Numerical methods for high-dimensional probability density function equations

Cho, Heyrim; Venturi, Daniele; Karniadakis, George

doi:10.1016/j.jcp.2015.10.030

Cited by 51 publications

(39 citation statements)

References 63 publications

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“…Among the various methods for UQ in PDEs, stochastic Galerkin (SG) methods based on generalized polynomial chaos (gPC) expansions are very attractive thanks to the spectral convergence property with respect to the random input [32,59,63,67,77,78,87,89]. On the other hand, their intrusive nature forces a complete reformulation of the problem and standard schemes for the corresponding deterministic problem cannot be used in a straightforward way.…”

Section: Stochastic Galerkin Methodsmentioning

confidence: 99%

Uncertainty Quantification for Kinetic Models in Socio–Economic and Life Sciences

Dimarco

Pareschi

Zanella

2017

SEMA SIMAI Springer Series

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Kinetic equations play a major rule in modeling large systems of interacting particles. Recently the legacy of classical kinetic theory found novel applications in socio-economic and life sciences, where processes characterized by large groups of agents exhibit spontaneous emergence of social structures. Well-known examples are the formation of clusters in opinion dynamics, the appearance of inequalities in wealth distributions, flocking and milling behaviors in swarming models, synchronization phenomena in biological systems and lane formation in pedestrian traffic. The construction of kinetic models describing the above processes, however, has to face the difficulty of the lack of fundamental principles since physical forces are replaced by empirical social forces. These empirical forces are typically constructed with the aim to reproduce qualitatively the observed system behaviors, like the emergence of social structures, and are at best known in terms of statistical information of the modeling parameters. For this reason the presence of random inputs characterizing the parameters uncertainty should be considered as an essential feature in the modeling process. In this survey we introduce several examples of such kinetic models, that are mathematically described by nonlinear Vlasov and Fokker-Planck equations, and present different numerical approaches for uncertainty quantification which preserve the main features of the kinetic solution.

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Section: Stochastic Galerkin Methodsmentioning

confidence: 99%

Uncertainty Quantification for Kinetic Models in Socio–Economic and Life Sciences

Dimarco

Pareschi

Zanella

2017

SEMA SIMAI Springer Series

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“…where I is the identity matrix, f n (z) = f (z, t n ), and τ n+1 (z) is the local truncation error of the Crank-Nicolson method at time t n+1 [63]. Rewriting this in a more compact notation yields [23] A…”

Section: Tensor Methods With Implicit Time Steppingmentioning

confidence: 99%

Parallel tensor methods for high-dimensional linear PDEs

Boelens

Venturi

Tartakovsky

2018

Journal of Computational Physics

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High-dimensional partial-differential equations (PDEs) arise in a number of fields of science and engineering, where they are used to describe the evolution of joint probability functions. Their examples include the Boltzmann and Fokker-Planck equations. We develop new parallel algorithms to solve high-dimensional PDEs. The algorithms are based on canonical and hierarchical numerical tensor methods combined with alternating least squares and hierarchical singular value decomposition. Both implicit and explicit integration schemes are presented and discussed. We demonstrate the accuracy and efficiency of the proposed new algorithms in computing the numerical solution to both an advection equation in six variables plus time and a linearized version of the Boltzmann equation.

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“…It arises in diverse areas of sciences as biology, finances, mechanics, and physics; see e.g. [12,13,15,24,19,16,25].…”

Section: The Liouville Equation and A Control Mechanismmentioning

confidence: 99%

A theoretical investigation of Brockett’s ensemble optimal control problems

et al. 2019

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This paper is devoted to the analysis of problems of optimal control of ensembles governed by the Liouville (or continuity) equation. The formulation and study of these problems have been put forward in recent years by R.W. Brockett, with the motivation that ensemble control may provide a more general and robust control framework.Following Brockett's formulation of ensemble control, a Liouville equation with unbounded drift function, and a class of cost functionals that include tracking of ensembles and different control costs is considered. For the theoretical investigation of the resulting optimal control problems, a well-posedness theory in weighted Sobolev spaces is presented for the Liouville and transport equations. Then, a class of non-smooth optimal control problems governed by the Liouville equation is formulated and existence of optimal controls is proved. Furthermore, optimal controls are characterised as solutions to optimality systems; such a characterisation is the key to get (under suitable assumptions) also uniqueness of optimal controls.

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Numerical methods for high-dimensional probability density function equations

Cited by 51 publications

References 63 publications

Uncertainty Quantification for Kinetic Models in Socio–Economic and Life Sciences

Uncertainty Quantification for Kinetic Models in Socio–Economic and Life Sciences

Parallel tensor methods for high-dimensional linear PDEs

A theoretical investigation of Brockett’s ensemble optimal control problems

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