Research On Moment Independent Importance And Neighborhood Importance Theoretics And Application

Aircraft design is a composite project relating to many different disciplines, andthere exists a vast amount of uncertainty factors during the design process. Therefore,the uncertainty analysis plays an important role in all the steps by which aircraft isdesigned. In order to obtain a better result of uncertainty analysis for aircraft designprocess, this thesis deals with uncertainty quantifcation and computation at frst. Thenthe idea of regional importance analysis is applied to estimate a kind of nonlinearpartial differential equation.Firstly, Borgonovo’s moment-independent importance measure, which involvesthe entire output distribution, is an useful indictor in uncertainty analysis. Then aseries of works based on this importance measure are developed.(1) We analyzed the computational diffculty of Borgonovo’s moment-independentindex.(2) Since the probability density function (PDF) integral of Borgonovo’s moment-independent importance measure is hard to calculate, a series of computationalstrategies are proposed. First of all, the concept of norm is introduced into theuncertainty importance analysis, and a new importance ranking computationalstrategy is developed, for which some equivalent norms are selected.(3) Because the PDF estimate is an ill-posed problem and its convergence rate isquite slow, then Borgonovo’s moment-independent importance measure is ap-proximatively represented by the cumulative distribution function (CDF). TheCDF estimate is well-posed and its convergence rate is always faster than that ofthe PDF estimate. From the representation, a stable approach is proposed with anadaptive procedure. Besides, a computational strategy named asymptotic spaceintegration (ASI) is introduced to estimate the small probability multidimen-sional integral in the adaptive procedure.Secondly, two new moment independent importance measures are proposed andtheir mathematical properties are discussed. Both of them are dual representations ofBorgonovo’s moment-independent importance measure.(1) For the moment independent importance measure based on characteristic func-tion (CF), it represents the normalized expectation of the distance between theunconditional and conditional CFs of output. Then a kind of fast computationalmethod based on kernel estimate is introduced to improve the convergence rateof estimating the CF-based moment independent importance measure.(2) For the moment independent importance measure based on moment generatingfunction (Mgf), it represents a particular case of CF-based moment independent importance measure. Since the Mgf is defned on real feld R, the computationof Mgf is easier than that of CF. Besides, a kernel estimate method is developedto compute this importance measure.Thirdly, a series of improved works based on the defnition of Borgonovo’smoment-independent importance measure are developed.(1) A new uncertainty importance measure is proposed. This proposed measurecombines the moment-independent approach with the variance-based method,which looks at the average influence of input uncertainty on the entire outputdistribution and its deviation, as well as avoids taking a sole variance as an eval-uation. It increases the information of the decision-making state, but would notlead the decisionmaker to noninformative conclusions to a great extent.(2) Probability and compound uncertainty importance measures are proposed by im-itating the failure probability in the reliability analysis. These indicators look atthe input influence on the shift trend and shift degree of the output distributionshape with the the model output skewness and kurtosis.Lastly, the uncertainty analysis is an important part of the modeling and risk as-sessment problems in many areas, and in this thesis, we consider using the idea ofregional importance analysis to the numerical solution of the nonlinear partial differ-ential equation.(1) We start with a one-dimensional function approximation, and discuss the re-lationship between the accuracy of kernel interpolating function and its nodaldistribution.(2) We obtain an optimal nodal distribution of kernel interpolation for a given er-ror, and defne a neighborhood importance measure (NIM) of any node. NIMindicates the importance degree of the any node over the whole region.(3) According to NIM, a predictor based method is introduced to obtain an optimalkernel interpolation.(4) The proposed method is applied to solve a kind of2-dimensional nonlinear hy-perbolic partial differential equation. The time derivatives are approximated byboth the fnite difference method and the Crank-Nicoison scheme. Then, theaccuracy of the proposed method is demonstrated by several examples. The nu-merical results are found to be in good agreement with the exact solutions andthe numerical solutions in existing literature.(5) The predictor based method is applied to the numerical solution of the nonlin-ear partial differential equation for an arbitrary number of dimensions. Fromnumerical examples, it is shown that the proposed method is very helpful forsimulating the partial differential equation.