Uncertainty Quantification in Forward Problems: Balancing Accuracy and Robustness Using CWENO Interpolations

TL;DR

This paper introduces a CWENO7-based surrogate modeling approach for uncertainty quantification in forward problems, effectively balancing accuracy and robustness, especially near discontinuities, outperforming traditional spectral methods like gPC.

Contribution

The paper presents a novel CWENO7 interpolation method integrated into stochastic collocation, offering improved non-oscillatory behavior and high-order accuracy for UQ in nonsmooth problems.

Findings

CWENO7 achieves high-order accuracy in smooth regions.

CWENO7 reduces oscillations near discontinuities.

The method is efficient and scalable for high-dimensional problems.

Abstract

In this paper, we study uncertainty quantification (UQ) in forward problems. Our objective is to construct accurate and robust surrogate models by incorporating the seventh-order central weighted essentially non-oscillatory (CWENO7) scheme into the stochastic collocation framework. A key focus is on mitigating the oscillatory behavior often encountered in traditional spectral methods while retaining high-order accuracy in smooth regions. We present a systematic comparison between CWENO7-based and generalized polynomial chaos (gPC)-based approaches. Although gPC methods achieve spectral convergence, they are prone to Gibbs-type oscillations in nonsmooth settings. By contrast, CWENO7 utilizes local stencils to achieve a balance: non-oscillatory behavior near discontinuities and high-order convergence in smooth regions. To validate the approach, we conduct numerical experiments on a range of one- and two-dimensional smooth and nonsmooth problems, including shallow water equations with random inputs. The results demonstrate that CWENO7 interpolation provides accurate estimates of probability density functions, mean values, and standard deviations, particularly in regimes where gPC expansions exhibit strong oscillations. Furthermore, computational tests confirm that CWENO7 interpolation is efficient and scalable, establishing it as a reliable alternative to conventional stochastic collocation techniques for UQ in the presence of discontinuities.