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We present a finite difference-(multi-level) Monte Carlo algorithm to efficiently compute statistical solutions of the two-dimensional incompressible Navier-Stokes equations (NSE), with periodic boundary conditions and for arbitrary high Reynolds number. We propose a reformulation of statistical solutions in the sense of Foiaş and Prodi in the vorticity-stream function form. The vorticity-stream function formulation of the NSE in two-space dimensions is discretized with a finite difference scheme. We obtain a convergence rate error estimate for this approximation which is explicit in the viscosity parameter , under realistic assumptions on the solution regularity. We also prove convergence and complexity estimates, for the (multi-level) Monte Carlo finite difference algorithm to compute statistical solutions. Numerical experiments illustrating the validity of our estimates are presented. They show that the multi-level Monte Carlo algorithm can significantly accelerate the computation of statistical solutions in the sense of Foiaş and Prodi, even for very high Reynolds numbers. [ABSTRACT FROM AUTHOR]

We combine the spectral (viscosity) method and ensemble averaging to propose an algorithm that computes admissible measure-valued solutions of the incompressible Euler equations. The resulting approximate young measures are proved to converge (with increasing numerical resolution) to a measure-valued solution. We present numerical experiments demonstrating the robustness and efficiency of the proposed algorithm, as well as the appropriateness of measure-valued solutions as a solution framework for the Euler equations. Furthermore, we report an extensive computational study of the two-dimensional vortex sheet, which indicates that the computed measure-valued solution is non-atomic and implies possible non-uniqueness of weak solutions constructed by Delort. [ABSTRACT FROM AUTHOR]