不同的是,我们建立了随机意义下的交换子估计,进而得到了薛定谔-泊松方程组整体解的存在性和唯一性。在研究随机薛定谔方程适定性的过程中我们发现,方程中的参数对解的动力学性质产生了重要影响,甚至不同参数会导致方程具有完全不同的动力学行为。这就提示我们在应用微分方程的数学理论之前,应当首先确定微分方程中的参数.为此,本文第二部分提出了一种非参数统计方法——贝叶斯惩罚B样条法,根据观测数据去估计微分方程模型中的参数,这其中包括估计常值参数和时变参数两种情形。我们在第四章中介绍了贝叶斯惩罚B样条法的一般理论,并且考虑了对于2×2的线性方程组及非线性方程组(Lotka-Volterra模型),在所有状态变量的观测数据均已知的情形下,用贝叶斯惩罚B样条法,对模型中含有的参数进行估计,模拟结果表明该方法对模型中的参数估计有效。流行病模型是微分方程中应用较多且与现实生活关系较为密切的一类模型,在本文的第五章我们考虑了流行病模型中参数的估计问题。估计此类模型中的参数与第四章中参数的估计最大的不同是：对于流行病模型,我们通常只有部分状态变量甚至只有一个状态变量的观测数据。本文在只有一个状态变量的观测数据情形下,首先通过数值模拟对Kermack-Mckendrick模型中的参数进行估计,发现此时贝叶斯惩罚B样条法仍然有效,并且比最小二乘法的估计效果好,其次,我们还做了一个实例研究。即,利用国家卫生和计划生育委员会公布的中国大陆从2004年1月到2014年12月共132个月的患丙肝疾病的人数的数据,对Zhang和Zhou在2012年针对中国大陆丙肝疾病提出的丙肝(HCV)流行病模型中的参数进行估计,估计结果表明贝叶斯惩罚B样条方法在只有部分状态变量的数据可观测时,对模型中的参数估计仍然有效。众所周知,微分方程中含有的参数通常会随着时间的变化而变化,我们称之为时变参数。我们在上述研究的基础上,在本文的第六章,又进一步考虑了用贝叶斯惩罚B样条法对模型中的时变参数进行估计,与常值参数估计的不同之处是,我们需要首先将待估的时变参数利用B样条进行展开,将时变参数转化为常值参数,然后再进行估计。本文通过模拟对HIV模型中的时变参数进行估计,说明了贝叶斯惩罚B样条法对微分方程中时变参数的估计仍然有效,同时还通过对Hong和Lian文中模型的时变参数进行估计,说明了对于该模型贝叶斯惩罚B样条法较比两阶段局部多项式法有较高的估计精度。
[Abstract]:This paper mainly studies the following two aspects. One is to discuss the well posedness of the solution of the stochastic Schrodinger equation, including the blow up properties of solutions and the existence and uniqueness of the solution; two is to punish B like Bayesian method gives a parameter model of several kinds of ordinary differential equations (constant parameter and time variable parameter estimation). On these issues research background and motivation we introduced in the first chapter. Research on mathematical theory of differential equations in physics, medicine, biology, finance plays an important role in Applied Science. Schrodinger equation is a kind of special differential equations, the atom, molecule, solid state physics nuclear physics, chemistry and other fields, has been widely used. However, in real life, many things are uncertain, is affected by random factors, in the first part of this article is considered under the influence of the noise of Schrodinger The random equation solution of Schrodinger equation dynamics. Specifically, in the second chapter, we discuss the additive noise and the two potential dual role, blasting properties of random Schrodinger equation, we get whatever potential is repulsive or attractive, the initial value of any finite energy may produce explosion crack, and blasting time can be arbitrarily small. This is different from the deterministic Schrodinger equation, the deterministic equation, repulsive potential has prevented the effect of blasting solution. Therefore, the results show that the effect of noise on the dynamics of Schrodinger Fang Chengjie than the effect of potential stronger. Corresponding and blow, we discuss the effect of Stratonovich type product noise under the Schrodinger Poisson equations of overall well posedness in the third chapter. And determine the type of Poisson and Schrodinger equations are different, we set up with Exchange sub machine under estimation, then obtained the Schrodinger Poisson equations existence and uniqueness of the global solution. In the study of stochastic Schrodinger equation posed in the process we found that the parameters in the equation has an important influence on the dynamic properties of the solutions, even different parameters will lead to equations with completely different dynamic behavior. This suggests that before the application of mathematical theory of differential equations, differential equations of the parameters should be determined first. Therefore, the second part of this paper proposes a nonparametric statistical method, Bayesian penalized spline B method to estimate the parameters of the differential equation in the model according to the observation data, including the constant parameter estimation and the time-varying parameters of two kinds of situations. We introduce the general theory of Bayesian penalized B spline method in the fourth chapter, and considering the linear equations and nonlinear 2 x 2 The process group (Lotka-Volterra model), in all the state variables of the observation data are known under the circumstances, Bias penalty B spline method, the model parameters are estimated to contain, simulation results show that the method is effective to estimate the parameters in the model. The epidemic model is a kind of model of differential equation and the relationship with the application of more real life more closely, in the fifth chapter of this paper we consider the parameter estimation problem of epidemic model. The model parameters in the parameter estimation and estimation in Chapter fourth of the biggest difference is: the epidemic model, we usually only part of state variables or even only one state variable in only one observation data. Observation data of state variables under the circumstances, based on Kermack-Mckendrick model parameter estimation numerical simulation, found at this time Bias penalty B spline method is However, effective, and estimated effect than the least squares method, secondly, we also do a case study. That is, the number of the national health and Family Planning Commission announced Chinese, suffering from hepatitis C disease from January 2004 to December 2014 a total of 132 months of data of Zhang and Zhou in 2012 for the Chinese, hepatitis C disease the hepatitis C virus (HCV) epidemic model to estimate the parameter estimation, the results show that the Bayesian penalized B spline method in only partial state variables can be observed when the data is still valid, estimate the parameters of the model. As everyone knows, parameter contained in differential equations usually will change over time, we call it we have variable parameters. Based on the above research, in the sixth chapter of this article, and further consider using Bayesian penalized spline method B variable parameters in the model is to estimate, and The constant difference of parameter estimation, we need to use the B spline of the first variable parameters to be estimated when the time-varying parameters into constant parameters, then estimation. Through the simulation to estimate HIV model with time-varying parameters, the Bayesian penalty B spline method the differential equation is still valid estimation of time-varying parameters, but also through the estimation of the model of Hong and Lian in this paper, the time-varying parameters showed that the estimation accuracy of the model B Bayesian penalized spline method than the two stage local polynomial method is high.

【学位授予单位】：东北师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175;O211.63

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