We present a finite difference-(Multi-level) Monte Carlo algorithm to efficiently compute statistical solutions of the two dimensional Navier-Stokes equations, with periodic bound- ary conditions and for arbitrarily high Reynolds number. We propose a reformulation of statistical solutions in the vorticity-stream function form. The vorticity-stream function for- mulation is discretized with a finite difference scheme. We obtain a convergence ...

We present a novel algorithmic approach and an error analysis leveraging Quasi-Monte Carlo points for training deep neural network (DNN) surrogates of Data-to-Observable (DtO) maps in engineering design. Our analysis reveals higher-order consistent, deterministic choices of training points in the input data space for deep and shallow Neural Networks with holomorphic activation functions such as tanh. These novel training points are proved ...

We extend the Multi-Level Monte Carlo (MLMC) algorithm of [19] in order to quantify uncertainty in the solutions of multi-dimensional hyperbolic systems of conservation laws with uncertain initial data. The algorithm is presented and several issues arising in the massively parallel numerical implementation are addressed. In particular, we present a novel load balancing procedure that ensures scalability of the MLMC algorithm on massively ...

We consider stochastic multi-dimensional linear hyperbolic systems of conservation laws. We prove existence and uniqueness of a random weak solution, provide estimates for the regularity of the solution in terms of regularities of input data, and show existence of statistical moments. Bounds for mean square error vs. expected work are proved for the Multi-Level Monte Carlo Finite Volume algorithm which is used to approximate the moments ...