%0 Conference Paper %1 conf/mcqmc/BarthSS14 %A Barth, Andrea %A Schwab, Christoph %A Sukys, Jonas %B Monte Carlo and Quasi-Monte Carlo Methods. Springer Proceedings in Mathematics & Statistics %D 2014 %E Cools, Ronald %E Nuyens, Dirk %I Springer International Publishing %K 2014 B07 sfbtrr161 %P 209-227 %R 10.1007/978-3-319-33507-0_8 %T Multilevel Monte Carlo Simulation of Statistical Solutions to the Navier-Stokes Equations %U https://doi.org/10.1007/978-3-319-33507-0_8 %V 163 %X 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⊂Rd, 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). %@ 978-3-319-33507-0

@inproceedings{conf/mcqmc/BarthSS14, 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⊂Rd, 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).}, added-at = {2020-03-05T13:58:33.000+0100}, author = {Barth, Andrea and Schwab, Christoph and Sukys, Jonas}, biburl = {https://puma.ub.uni-stuttgart.de/bibtex/2c91ceee17d29065595663f07c8a68bb4/leonkokkoliadis}, booktitle = {Monte Carlo and Quasi-Monte Carlo Methods. Springer Proceedings in Mathematics & Statistics}, doi = {10.1007/978-3-319-33507-0_8}, editor = {Cools, Ronald and Nuyens, Dirk}, ee = {https://doi.org/10.1007/978-3-319-33507-0_8}, interhash = {efe57847ad23d4e827441df5f4c581bb}, intrahash = {c91ceee17d29065595663f07c8a68bb4}, isbn = {978-3-319-33507-0}, keywords = {2014 B07 sfbtrr161}, pages = {209-227}, publisher = {Springer International Publishing}, timestamp = {2020-03-05T12:58:33.000+0100}, title = {Multilevel Monte Carlo Simulation of Statistical Solutions to the Navier-Stokes Equations}, url = {https://doi.org/10.1007/978-3-319-33507-0_8}, volume = 163, year = 2014 }