论文标题
Navier-Stokes方程的准蒙特卡洛有限元近似，其初始数据由对数正态随机字段建模
Quasi-Monte Carlo finite element approximation of the Navier-Stokes equations with initial data modeled by log-normal random fields
论文作者
论文摘要
在本文中，我们分析了$ \ mathbb {r}^2 $上有限的多边形域上Navier-Stokes问题的数值近似，其中初始条件是由对数符号正常随机字段建模的。这个问题通常是在不确定性定量领域出现的。我们旨在计算解决方案的解决方案线性功能的期望值，并对问题进行严格的错误分析。特别是，我们的方法包括有限元，完全差异化的离散化，截短的karhunen-loéve扩展以实现初始条件，以及基于晶状体的准蒙特卡（QMC）方法，以估计参数空间上的预期值。 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.
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.
