Random Least Squares Method Based On Generalized Polynomial Chaos Expansion

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.