Multilevel Monte Carlo Simulation of Statistical Solutions to the Navier–Stokes Equations

Abstract

We propose Monte Carlo (MC), single level Monte Carlo (SLMC) and multilevel Monte Carlo (MLMC) methods for the numerical approximation of statistical solutions to the viscous, incompressible Navier-Stokes equations (NSE) on a bounded, connected domain D ⊂ R d , d = 1, 2 with no-slip or periodic boundary conditions on the boundary ∂D. The MC convergence rate of order 1/2 is shown to hold independently of the Reynolds number with constant depending only on the mean kinetic energy of the initial velocity ensemble. We discuss the effect of space-time discretizations on the MC convergence. We propose a numerical MLMC estimator, based on finite samples of numerical solutions with finite mean kinetic energy in a suitable function space and give sufficient conditions for mean-square convergence to a (generalized) moment of the statistical solution. We provide in particular error bounds for MLMC approximations of statistical solutions to the viscous Burgers equation in space dimension d = 1 and to the viscous, incompressible Navier-Stokes equations in space dimension d = 2 which are uniform with respect to the viscosity parameter. For a more detailed presentation and proofs we refer the reader to Barth et al. (Multilevel Monte Carlo approximations of statistical solutions of the Navier-Stokes equations, 2013, [6]).

Multilevel Monte Carlo applied to chemical engineering systems subject to uncertainty

“…23 However, MLMC has been shown to converge and perform well in numerous applications. 23,36,39,40 Another limitation is that the MLMC Theorem requires many constants to be known or approximated, 23 which adds the need for preliminary sampling when these constants are not available. However, the computational time for MLMC has been shown to be orders of magnitude less than for standard MC sampling, 22,23 which compensates for the time spent on preliminary sampling.…”

“…The main limitation of the MLMC method is that the algorithm is heuristic and is not guaranteed to converge . However, MLMC has been shown to converge and perform well in numerous applications . Another limitation is that the MLMC Theorem requires many constants to be known or approximated, which adds the need for preliminary sampling when these constants are not available.…”

“…Since the original publication, 22 the MLMC method has been expanded to estimating the expected values of various types of ordinary differential equations (ODEs), 23,24 continuous time Markov chains, 25 computing mean exit times, 26 central statistical moments of arbitrary order, 27 and even probability density functions. [28][29][30][31] Furthermore, MLMC has been applied to uncertainty analysis in flows through porous media, [31][32][33][34][35][36][37][38] statistical solutions of the Navier-Stokes equations, 39 reliability analysis of engineered systems, 40 and surfactant/polymer enhanced-oil-recovery. 41 In all cases, improvements in computational costs associated with obtaining accurate results have been reported.…”

Seepage Time Soft Sensor Model of Nonwoven Fabric Based on the Extreme Learning Machine Integrating Monte Carlo

“…The MC algorithm is a random simulation method based on probability and statistical theory. It aims to establish a probability model or random process whose parameters are equal to the solution of the problem, and then use a computer to complete a statistical simulation or sampling, calculate the statistical characteristics of the required parameters, and then obtain the approximate solution to the problem [28]. The reliability parameters of the system are expressed as Equation (5).…”