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Multilevel Monte Carlo approximations of statistical solutions to the Navier-Stokes equations

by A. Barth and Ch. Schwab and J. Sukys

(Report number 2013-33)

Abstract
We present Monte Carlo and multilevel Monte Carlo discretizations for the numerical approximation of the statistical solution to the viscous, incompressible Navier--Stokes equation in a bounded domain \(D\subset \mathbb{R}^d\). We prove that Monte Carlo sampling produces sequences of sample averages of (Leray-Hopf) solutions to the Navier--Stokes equations which converge to a (generalized) moment of a (in two space dimensions unique) statistical solution (in the sense of Foias} and Prodi), at the rate \(M^{-1/2}\) in terms of the number of samples \(M\in\mathbb{N}\). The convergence rate \(M^{-1/2}\) is shown to hold independently of the Reynolds number, with constant depending only on the mean kinetic energy of the initial velocity. We discuss the effect of a space-time discretization on the Monte Carlo convergence with particular attention on the kinematic viscosity \(\nu\) resp. on the Reynolds number. For a multilevel Monte Carlo estimator, composed of ensembles of solutions with finite mean kinetic energy in \(L^2(D)\), we establish robust mean-square convergence to a (generalized) moment of the statistical solution. It is concluded that robust (i.e. Reynolds number independent) convergence rates are possible for multilevel Monte Carlo sample averages provided that solution samples on coarse discretization levels are computed with turbulence models which deliver mean-square consistent bulk properties of the turbulent flow.

Keywords: Navier--Stokes equation, statistical solutions, multilevel Monte Carlo, turbulence modeling

@Techreport{BSS13_530,
  author = {A. Barth and Ch. Schwab and J. Sukys},
  title = {Multilevel Monte Carlo approximations of statistical solutions to the Navier-Stokes equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-33},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-33.pdf },
  year = {2013}
}

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