CWENO Interpolation for Non-Oscillatory Stochastic Collocation in Uncertainty Quantification Problems

TL;DR

Uncertainty quantification (UQ) for PDE models is addressed using stochastic collocation (SC) methods that compare gPC, B-splines, SP splines, and CWENO interpolation for reconstructing PDFs $p(U)$ and estimating moments such as $E[U]$ and ${\rm Var}(U)$. The study demonstrates that gPC and interpolation B-splines perform well for smooth inputs but exhibit Gibbs-type oscillations near discontinuities, while approximation B-splines smear features and SP splines converge slowly. CWENO interpolation provides high accuracy in smooth regions while remaining non-oscillatory near sharp gradients, yielding robust PDFs across both smooth and discontinuous data. Overall, CWENO emerges as a versatile SC approach for UQ, with future work focusing on non-negativity constraints and multidimensional extensions.

Abstract

Uncertainty quantification (UQ) in mathematical models is essential for accurately predicting system behavior under variability. This study provides guidance on method selection for reliable UQ across varied functional behaviors in engineering applications. Specifically, we compare several interpolation and approximation methods within a stochastic collocation (SC) framework, namely: generalized polynomial chaos (gPC), B-splines, shape-preserving (SP) splines, and central weighted essentially nonoscillatory (CWENO) interpolation, to reconstruct probability density functions (PDFs) and estimate statistical moments. These methods are assessed for both smooth and discontinuous functions, as well as for the solution of the 1-D Euler and shallow water equations. While gPC and interpolation B-splines perform well with smooth data, they produce oscillations near discontinuities. Approximation B-splines and SP splines, while avoiding oscillations, converge more slowly. In contrast, CWENO interpolation demonstrates high robustness, effectively capturing sharp gradients without oscillations, making it suitable for complex, discontinuous data. Overall, CWENO interpolation emerges as a versatile and effective approach for SC, particularly in handling discontinuities in UQ.

Figures (12)

• Figure 1: Example 1, Test 1: PDF of $U(\xi)$ for (\ref{['fig:3.1.1a']}) $N=7$, (\ref{['fig:3.1.1b']}) $N=12$ and (\ref{['fig:3.1.1c']}) $N=16$: 1 -- reference, 2 -- gPC, 3 -- interpolation B-spline, 4 -- approximation B-spline, 5 -- SP spline, 6 -- CWENO interpolation.
• Figure 2: Example 1, Test 1: L$^1$ error of PDF reconstruction as a function of $N$, and the corresponding power-law fit (solid lines).
• Figure 3: Example 1, Test 1: Error of (\ref{['fig:3.1.3a']}) mean and (\ref{['fig:3.1.3b']}) standard deviation estimates for different $N$.
• Figure 4: Example 1, Test 2: L$^1$ error of PDF reconstruction as a function of $N$, and the corresponding power-law fit (solid lines).
• Figure 5: Example 2: PDF of $U$ for (\ref{['fig:3.2.1a']}) $N=7$, (\ref{['fig:3.2.1b']}) $N=12$, and (\ref{['fig:3.2.1c']}) $N=16$.