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The Hessian Riemannian flow and Newton’S method for effective Hamiltonians and Mather measures

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arXiv 2018年

作者： Gomes, Diogo A. Yang, Xianjin CEMSE Division Thuwal23955-6900 Saudi Arabia KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering

Effective Hamiltonians arise in multiple problems, including homogenization of Hamilton-Jacobi equations, nonlinear control systems, Hamiltonian dynamics, and Aubry-Mather theory. In Aubry-Mather theory, related objects, Mather measures, are also of great importance. Here, we combine ideas from mean-field games with the Hessian Riemannian flow to compute effective Hamiltonians and Mather measures simultaneously. We prove the existence and convergence of the Hessian Riemannian flow. In addition, we explore the relation between the Hessian Riemannian flow and a variant of Newton’s method that greatly improves the performance of our numerical methods. In our numerical experiments, we see that our algorithms preserve the non-negativity of Mather measures and are more stable than previous methods in problems that are close to singular. Copyright © 2018, The Authors. All rights reserved.

关键词： Numerical methods

Multilevel monte carlo acceleration of seismic wave propagation under uncertainty

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arXiv 2018年

作者： Ballesio, Marco Beck, Joakim Pandey, Anamika Parisi, Laura von Schwerin, Erik Tempone, Raúl Thuwal23955-6900 Saudi Arabia KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering Thuwal23955-6900 Saudi Arabia Alexander von Humboldt Professor in Mathematics for Uncertainty Quantification RWTH Aachen University Germany

We interpret uncertainty in a model for seismic wave propagation by treating the model parameters as random variables, and apply the Multilevel Monte Carlo (MLMC) method to reduce the cost of approx-imating expected values of selected, physically relevant, quantities of interest (QoI) with respect to the random variables. Targeting source inversion problems, where the source of an earthquake is inferred from ground motion recordings on the Earth's surface, we consider two QoIs that measure the discrepancies between computed seismic signals and given reference signals: One QoI, QE, is defined in terms of the L2-misfit, which is directly related to maximum likelihood estimates of the source parameters;the other, QW, is based on the quadratic Wasserstein distance between probability distributions, and represents one possible choice in a class of such misfit functions that have become increasingly popular to solve seismic inversion in recent years. We simulate seismic wave propagation, including seismic attenuation, using a publicly available code in widespread use, based on the spectral element method. Using random coefficients and deterministic initial and boundary data, we present benchmark numerical experiments with synthetic data in a two-dimensional physical domain and a one-dimensional velocity model where the assumed parameter uncertainty is motivated by realistic Earth models. Here, the computational cost of the standard Monte Carlo method was reduced by up to 97% for QE, and up to 78% for QW, using a relevant range of tolerances. Shifting to three-dimensional domains is straight-forward and will further increase the relative computational work reduction. Copyright © 2018, The Authors. All rights reserved.

关键词： Monte Carlo methods

A multifidelity multilevel Monte Carlo method for uncertainty propagation in aerospace applications 19th

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A multifidelity multilevel Monte Carlo method for uncertaint...

19th AIAA Non-Deterministic Approaches Conference, 2017

作者： Gianluca, Geraci Eldred, Michael S. Iaccarino, Gianluca Flow Physics and Computational Engineering Stanford University Optimization and Uncertainty Quantification Department Sandia National Laboratories StanfordCA94305 United States Computer Science Research Institute Sandia National Laboratories Optimization and Uncertainty Quantification Department AlbuquerqueNM87185 United States Flow Physics and Computational Engineering Stanford University Mechanical Engineering Department StanfordCA94305 United States

ISBN: (纸本)9781624104527

The accurate evaluation of the performance of complex engineering devices needs to rely on high-fidelity numerical simulations and the systematic characterization and propagation of uncertainties. Several sources of uncertainty may impact the performance of an engineering device through operative conditions, manufacturing tolerances, and even physical models. In the presence of multiphysics systems the number of the uncertain parameters can be fairly large and their propagation through the numerical codes still remains prohibitive because the overall computational budget often allows for only an handful of such high-fidelity realizations. On the other side, common engineering practice can take advantage from a solid history of development and assessment of so called low-fidelity models which albeit less accurate are often capable to at least capture overall trends and parameter dependencies of the system. In this contribution we address the forward propagation of uncertainty parameters relying on statistical estimators built on sequences of numerical and physical discretizations which are provably convergent to the high-fidelity statistics, while exploiting low-fidelity computational models to increase the reliability and confidence in the numerical predictions. The performances of the approaches are presented by means of two fairly complicated aerospace problems, namely the aero-thermo-structural analysis of a turbofan engine nozzle and a flow through a scramjet-like device. © 2017, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved.

关键词： Turbofan engines

First-order, stationary mean-field games with congestion

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arXiv 2017年

作者： Evangelista, David Ferreira, Rita Gomes, Diogo A. Nurbekyan, Levon Voskanyan, Vardan CEMSE Division Thuwal23955-6900 Saudi Arabia KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering

Mean-field games (MFGs) are models for large populations of competing rational agents that seek to optimize a suitable functional. In the case of congestion, this functional takes into account the difficulty of moving in high-density areas. Here, we study stationary MFGs with congestion with quadratic or power-like Hamiltonians. First, using explicit examples, we illustrate two main difficulties: the lack of classical solutions and the existence of areas with vanishing density. Our main contribution is a new variational formulation for MFGs with congestion. This formulation was not previously known, and, thanks to it, we prove the existence and uniqueness of solutions. Finally, we consider applications to numerical methods.35J47, 35A01 Copyright © 2017, The Authors. All rights reserved.

关键词： Numerical methods

On the uniqueness of minimizers for a class of variational problems with polyconvex integrand

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arXiv 2017年

作者： Awi, Romeo Sedjro, Marc Institute for Mathematics and Its Applications 207 Church Street MinneapolisMN55455 United States CEMSE Division Thuwal23955-6900 Saudi Arabia KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering

We prove existence and uniqueness of minimizers for a family of energy functionals that arises in Elasticity and involves polyconvex integrands over a certain subset of displacement maps. This work extends previous results by Awi and Gangbo to a larger class of integrands. First, we study these variational problems over displacements for which the determinant is positive. Second, we consider a limit case in which the functionals are degenerate. In that case, the set of admissible displacements reduces to that of incompressible displacements which are measure preserving maps. Finally, we establish that the minimizer over the set of incompressible maps may be obtained as a limit of minimizers corresponding to a sequence of minimization problems over general displacements provided we have enough regularity on the dual problems. We point out that these results defy the direct methods of the calculus of variations.35B27, 39J40 Copyright © 2017, The Authors. All rights reserved.

关键词： Variational techniques

Monotone numerical methods for finite-state mean-field games

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arXiv 2017年

作者： Gomes, Diogo A. Saúde, João CEMSE Division Thuwal23955-6900 Saudi Arabia KAUST SRI Uncertainty Quantification Center in Computational Science and Engineering Carnegie Mellon University Electrical and Computer Engineering department 5000 Forbes Avenue PittsburghPA15213-3890 United States

Here, we develop numerical methods for finite-state mean-field games (MFGs) that satisfy a monotonicity condition. MFGs are determined by a system of differential equations with initial and terminal boundary conditions. These non-standard conditions are the main difficulty in the numerical approximation of solutions. Using the monotonicity condition, we build a flow that is a contraction and whose fixed points solve the MFG, both for stationary and time-dependent problems. We illustrate our methods in a MFG modeling the paradigm-shift problem.91A13, 91A10, 49M30. Copyright © 2017, The Authors. All rights reserved.

关键词： Numerical methods

Application of Bayesian networks for estimation of individual psychological characteristics

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arXiv 2017年

作者： Litvinenko, Alexander Litvinenko, Natalya Mamyrbayev, Orken Extreme Computing Research Center Center for Uncertainty Quantification in Computational Science & Engineering King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia Applied research department Institute of Mathematics and Mathematical Modeling CS MES RK Almaty Kazakhstan Institute of information and computational technologies CS MES RK Almaty Kazakhstan

An accurate qualitative and comprehensive assessment of human potential is one of the most important challenges in any company or collective. We apply Bayesian networks for developing more accurate overall estimations of psychological characteristics of an individual, based on psychological test results, which identify how much an individual possesses a certain trait. Examples of traits could be a stress resistance, the readiness to take a risk, the ability to concentrate on certain complicated work. The most common way of studying psychological characteristics of each individual is testing. Additionally, the overall estimation is usually based on personal experiences and the subjective perception of a psychologist or a group of psychologists about the investigated psychological personality traits.62C10, 62C12 Copyright © 2017, The Authors. All rights reserved.

关键词： Bayesian networks

Advanced multilevel Monte Carlo methods

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arXiv 2017年

作者： Jasra, Ajay Law, Kody Suciu, Carina Department of Statistics & Applied Probability National University of Singapore Singapore117546 Singapore Computer Science and Mathematics Division Oak Ridge National Laboratory Oak RidgeTN37934 United States Center for Uncertainty Quantification in Computational Science & Engineering King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

This article reviews the application of advanced Monte Carlo techniques in the context of Multilevel Monte Carlo (MLMC). MLMC is a strategy employed to compute expectations which can be biased in some sense, for instance, by using the discretization of a associated probability law. The MLMC approach works with a hierarchy of biased approximations which become progressively more accurate and more expensive. Using a telescoping representation of the most accurate approximation, the method is able to reduce the computational cost for a given level of error versus i.i.d. sampling from this latter approximation. All of these ideas originated for cases where exact sampling from couples in the hierarchy is possible. This article considers the case where such exact sampling is not currently possible. We consider Markov chain Monte Carlo and sequential Monte Carlo methods which have been introduced in the literature and we describe different strategies which facilitate the application of MLMC within these methods. Copyright © 2017, The Authors. All rights reserved.

关键词： Monte Carlo methods

Multilevel monte carlo in approximate bayesian computation

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arXiv 2017年

作者： Jasra, Ajay Jo, Seongil Nott, David Shoemaker, Christine Tempone, Raul Department of Statistics & Applied Probability & Operations Research Cluster National University of Singapore Singapore117546 Singapore Department of Civil & Environmental Engineering & Operations Research Cluster National University of Singapore Singapore119260 Singapore Center for Uncertainty Quantification in Computational Science & Engineering King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

In the following article we consider approximate Bayesian computation (ABC) inference. We introduce a method for numerically approximating ABC posteriors using the multilevel Monte Carlo (MLMC). A sequential Monte Carlo version of the approach is developed and it is shown under some assumptions that for a given level of mean square error, this method for ABC has a lower cost than i.i.d. sampling from the most accurate ABC approximation. Several numerical examples are given. Copyright © 2017, The Authors. All rights reserved.

关键词： Bayesian networks