Conjugate flow action functionals

Venturi, Daniele

doi:10.1063/1.4827679

Cited by 9 publications

(15 citation statements)

References 44 publications

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“…Such derivatives come from the exponential operators e sL 0 within the time convolution term. Note that such convolution can be also expressed as a functional derivative [133] of the exponential operator along Ä 1 .x/, by using an identity of Feynman (see [38], Eq. (6) or [151]).…”

Section: Stochastic Advection-reactionmentioning

confidence: 99%

Mori-Zwanzig Approach to Uncertainty Quantification

Venturi¹,

Cho²,

Karniadakis³

2016

Handbook of Uncertainty Quantification

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Determining the statistical properties of nonlinear random systems is a problem of major interest in many areas of physics and engineering. Even with recent theoretical and computational advancements, no broadly applicable technique has yet been developed for dealing with the challenging problems of high dimensionality, low regularity and random frequencies often exhibited by the system. The Mori-Zwanzig and the effective propagator approaches discussed in this chapter have the potential of overcoming some of these limitations, in particular the curse of dimensionality and the lack of regularity. The key idea stems from techniques of irreversible statistical mechanics, and it relies on developing exact evolution equations and corresponding numerical methods for quantities of interest, e.g., functionals of the solution to stochastic ordinary and partial differential equations. Such quantities of interest could be lowdimensional objects in infinite-dimensional phase spaces, e.g., the lift of an airfoil in a turbulent flow, the local displacement of a structure subject to random loads (e.g., ocean waves loading on an offshore platform), or the macroscopic properties of materials with random microstructure (e.g., modeled atomistically in terms of particles). We develop the goal-oriented framework in two different, although related, mathematical settings: the first one is based on the Mori-Zwanzig projection operator method, and it yields exact reduced-order equations for the quantity of interest. The second approach relies on effective propagators, i.e., integrals of exponential operators with respect to suitable distributions. Both methods can be applied to nonlinear systems of stochastic ordinary and partial differential equations subject to random forcing terms, random boundary conditions, or random initial conditions.

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Section: Stochastic Advection-reactionmentioning

confidence: 99%

Mori-Zwanzig Approach to Uncertainty Quantification

Venturi¹,

Cho²,

Karniadakis³

2016

Handbook of Uncertainty Quantification

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“…The quantity dF η ([θ ]) is known as Gâteaux differential of F in the direction of η [83,90]. Under rather mild conditions (see, e.g., [90, p. 37]), such differential can be represented as a linear operator acting on η [65,92]. Such linear operator is known as the Gâteaux derivative of F at θ , and it will be denoted by…”

Section: Differentials and Derivatives Of Nonlinear Functionalsmentioning

confidence: 99%

“…In fact, nonlinear functionals were used, for example, by Wiener to describe Brownian motion mathematically [100], by Hohenberg and Kohn [47] to reduce the dimensionality of the Schrödinger equation in many-body quantum systems [56,67], by Hopf to describe the statistical properties of turbulence [2,48,63], and by Bogoliubov to model systems of interacting bosons in superfluid liquid helium [12,84]. Applications of nonlinear functionals to other areas of mathematical physics can be found in [3,37,52,54,59,92].…”

Section: Introductionmentioning

confidence: 99%

Spectral methods for nonlinear functionals and functional differential equations

Venturi

Dektor

2021

Res Math Sci

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We present a rigorous convergence analysis for cylindrical approximations of nonlinear functionals, functional derivatives, and functional differential equations (FDEs). The purpose of this analysis is twofold: First, we prove that continuous nonlinear functionals, functional derivatives, and FDEs can be approximated uniformly on any compact subset of a real Banach space admitting a basis by high-dimensional multivariate functions and high-dimensional partial differential equations (PDEs), respectively. Second, we show that the convergence rate of such functional approximations can be exponential, depending on the regularity of the functional (in particular its Fréchet differentiability), and its domain. We also provide necessary and sufficient conditions for consistency, stability and convergence of cylindrical approximations to linear FDEs. These results open the possibility to utilize numerical techniques for high-dimensional systems such as deep neural networks and numerical tensor methods to approximate nonlinear functionals in terms of high-dimensional functions, and compute approximate solutions to FDEs by solving high-dimensional PDEs. Numerical examples are presented and discussed for prototype nonlinear functionals and for an initial value problem involving a linear FDE.

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“…The quantity dF η ([θ]) is known as Gâteaux differential of F in the direction of η [66,60]. Under rather mild conditions (see, e.g., [66, p. 37]) such differential can be represented as a linear operator acting on η [44,68] . Such linear operator is known as the Gâteaux derivative of F and and it will be denoted by F ([θ])…”

Section: Differentials and Derivatives Of Nonlinear Functionalsmentioning

confidence: 99%

“…For instance, functionals were used by Wiener for the description of Brownian motion [76], by Hohenberg and Kohn [30] to reduce the dimensionality of the Schrödinger equation in many-body quantum systems (density functional theory [46? ]), by Hopf to describe the statistical properties of turbulence [31,1,43,41], and by Bogoliubov to model systems of interacting bosons in superfluid liquid helium [8,61]. Other applications of nonlinear functionals can be found in [68,15,47,46,36,64,48,75,35,40].…”

Section: Introductionmentioning

confidence: 99%

The numerical approximation of nonlinear functionals and functional differential equations

Venturi

2018

Physics Reports

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We develop a rigorous convergence analysis for finite-dimensional approximations of nonlinear functionals, functional derivatives, and functional differential equations (FDEs) defined on compact subsets of real separable Hilbert spaces. The purpose this analysis is twofold: first, we prove that under rather mild conditions, nonlinear functionals, functional derivatives and FDEs can be approximated uniformly by high-dimensional multivariate functions and high-dimensional partial differential equations (PDEs), respectively. Second, we prove that such functional approximations can converge exponentially fast, depending on the regularity of the functional (in particular its Fréchet differentiability), and its domain. We also provide sufficient conditions for consistency, stability and convergence of functional approximation schemes to compute the solution of FDEs, thus extending the Lax-Richtmyer theorem from PDEs to FDEs. Numerical applications are presented and discussed for prototype nonlinear functional approximation problems, and for linear FDEs.

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Conjugate flow action functionals

Cited by 9 publications

References 44 publications

Mori-Zwanzig Approach to Uncertainty Quantification

Mori-Zwanzig Approach to Uncertainty Quantification

Spectral methods for nonlinear functionals and functional differential equations

The numerical approximation of nonlinear functionals and functional differential equations

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