【作者】 徐岩；

【导师】 孙杰宝；

【作者基本信息】 哈尔滨工业大学 ， 计算数学， 2021， 硕士

【摘要】 工程问题中存在许多不确定的因素,对工程问题中的不确定因素进行不确定分析可以提高系统的可靠性,所以,近年来,不确定分析受到了广泛的关注并得到了飞速的发展。不确定分析的实质就是分析系统中存在的不确定性因素,量化系统输出不确定的过程。随着工程问题的复杂化,不确定性分析的方法也层出不穷,目前应用最为广泛的是概率不确定性分析,即已知系统中不确定因素的统计信息,应用概率理论进行不确定量化分析的方法。本文在概率框架下,考虑广义多项式混沌展开理论,研究在该情况下,如何利用随机最小二乘方法求解最佳逼近系数。以下是本文的主要工作内容:首先,基于不确定量化分析,本文先介绍广义多项式混沌展开理论,并给出针对变量的不同维数,正交多项式对应的选取方式。在上述基础上,介绍以广义正交多项式为基底的最小二乘方法,通过实际算例选取不同的多项式,对比最小二乘方法的逼近效果,验证不同的随机输入,有不同的最优正交多项式与之对应的结论,并给出误差比较。同时,对于最小二乘方法,介绍加权策略,并通过实验,给出加权与不加权的误差分析。采样方式,对于最小二乘方法的稳定性有极其重要的影响,所以本文针对不同样本点的选取,研究对最小二乘稳定性的影响。进一步为了解决不确定量化中出现的维数灾难,本文针对不同维数问题,利用不同的多重指标排序方法,以避免维数灾难。其次,在工程中,可能面临已知不确定的因素给出的信息较少的实际问题。本文利用Bootstrap法采样,将较少的随机输入样本点,转化为大样本数据。在此基础上,与本文介绍矩方法相结合,更准确根据随机输入信息构造统计矩,并将其与加权最小二乘方法相结合,求解展开系数,并对该方法进行稳定性和误差分析。最后,在上述理论的基础上,将基于广义多项式混沌展开的最小二乘方法应用于工程问题中,并给出具体算例分析。更多还原

【Abstract】 There are many uncertain factors in engineering problems.Analyzing the uncertain factors in engineering problems can improve the reliability of the system.Therefore,in recent years,uncertainty analysis has been widely concerned and developed rapidly.The essence of uncertainty analysis is to analyze the uncertainty factors in the system and quantify the process of the output uncertainty of the system.At the same time,with the gradual complexity of engineering problems,methods of uncertainty analysis also emerge in an endless stream.At present,probabilistic uncertainty analysis is widely used,that is,for statistics of known uncertainties in the system,using probability theory to quantify the uncertainty.In this dissertation,under the probabilistic framework,considering the generalized polynomial chaos expansion theory,how to use the random least squares method to solve the best approximation coefficient under this condition is studied.The main work content of this dissertation is as follows:Firstly,this dissertation first introduces the chaos expansion theory of generalized polynomials,and gives the corresponding selection methods of orthogonal polynomials for different dimensions of variables.On the basis of the above,the least squares method based on generalized orthogonal polynomials is introduced,and different polynomials are selected by practical examples to compare the approximation effect of the least square method,and the conclusion that different random inputs have different optimal orthogonal polynomials corresponding to them is obtained,and the error comparison is verified.At the same time,for the least square method,the weighted strategy is introduced,and the error analysis of weighted and unweighted is given through the experiment.The different sampling methods have an extremely important influence on the stability of the least square method.Therefore,this dissertation studies the influence on the stability of the least square method by selecting different sample points.Further,in order to solve the dimensional disaster in uncertainty quantification,this dissertation uses different sorting methods of multiple indexes for different dimension problems.Secondly,in the engineering,the actual problem may be faced with less information given by known uncertain factors.In this dissertation,the Bootstrap sampling method is used to convert fewer random input sample points into large sample data.On this basis,combining with the moment method proposed in this dissertation,the statistical moment is constructed more accurately from the random input information,and combining with the weighted least square method,the expansion coefficient is solved,and the stability and error of this method are analyzed.Finally,on the basis of the above theory,random least squares method based on generalized polynomial chaos expansion is applied to engineering problems,and some concrete examples are given.更多还原