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Goal-Oriented Adaptive Finite Element Multilevel Monte Carlo with Convergence Rates

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arXiv 2022年

作者： Beck, Joakim Liu, Yang von Schwerin, Erik Tempone, Raúl Thuwal23955-6900 Saudi Arabia KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering Saudi Arabia Alexander von Humboldt Professor in Mathematics for Uncertainty Quantification RWTH Aachen University Aachen52062 Germany

In this study, we present an adaptive multilevel Monte Carlo (AMLMC) algorithm for approximating deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity coefficient and geometric singularities in bounded domains of Rd. Our AMLMC algorithm is built on the results of the weak convergence rates in the work [Moon et al., BIT Numer. Math., 46 (2006), pp. 367-407] for an adaptive algorithm using isoparametric d-linear quadrilateral finite element approximations and the dual weighted residual error representation in a deterministic setting. Designed to suit the geometric nature of the singularities in the solution, our AMLMC algorithm uses a sequence of deterministic, non-uniform auxiliary meshes as a building block. The above-mentioned deterministic adaptive algorithm generates these meshes, corresponding to a geometrically decreasing sequence of tolerances. In particular, for a given realization of the diffusivity coefficient and accuracy level, AMLMC constructs its approximate sample using the first mesh in the hierarchy that satisfies the corresponding bias accuracy constraint. This adaptive approach is particularly useful for the lognormal case treated here, which lacks uniform coercivity and thus produces functional outputs that vary over orders of magnitude when sampled. Furthermore, we discuss iterative solvers and compare their efficiency with direct ones. To reduce computational work, we propose a stopping criterion for the iterative solver with respect to the quantity of interest, the realization of the diffusivity coefficient, and the desired level of AMLMC approximation. From the numerical experiments, based on a Fourier expansion of the diffusivity coefficient field, we observe improvements in efficiency compared with both standard Monte Carlo (MC) and standard MLMC (SMLMC) for a problem with a singularity similar to that at the tip of a slit modeling a *** Codes 65C05, 65N

关键词： Adaptive algorithms

Particle approximation of one-dimensional mean-field-games with local interactions

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arXiv 2021年

作者： Di Francesco, Marco Duisembay, Serikbolsyn Gomes, Diogo Aguiar Ribeiro, Ricardo Via Vetoio 1 Coppito L’AquilaI-67100 Italy CEMSE Division Thuwal23955-6900 Saudi Arabia KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering

We study a particle approximation for one-dimensional first-order Mean-Field-Games (MFGs) with local interactions with planning conditions. Our problem comprises a system of a Hamilton-Jacobi equation coupled with a transport equation. As we deal with the planning problem, we prescribe initial and terminal distributions for the transport equation. The particle approximation builds on a semi-discrete variational problem. First, we address the existence and uniqueness of a solution to the semi-discrete variational problem. Next, we show that our discretization preserves some previously identified conserved quantities. Finally, we prove that the approximation by particle systems preserves displacement convexity. We use this last property to establish uniform estimates for the discrete problem. We illustrate our results for the discrete problem with numerical *** Codes 49M25, 91A06, 91A16 Copyright © 2021, The Authors. All rights reserved.

关键词： Variational techniques

A Wasserstein Coupled Particle Filter for Multilevel Estimation

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arXiv 2020年

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

In this paper, we consider the filtering problem for partially observed diffusions, which are regularly observed at discrete times. We are concerned with the case when one must resort to time-discretization of the diffusion process if the transition density is not available in an appropriate form. In such cases, one must resort to advanced numerical algorithms such as particle filters to consistently estimate the filter. It is also well known that the particle filter can be enhanced by considering hierarchies of discretizations and the multilevel Monte Carlo (MLMC) method, in the sense of reducing the computational effort to achieve a given mean square error (MSE). A variety of multilevel particle filters (MLPF) have been suggested in the literature, e.g., in Jasra et al., SIAM J, Numer. Anal., 55, 3068–3096. Here we introduce a new alternative that involves a resampling step based on the optimal Wasserstein coupling. We prove a central limit theorem (CLT) for the new method. On considering the asymptotic variance, we establish that in some scenarios, there is a reduction, relative to the approach in the aforementioned paper by Jasra et al., in computational effort to achieve a given MSE. These findings are confirmed in numerical examples. We also consider filtering diffusions with unstable dynamics;we empirically show that in such cases a change of measure technique seems to be required to maintain our *** Codes 65C05 (Primary) 65C20, 65C30 (Secondary) Copyright © 2020, The Authors. All rights reserved.

关键词： Monte Carlo methods

The selection problem for some first-order stationary mean-field games

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arXiv 2019年

作者： Gomes, Diogo A. Mitake, Hiroyoshi Terai, Kengo CEMSE Division Thuwal23955-6900 Saudi Arabia KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering Graduate School of Mathematical Sciences University of Tokyo 3-8-1 Komaba Meguro-ku Tokyo153-8914 Japan

Here, we study the existence and the convergence of solutions for the vanishing discount MFG problem with a quadratic Hamiltonian. We give conditions under which the discounted problem has a unique classical solution and prove convergence of the vanishing-discount limit to a unique solution up to constants. Then, we establish refined asymptotics for the limit. When those conditions do not hold, the limit problem may not have a unique solution and its solutions may not be smooth, as we illustrate in an elementary example. Finally, we investigate the stability of regular weak solutions and address the selection problem. Using ideas from Aubry-Mather theory, we establish a selection criterion for the *** Codes 35A01, 49L25, 91A13 Copyright © 2019, The Authors. All rights reserved.

关键词： Mechanics

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

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