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Inspired by the novel ensemble method for the Navier-Stokes equations (NSE)in 2014 \cite{Nan-Layton-ensemble}, and the work of\cite{YanLuo-ZhuWang-ensemble}, this report develops an ensemble penalty methodfor the NSE. In addition, the method is extended to Monte Carlo randomsampling. The combination allows greater ensemble sizes with reduced complexityand thus gives a longer predictability horizon.

Penalty Ensembles for Navier-Stokes with Random Initial Conditions & Forcing