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

TL;DR

This work presents a seventh-order central weighted essentially non-oscillatory (CWENO7) interpolation-based surrogate for forward uncertainty quantification within stochastic collocation. By contrast with generalized polynomial chaos (gPC), CWENO7 achieves non-oscillatory behavior near discontinuities while preserving high-order accuracy in smooth regions, enabling accurate estimates of means, variances, and full PDFs even in nonsmooth regimes. The methodology supports both single and multi-dimensional random inputs via a dimension-by-dimension approach and demonstrates stability, efficiency, and scalability on smooth and nonsmooth test problems, including shallow-water equations with random inputs. The findings suggest CWENO7 as a reliable alternative to conventional stochastic collocation techniques in the presence of discontinuities, with potential extensions to higher-order schemes and 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.

Figures (22)

• Figure 2.1: CWENO7 stencil structure for degree 6 polynomial $P_{\rm opt}$ and four sub-stencils for cubic polynomials $P_1$, $P_2$, $P_3$, $P_4$.
• Figure 3.1: Example 1, Test 1: Interpolations obtained for $L=7$ using CWENO3 (left), CWENO5 (middle), and CWENO7 (right). Green dots represent the output data being interpolated.
• Figure 3.2: Example 1, Test 1: $L^1$-errors for the gPC expansion and CWENO7 interpolation as functions of $L$ and the corresponding power-law fits (solid lines).
• Figure 3.3: Example 1, Test 1: Errors in $\mu$ (left) and $\sigma$ (right) for the gPC expansion and CWENO7 interpolation as functions of $L$ and the corresponding power-law fits (solid lines).
• Figure 3.4: Example 1, Test 1: Estimated PDFs for the gPC and CWENO7 interpolations $\widetilde{U}$ together with the reference PDF, reconstructed from $U$ for $L=7$ (left), $9$ (middle), and $15$ (right).

Theorems & Definitions (1)

• Remark 2.1