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Higher-order Quasi-Monte Carlo Training of Deep Neural Networks
(2020)SAM Research ReportWe 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 ...Report -
Monte-Carlo Finite-Volume Methods in Uncertainty Quantification for Hyperbolic Conservation Laws
(2017)SAM Research ReportReport -
Numerical approximation of statistical solutions of incompressible flow
(2015)Research ReportWe 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 ...Report -
Multi-Level Monte Carlo Finite Volume methods for uncertainty quantification of acoustic wave propagation in random heterogeneous layered medium
(2014)Research ReportWe consider the very challenging problem of efficient uncertainty quantification for acoustic wave propagation in a highly heterogeneous, possibly layered, random medium, characterized by possibly anisotropic, piecewise log-exponentially distributed Gaussian random fields. A multi-level Monte Carlo finite volume method is proposed, along with a novel, bias-free upscaling technique that allows to represent the input random fields, generated ...Report -
Numerical solution of scalar conservation laws with random flux functions
(2012)Research ReportWe consider scalar hyperbolic conservation laws in several space dimensions, with a class of random (and parametric) flux functions. We propose a Karhunen–Loève expansion on the state space of the random flux. For random flux functions which are Lipschitz continuous with respect to the state variable, we prove the existence of a unique random entropy solution. Using a Karhunen–Loève spectral decomposition of the random flux into principal ...Report -
Numerical solution of scalar conservation laws with random flux functions
(2012)SAM Research ReportWe consider scalar hyperbolic conservation laws in several space dimensions, with a class of random (and parametric) flux functions. We propose a Karhunen-Loève expansion on the state space of the random flux. For random flux functions which are Lipschitz continuous with respect to the state variable, we prove the existence of a unique random entropy solution. Using a Karhunen-Loève spectral decomposition of the random flux into principal ...Report -
Multi-level Monte Carlo finite difference and finite volume methods for stochastic linear hyperbolic systems
(2012)SAM Research ReportWe 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 ...Report -
Multi-level Monte Carlo finite volume methods for uncertainty quantification in nonlinear systems of balance laws
(2012)Research ReportReport -
Multi-level Monte Carlo finite volume methods for shallow water equations with uncertain topography in multi-dimensions
(2011)SAM Research ReportThe initial data and bottom topography, used as inputs in shallow water models, are prone to uncertainty due to measurement errors. We model this uncertainty statistically in terms of random shallow water equations. We extend the Multi-Level Monte Carlo (MLMC) algorithm to numerically approximate the random shallow water equations efficiently. The MLMC algorithm is suitably modified to deal with uncertain (and possibly uncorrelated) data ...Report -
Static load balancing for multi-level Monte Carlo finite volume solvers
(2011)SAM Research ReportThe Multi-Level Monte Carlo finite volumes (MLMC-FVM) algorithm was shown to be a robust and fast solver for uncertainty quantification in the solutions of multi- dimensional systems of stochastic conservation laws. A novel load balancing procedure is used to ensure scalability of the MLMC algorithm on massively parallel hardware. We describe this procedure together with other arising challenges in great detail. Finally, numerical experiments ...Report