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Quasi-Monte Carlo finite element approximation of the Navier-Stokes equations with initial data modeled by log-normal random fields

Seungchan Ko, Guanglian Li, Yi Yu

TL;DR

This work addresses uncertainty quantification for the Navier–Stokes equations in a 2D bounded domain with log-normal random initial data by integrating a fully discrete FEM, truncated Karhunen–Loève expansion, and quasi-Monte Carlo (QMC) integration. The authors derive an RMSE bound that decomposes into finite element discretization, KL truncation, and QMC quadrature errors, and they prove dimension-robust convergence for randomly shifted lattice rules using a weighted Sobolev space framework. A key contribution is new regularity estimates for the parametric dependence of the nonlinear Navier–Stokes solution, enabling nearly optimal QMC rates that are independent of stochastic dimension under appropriate weight choices. Numerical experiments with Matérn-type random fields demonstrate that QMC substantially outperforms standard Monte Carlo, validating the method's efficiency for nonlinear PDEs with random initial data and highlighting avenues for further generalization and multilevel extensions.

Abstract

In this paper, we analyze the numerical approximation of the Navier-Stokes problem over a bounded polygonal domain in $\mathbb{R}^2$, where the initial condition is modeled by a log-normal random field. This problem usually arises in the area of uncertainty quantification. We aim to compute the expectation value of linear functionals of the solution to the Navier-Stokes equations and perform a rigorous error analysis for the problem. In particular, our method includes the finite element, fully-discrete discretizations, truncated Karhunen-Loéve expansion for the realizations of the initial condition, and lattice-based quasi-Monte Carlo (QMC) method to estimate the expected values over the parameter space. Our QMC analysis is based on randomly-shifted lattice rules for the integration over the domain in high-dimensional space, which guarantees the error decays with $\mathcal{O}(N^{-1+δ})$, where $N$ is the number of sampling points, $δ>0$ is an arbitrary small number, and the constant in the decay estimate is independent of the dimension of integration.

Quasi-Monte Carlo finite element approximation of the Navier-Stokes equations with initial data modeled by log-normal random fields

TL;DR

This work addresses uncertainty quantification for the Navier–Stokes equations in a 2D bounded domain with log-normal random initial data by integrating a fully discrete FEM, truncated Karhunen–Loève expansion, and quasi-Monte Carlo (QMC) integration. The authors derive an RMSE bound that decomposes into finite element discretization, KL truncation, and QMC quadrature errors, and they prove dimension-robust convergence for randomly shifted lattice rules using a weighted Sobolev space framework. A key contribution is new regularity estimates for the parametric dependence of the nonlinear Navier–Stokes solution, enabling nearly optimal QMC rates that are independent of stochastic dimension under appropriate weight choices. Numerical experiments with Matérn-type random fields demonstrate that QMC substantially outperforms standard Monte Carlo, validating the method's efficiency for nonlinear PDEs with random initial data and highlighting avenues for further generalization and multilevel extensions.

Abstract

In this paper, we analyze the numerical approximation of the Navier-Stokes problem over a bounded polygonal domain in , where the initial condition is modeled by a log-normal random field. This problem usually arises in the area of uncertainty quantification. We aim to compute the expectation value of linear functionals of the solution to the Navier-Stokes equations and perform a rigorous error analysis for the problem. In particular, our method includes the finite element, fully-discrete discretizations, truncated Karhunen-Loéve expansion for the realizations of the initial condition, and lattice-based quasi-Monte Carlo (QMC) method to estimate the expected values over the parameter space. Our QMC analysis is based on randomly-shifted lattice rules for the integration over the domain in high-dimensional space, which guarantees the error decays with , where is the number of sampling points, is an arbitrary small number, and the constant in the decay estimate is independent of the dimension of integration.
Paper Structure (13 sections, 17 theorems, 125 equations, 3 figures, 4 tables)

This paper contains 13 sections, 17 theorems, 125 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Weak formulation of the parametric problem
  3. Finite element approximation
  4. Truncation of the infinite-dimensional problem
  5. Quasi-Monte Carlo integration
  6. Solution regularity in the stochastic variables
  7. A function space setting in $\mathbb{R}^s$
  8. Error bound for randomly-shifted lattice rules
  9. Estimate for the weighted Sobolev norm
  10. Choice of weight parameters $\gamma_{\boldsymbol{\mathfrak{u}}}$
  11. Numerical experiments
  12. Conclusion
  13. Useful inequalities

Key Result

Theorem 2.1

For any given $\boldsymbol{u}^0\in H$, there exists a unique $\boldsymbol{u}\in L^{\infty}(0,T;H)\cap L^2(0,T;V)$ such that Furthermore, the following energy estimate holds

Figures (3)

  • Figure 1: Log-log plot of $\|\xi_j\|_{\mathbb{C}(\overline{D})}$, $\|\nabla \xi_j\|_{\mathbb{C}(\overline{D})}$ and $b_j$ against $j$ for the Matérn covariance with $\nu=2.5$, $\sigma^2=1$ and $\lambda_{\mathrm{C}}=1$.
  • Figure 2: The standard errors of $e_1$ with various Matérn covariance parameters for QMC and MC plotted versus the number of sampling points $N$.
  • Figure 3: The standard errors of $e_2$ with various Matérn covariance parameters for QMC and MC plotted versus the number of sampling points $N$.

Theorems & Definitions (26)

  • Theorem 2.1
  • Proposition 3.1
  • Theorem 3.2
  • Theorem 4.1
  • proof
  • Remark 4.2
  • Lemma 5.1
  • proof
  • Remark 5.2
  • Theorem 5.3
  • ...and 16 more