A fully symmetric nonlinear biorthogonal decomposition theory for random fields

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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“…, z 6 ). We first split the variables as (z 1 , z 2 , z 3 ) and (z 4 , z 5 , z 6 ) through one Schmidt decomposition [46] as…”

Section: Hierarchical Tensor Methodsmentioning

confidence: 99%

Parallel tensor methods for high-dimensional linear PDEs

Boelens

Venturi

Tartakovsky

2018

Journal of Computational Physics

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“…[75,76]). 7 The proof is simple, and it relies on the limits In fact, these equations allow us to conclude that Figure 5. Suppression of stochastic resonance with increasing noise colour.…”

Section: U(x)mentioning

confidence: 97%

“…For example, Faetti et al [13] have shown that for = 0 the correction to (3.7) due to the fourth-order cumulant is O(σ 4 τ 2 ) for exponentially correlated Gaussian noise (see eqn (4.18) in [13] time τ goes to zero (white-noise limit), then equation (3.7), with C(t, s) defined in (3.4), consistently reduces to the classical Fokker-Planck equation. 7 Next, we study the transient dynamics of px(t) within the time interval [0,3]. To this end, we consider the following set of parameters μ = 1, ν = 1, Ω = 10 and = 0.5, leading to a slow relaxation to statistical equilibrium.…”

Section: U(x)mentioning

confidence: 99%

“…Well-known approaches are polynomial chaos [1][2][3], multi-element and sparse adaptive probabilistic collocation [4,5], high-dimensional model representations [6], stochastic biorthogonal expansions [7] and separated representations [8,9]. These techniques can provide considerable speed-up in computational time when compared with Monte Carlo (MC) or quasi-MC methods, for low to moderate number of stochastic dimensions.…”

Section: Introductionmentioning

confidence: 99%

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Convolutionless Nakajima–Zwanzig equations for stochastic analysis in nonlinear dynamical systems

Venturi

Karniadakis

2014

Proc. R. Soc. A.

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Determining the statistical properties of stochastic nonlinear systems is of major interest across many disciplines. Currently, there are no general efficient methods to deal with this challenging problem that involves high dimensionality, low regularity and random frequencies. We propose a framework for stochastic analysis in nonlinear dynamical systems based on goal-oriented probability density function (PDF) methods. The key idea stems from techniques of irreversible statistical mechanics, and it relies on deriving evolution equations for the PDF of quantities of interest, e.g. functionals of the solution to systems of stochastic ordinary and partial differential equations. Such quantities could be low-dimensional objects in infinite dimensional phase spaces. We develop the goal-oriented PDF method in the context of the time-convolutionless Nakajima-Zwanzig-Mori formalism. We address the question of approximation of reduced-order density equations by multi-level coarse graining, perturbation series and operator cumulant resummation. Numerical examples are presented for stochastic resonance and stochastic advection-reaction problems.

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“…In this section we propose a different stochastic modeling approach for f (t) based on bi-orthogonal representations random processes [57,61,56,3,2]. To illustrate the method, we study the case where the observable u(t) is real valued (one-dimensional) and square integrable.…”

Section: Stochastic Low-dimensional Modelingmentioning

confidence: 99%

Generalized Langevin Equations for Systems with Local Interactions

Zhu

Venturi

2020

J Stat Phys

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We present a new method to approximate the Mori-Zwanzig (MZ) memory integral in generalized Langevin equations (GLEs) describing the evolution of observables in high-dimensional nonlinear systems with local interactions. Building upon the Faber operator series we recently developed for the orthogonal dynamics propagator, and an exact combinatorial algorithm that allows us to compute memory kernels to any desired accuracy from first principles, we demonstrate that the proposed method is effective in computing auto-correlation functions, intermediate scattering functions and other important statistical properties of the observables. We also develop a new stochastic process representation that combines MZ memory kernels and Karhunen-Loève (KL) series expansions to build reduced-order models of observables in statistical equilibrium. Numerical applications are presented for the Fermi-Pasta-Ulam β-model, and for random wave propagation in homegeneous media.

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A fully symmetric nonlinear biorthogonal decomposition theory for random fields

Cited by 31 publications

References 66 publications

Parallel tensor methods for high-dimensional linear PDEs

Parallel tensor methods for high-dimensional linear PDEs

Convolutionless Nakajima–Zwanzig equations for stochastic analysis in nonlinear dynamical systems

Generalized Langevin Equations for Systems with Local Interactions

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