Chair participation in UNCECOMP 2015

Sparse polynomial chaos expansions for solving high-dimensional UQ problems

Abstract

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In the last decade polynomial chaos (PC) expansions have become a key tool for solving uncertainty quantification (UQ) problems in many fields of applied mathematics and engineering. The general set up of UQ assumes that the governing equations of a system of interest have uncertain parameters modelled by random variables (resp.  fields). Once the probability distribution of these input parameters is defined, uncertainty propagation aims at quantifying some statistics of the response of the computational model, e.g., the probability density function (PDF), statistical moments, quantiles and probability of threshold exceedance. Global sensitivity indices may be also be computed to determine which ``important'' input parameters drive the uncertainty of the model response.

Monte Carlo simulation is a well-known tool for uncertainty propagation. Although universal, the approach is rather inefficient, typically requiring tens of thousands to millions of samples to get accurate results. In practice it cannot be used in conjunction with advanced computational models such as, e.g., finite element models in engineering sciences.  In contrast, polynomial chaos-based methods cast the model response as a series expansion on a basis of multivariate polynomials in the input parameters, thus leading to an intrinsic representation of the random response.

In this lecture the basics of polynomial chaos expansions and the challenges posed by the curse of dimensionality will be first introduced. A non-intrusive approach based on least-square minimization and model selection (least-angle regression algorithm) is then introduced to derive sparse PC expansions to solve this issue. In this setup, the PC expansions is viewed as a surrogate model calibrated on a small number of runs of the computational model, the so-called experimental design. A posteriori error estimators are discussed.

The remarkable efficiency of sparse PC expansions will be illustrated on different environmental and engineering problems: a global sensitivity analysis on the hydro-dispersive parameters of a multi-layered geological model, a geotechnical problem of a foundation lying on uncertain multi-layered soil mass and a time-dependent analysis of non-linear oscillators.