Sparse polynomial surrogates for aerodynamic computations with random inputs

TL;DR

This paper develops sparse polynomial surrogate models for aerodynamic uncertainty quantification, utilizing specialized sampling strategies and compressed sensing to efficiently handle multiple random inputs with weak interactions.

Contribution

It introduces sparsity-based sampling rules and reconstruction techniques for polynomial chaos surrogates in aerodynamic UQ, emphasizing low-order effects and compressed sensing.

Findings

Sparse multi-dimensional cubature rules effectively handle compact support distributions.

Surrogates based on sparsity principles require fewer samples for accurate modeling.

Results confirm the weak dependence of outputs on high-order interactions.

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.