Mori-Zwanzig Approach to Uncertainty Quantification

Venturi, Daniele; Cho, Heyrim; Karniadakis, George Em

doi:10.1007/978-3-319-11259-6_28-1

Cited by 15 publications

(24 citation statements)

References 134 publications

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“…Now we treat the exact 'local' system d u/dt = L u + r as non-autonomously 'forced' by coupling to all cross-sections in X through the remainder (aka Mori-Zwanzig transformation, e.g., Venturi et al 2015). There are two justifications, both a simple and a rigorous, for being able to project such 'forcing' onto the local model.…”

Section: Account For the Coupling Remaindermentioning

confidence: 99%

Rigorous modelling of nonlocal interactions determines a macroscale advection-diffusion PDE

Roberts¹

2020

Preprint

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A slowly-varying or thin-layer multiscale assumption empowers macroscale understanding of many physical scenarios from dispersion in pipes and rivers, including beams, shells, and the modulation of nonlinear waves, to homogenisation of micro-structures. Here we begin a new exploration of the scenario where the given physics has non-local microscale interactions. We rigorously analyse the dynamics of a basic example of shear dispersion. Near each cross-section, the dynamics is expressed in the local moments of the microscale non-local effects. Centre manifold theory then supports the local modelling of the system's dynamics with coupling to neighbouring cross-sections as a non-autonomous forcing. The union over all cross-sections then provides powerful new support for the existence and emergence of a macroscale model advection-diffusion pde global in the large, finitesized, domain. The approach quantifies the accuracy of macroscale advection-diffusion approximations, and has the potential to open previously intractable multiscale issues to new insights. Contents 1 Introduction 2 2 Zappa shear dispersion 4

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Section: Account For the Coupling Remaindermentioning

confidence: 99%

Rigorous modelling of nonlocal interactions determines a macroscale advection-diffusion PDE

Roberts¹

2020

Preprint

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“…The most attracting feature of the MZ theory is that it allows us to systematically derive exact evolution equations, now known as the generalized Langevin equations (GLEs), for any quantities of interest based on the microscopic equations of motion. Such GLEs can be used as the ansatz for coarse-grained models which found applications in molecular dynamics [22,38,42,9,8,44,45], fluid mechanics [33,32], and, more generally, systems described by nonlinear partial differential equations (PDEs) [39,3,37,36,24,25].…”

Section: Introductionmentioning

confidence: 99%

A simple hypocoercivity analysis for the effective Mori-Zwanzig equation

Zhu¹

2021

Preprint

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We provide a simple hypocoercivity analysis for the effective Mori-Zwanzig equation governing the time evolution of noise-averaged observables in a stochastic dynamical system. Under the hypocoercivity framework mainly developed by Dolbeault, Mouhot and Schmeiser and further extended by Grothaus and Stilgenbauer, we prove that under the same conditions which lead to the geometric ergodicity of the Markov semigroup e tK , the Mori-Zwanzig orthogonal semigroup e tQKQ is also geometrically ergodic, provided that P = I − Q is a finite-rank, orthogonal projection operator in a certain Hilbert space. The result is applied to the widely used Mori-type effective Mori-Zwanzig equations in the coarse-grained modeling of molecular dynamics and leads to exponentially decaying estimates for the memory kernel and the fluctuation force.

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“…where x(t) ∈ R n is the system's state (random process), u(t) ∈ R m is the (deterministic) control, and x 0 (ω) ∈ R n is an random initial condition with prescribed probability density function p 0 (x). As is well known [25,24,22], uncertain parameters in f can always be transferred to the initial condition. The solution to the Cauchy problem (1) is a function of the initial state x 0 and a functional of the control u(t), i.e., x(t) = x(t, x 0 , [u]).…”

Section: Introductionmentioning

confidence: 99%

A new scalable algorithm for computational optimal control under uncertainty

Lambrianides,

Gong,

Venturi

2019

Preprint

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We address the design and synthesis of optimal control strategies for high-dimensional stochastic dynamical systems. Such systems may be deterministic nonlinear systems evolving from random initial states, or systems driven by random parameters or processes. The objective is to provide a validated new computational capability for optimal control which will be achieved more efficiently than current state-of-the-art methods. The new framework utilizes direct single or multi-shooting discretization, and is based on efficient vectorized gradient computation with adaptable memory management. The algorithm is demonstrated to be scalable to high dimensional nonlinear control systems with random initial condition and unknown parameters.

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Mori-Zwanzig Approach to Uncertainty Quantification

Cited by 15 publications

References 134 publications

Rigorous modelling of nonlocal interactions determines a macroscale advection-diffusion PDE

Rigorous modelling of nonlocal interactions determines a macroscale advection-diffusion PDE

A simple hypocoercivity analysis for the effective Mori-Zwanzig equation

A new scalable algorithm for computational optimal control under uncertainty

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