Sparse polynomial surrogates for aerodynamic computations with random inputs

Abstract

This paper deals with some of the methodologies used to construct polynomial surrogate models based on generalized polynomial chaos (gPC) expansions for applications to uncertainty quantification (UQ) in aerodynamic computations. A core ingredient in gPC expansions is the choice of a dedicated sampling strategy, so as to define the most significant scenarios to be considered for the construction of such metamodels. A desirable feature of the proposed rules shall be their ability to handle several random inputs simultaneously. Methods to identify the relative "importance" of those variables or uncertain data shall be ideally considered as well. The present work is more particularly dedicated to the development of sampling strategies based on sparsity principles. Sparse multi-dimensional cubature rules based on general one-dimensional Gauss-Jacobi-type quadratures are first addressed. These sets are non nested, but they are well adapted to the probability density functions with compact support for the random inputs considered in this study. On the other hand, observing that the aerodynamic quantities of interest (outputs) depend only weakly on the cross-interactions between the variable inputs, it is argued that only low-order polynomials shall significantly contribute to their surrogates. This "sparsity-of-effects" trend prompts the use of reconstruction techniques benefiting from the sparsity of the outputs, such as compressed sensing (CS). CS relies on the observation that one only needs a number of samples proportional to the compressed size of the outputs, rather than their uncompressed size, to construct reliable surrogate models. The results obtained with the test case considered in this work corroborate this expected feature.

Sparse polynomial surrogates for non-intrusive, high-dimensional uncertainty quantification of aeroelastic computations

“…Polynomial surrogates for aerodynamic computations have been outlined in e.g. [49]. We basically follow this presentation here.…”

“…In [49] it has been observed that such sparse rules typically become competitive with respect to tensor grids for dimensions D ≥ 4. Consequently, collocation techniques with sparse quadratures or adaptive regression strategies have been developed in order to circumvent the dimensionality concern [5,28,64].…”

Computation of higher‐order moments of generalized polynomial chaos expansions

“…Indeed, one has and Γ( p + 1) = p !, the usual factorial, if p is an integer. Jacobi polynomials arise in gPC expansions for random variables following beta distributions of the first kind; see, e.g., . The normalization constant γ n in then reads The linearization coefficients B n ( j , k ) in the general linearization problem are given in [, equation (12)] in terms of double hypergeometric functions (the so‐called Kampé de Fériet functions).…”

“…∶= Γ(z + 1) = ∫ +∞ 0 t z e −t dt and (p + 1) = p!, the usual factorial, if p is an integer. Jacobi polynomials arise in gPC expansions for random variables following beta distributions of the first kind; see, e.g., [3,13]. The normalization constant n in (3) then reads n = 2 + +1 (n + )!…”

“…The homogeneous and generalized homogeneous chaos expansions have recently received a broad attention in engineering sciences, where they are extensively used as a constructive tool for representing random vectors, matrices, tensors, or fields for the purpose of quantifying uncertainty in complex systems. Several applications are described in, e.g., [3][4][5][6][7][8][9][10][11][12][13][14][15][16] and references therein.…”