Research On High-dimensional Function Approximation Theory In Aircraft Design

This thesis focuses on issues such as (1) function approximation spaces,(2) ap-proximate operators,(3) estimations of Sobol indices and Borgonovo indices, and (4)evaluations of multivariate integrals according to the needs of construction of surfaces,solving of partial differential equations and meta model in aircraft design.(1) A new class of stellate subspaces is presented based on multiple orthogonalseries. The degrees of freedom of these spaces are independent of the dimension ofproblem with a fast rate of convergence. The main results are listed as follows:A class of stellate partial sum of multiple orthogonal series is built on the de-cay of its coeffcients, this makes that the corresponding degree of freedom isindependent of the dimension;Both the order and bound of the term number of a stellate partial sum are strictlybuilt;As an approximation of a given function, the error bound of a stellate partial sumis strictly built;A new idea of multivariate interpolation is discussed based on the summabilityof a stellate partial sum.(2) A polynomial based multi-resolution analysis is introduced with two polyno-mial jigsaw quasi-interpolation operators. It provides a representation for multivariatefunctions or data with a coarse-to-fne hierarchy. The main results are listed as follows:The proposed multi-resolution analysis can be easily extended to functions ondifferential manifolds according to the paracompactness of differential mani-folds;A family of jigsaw functions by univariate piecewise Hermite interpolating poly-nomial is introduced to construct a multilevel jigsaw partition algorithm;Two polynomial jigsaw quasi-interpolations are proposed with uniform errorbounds for multivariate scattered data;A local error estimate of the polynomial jigsaw quasi-interpolations is intro-duced to construct a polynomial jigsaw approximation of any function for agiven error. It is shown by an especial discussion on the constant jigsaw quasi-interpolation that there exists an optimal piecewise constant interpolation asso-ciated with the Voronoi graph and the Delaunay triangulation.(3) A series of computational methods are proposed for HDMR decomposition,Sobol indices and Borgonovo indices. The main results are listed as follows: The corresponding reproducing kernel is constructed to approximate the HDMRapproximations, then the Sobol indices can be further estimated;Pursuing the point of view of non-parametric smoothing, the kernel estimatorand high order kernel estimator are built for the computation of E(y|xI), thenthe HDMR decomposition and Sobol indices can be further estimated. And theasymptotic convergence rate is theoretically obtained for these estimators, It isshown that its convergence rate is always less than N1/2;Two associated improvements are suggested for the main problems of the com-putaion of Borgonovo indices. A kernel estimate is frst introduced to avoid theuse of two-loop MC, and a moment expansion of the associated norm which isnot simply obtained by using the Edgeworth series is introduced to avoid thedensity estimation. Then a fast computational method is proposed.(4) The notion of asymptotic space is introduced according to the helical line. Byusing an appropriate asymptotic curve, a d-dimensional integration can be reduced toa1-dimensional integration. The main results are listed as follows:It is proved that any d-dimensional integration can be estimated by a1-dimensional integration over an appropriate asymptotic curve;The corresponding error estimate is built, and it shows that the error can becontroled by the modulus of continuity;A special class of asymptotic curves over sphere is introduced. Together withGauss formula, it can be used for solving the weak form of three dimensionalpartial differential equations.