A Spline Dimensional Decomposition for Uncertainty Quantification in High Dimensions

TL;DR

This paper introduces a spline dimensional decomposition (SDD) method for efficient uncertainty quantification in high-dimensional, possibly nonsmooth, nonlinear functions, reducing computational complexity and improving accuracy over existing methods.

Contribution

The paper presents a novel SDD approach using measure-consistent orthonormalized B-splines, with proven convergence and polynomial complexity, advancing high-dimensional uncertainty analysis.

Findings

SDD converges in mean-square to the true output.

Low-order SDD achieves high accuracy for nonsmooth functions.

Demonstrated effectiveness in a 34-dimensional eigenvalue problem.

Abstract

This study debuts a new spline dimensional decomposition (SDD) for uncertainty quantification analysis of high-dimensional functions, including those endowed with high nonlinearity and nonsmoothness, if they exist, in a proficient manner. The decomposition creates an hierarchical expansion for an output random variable of interest with respect to measure-consistent orthonormalized basis splines (B-splines) in independent input random variables. A dimensionwise decomposition of a spline space into orthogonal subspaces, each spanned by a reduced set of such orthonormal splines, results in SDD. Exploiting the modulus of smoothness, the SDD approximation is shown to converge in mean-square to the correct limit. The computational complexity of the SDD method is polynomial, as opposed to exponential, thus alleviating the curse of dimensionality to the extent possible. Analytical formulae are proposed to calculate the second-moment properties of a truncated SDD approximation for a general output random variable in terms of the expansion coefficients involved. Numerical results indicate that a low-order SDD approximation of nonsmooth functions calculates the probabilistic characteristics of an output variable with an accuracy matching or surpassing those obtained by high-order approximations from several existing methods. Finally, a 34-dimensional random eigenvalue analysis demonstrates the utility of SDD in solving practical problems.