Quantification of uncertainty from high-dimensional scattered data via polynomial approximation

TL;DR

This paper presents a novel polynomial approximation method for quantifying uncertainty in high-dimensional scattered data, effectively handling limited samples and noise with scalable computational cost.

Contribution

It introduces a new approach combining High-Dimensional Model Representation with modified Least Angle Regression for basis selection and coefficient evaluation.

Findings

Accurately approximates high-dimensional random variables with as few as 3 samples per dimension.

Demonstrates robustness of the method against noisy data.

Shows linear scaling of computational cost with basis size and polynomial dependence on sample number.

Abstract

This paper discusses a methodology for determining a functional representation of a random process from a collection of scattered pointwise samples. The present work specifically focuses onto random quantities lying in a high dimensional stochastic space in the context of limited amount of information. The proposed approach involves a procedure for the selection of an approximation basis and the evaluation of the associated coefficients. The selection of the approximation basis relies on the a priori choice of the High-Dimensional Model Representation format combined with a modified Least Angle Regression technique. The resulting basis then provides the structure for the actual approximation basis, possibly using different functions, more parsimonious and nonlinear in its coefficients. To evaluate the coefficients, both an alternate least squares and an alternate weighted total least squares methods are employed. Examples are provided for the approximation of a random variable in a high-dimensional space as well as the estimation of a random field. Stochastic dimensions up to 100 are considered, with an amount of information as low as about 3 samples per dimension, and robustness of the approximation is demonstrated w.r.t. noise in the dataset. The computational cost of the solution method is shown to scale only linearly with the cardinality of the a priori basis and exhibits a (N_q)^s, 2 <= s <= 3, dependence with the number N_q of samples in the dataset. The provided numerical experiments illustrate the ability of the present approach to derive an accurate approximation from scarce scattered data even in the presence of noise.