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Stochastic Finite Volume Approximation with Clustering in the Parameter Space for Forward Uncertainty Quantification of PDEs with Random Parameters

Zhao Zhang, Na Ou

TL;DR

The paper addresses forward uncertainty quantification for PDEs with random parameters by introducing a clustering-based SFV (SFV-Cluster) framework that forms parameter-space cells through clustering MC samples rather than fixed grids. This unstructured, implicit-boundary discretization reduces the number of forward simulations required while preserving the semi-intrusive SFV structure, with error bounds showing how discretization, clustering, and sampling contribute to overall accuracy. Validation across elliptic, parabolic, and hyperbolic Darcy-flow models—including up to 10 uncertain dimensions—demonstrates faster convergence for mean and variance of QoIs compared with Monte Carlo, especially when forward runs are costly. The approach offers a scalable path for forward UQ in porous-media applications and suggests future work in dimension reduction or manifold learning for very high-dimensional parameter spaces.

Abstract

The uncertainty quantification (UQ) for partial differential equations (PDEs) with random parameters is important for science and engineering. Forward UQ quantifies the impact of random parameters on the solution or the quantity-of-interest (QoI). In the current study, we propose a new extension of the stochastic finite volume (SFV) method by clustering samples in the parameter space. Compared to classic SFV based on structured grid in the parameter space, the new scheme based on clustering extends SFV to parameter spaces of higher dimensions. This paper presents the construction of SFV schemes for typical parametric elliptic, parabolic and hyperbolic equations for Darcy flows in porous media, as well as the error analysis, demonstration and validation of the new extension using typical reservoir simulation test cases.

Stochastic Finite Volume Approximation with Clustering in the Parameter Space for Forward Uncertainty Quantification of PDEs with Random Parameters

TL;DR

The paper addresses forward uncertainty quantification for PDEs with random parameters by introducing a clustering-based SFV (SFV-Cluster) framework that forms parameter-space cells through clustering MC samples rather than fixed grids. This unstructured, implicit-boundary discretization reduces the number of forward simulations required while preserving the semi-intrusive SFV structure, with error bounds showing how discretization, clustering, and sampling contribute to overall accuracy. Validation across elliptic, parabolic, and hyperbolic Darcy-flow models—including up to 10 uncertain dimensions—demonstrates faster convergence for mean and variance of QoIs compared with Monte Carlo, especially when forward runs are costly. The approach offers a scalable path for forward UQ in porous-media applications and suggests future work in dimension reduction or manifold learning for very high-dimensional parameter spaces.

Abstract

The uncertainty quantification (UQ) for partial differential equations (PDEs) with random parameters is important for science and engineering. Forward UQ quantifies the impact of random parameters on the solution or the quantity-of-interest (QoI). In the current study, we propose a new extension of the stochastic finite volume (SFV) method by clustering samples in the parameter space. Compared to classic SFV based on structured grid in the parameter space, the new scheme based on clustering extends SFV to parameter spaces of higher dimensions. This paper presents the construction of SFV schemes for typical parametric elliptic, parabolic and hyperbolic equations for Darcy flows in porous media, as well as the error analysis, demonstration and validation of the new extension using typical reservoir simulation test cases.
Paper Structure (13 sections, 1 theorem, 51 equations, 11 figures, 2 tables)

This paper contains 13 sections, 1 theorem, 51 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Stochastic Finite Volume Schemes
  3. Basic Idea of SFV
  4. SFV scheme for elliptic steady-state single-phase Darcy flow
  5. SFV scheme for parabolic transient single-phase Darcy flow
  6. SFV scheme for hyperbolic transient two-phase Darcy flow
  7. A New SFV-Cluster Scheme
  8. Error analysis
  9. Test Cases
  10. Single-Phase Steady-State Darcy Flow with 2D Random Parameter
  11. Single-Phase Transient Darcy Flow with 10D Random Parameter
  12. Two-Phase Transient Darcy Flow with 10D Random Parameter
  13. Conclusions

Key Result

Theorem 4.1

The error of the expectation and variance are

Figures (11)

  • Figure 1: A realisation of the random permeability field for the single-phase steady-state Darcy flow example (a) and the corresponding steady-state pressure field (b).
  • Figure 2: Mean value for $p^{20}$ computed by MC over realisations and by SFV over cells in the parameter space in the steady-state test case. An isotropic Cartesian grid is built is the 2D parameter space for SFV.
  • Figure 3: Standard deviation for $p^{20}$ computed by MC over realisations and by SFV over cells in the parameter space in the steady-state test case.
  • Figure 4: Comparison between the convergence of mean $p^{20}$ for SFV schemes using Cartesian or sample clusters in the parameter space.
  • Figure 5: A realisation of the random permeability field for the single-phase transient Dary flow test case (a) and the corresponding pressure field at end of simulation (b).
  • ...and 6 more figures

Theorems & Definitions (2)

  • Theorem 4.1
  • proof