Noncommutative cumulants for stochastic differential equations and for generalized Dyson series

Hegerfeldt, Gerhard C.; Schulze, Henrik

doi:10.1007/bf01028479

Cited by 16 publications

(16 citation statements)

References 13 publications

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“…Many other equivalent forms of the Mori-Zwanzig equation can be constructed (see the Appendix in [136]), exactly for the same reason as why it is possible to represent an effective propagator of reduced-order dynamics in terms of generalized operator cumulants [55,73,74,97]. So far, everything that has been said is exact, and it led us to the equation of motion (22), which is linear and local in time.…”

Section: Time-convolutionless Form Of the Mori-zwanzig Equationmentioning

confidence: 99%

“…This is because we may be discarding terms growing like t k , if the norm of˝L k˛d oes not decay rapidly enough. A classical approach to overcome these limitations is to use operator cumulants [55,64,66,73,74]. For autonomous dynamical systems, we have the exact formula (see, e.g., [4])˝e tL˛D e he tL I i c ;…”

Section: Effective Propagatorsmentioning

confidence: 99%

“…Following Kubo [73,74], we emphasize that many different types of operator cumulants can be defined. Disregarding, for the moment, the specific prescription we use to construct such operator cumulants (see [55,74]), let us differentiate (29) with respect to time and take (35) into account. This yields the following exact reduced-order PDF equation…”

Section: Effective Propagatorsmentioning

confidence: 99%

“…By using the interaction representation of quantum mechanics [9,149], then it is quite straightforward to obtain from (22) and (27) an effective approximation (see [136]). One way to do so is to expand the operators (23) in a cumulant series, e.g., in terms of Kubo-Van Kampen operator cumulants [55,64,74,97], involving increasing powers of (coupling parameter). Any finite-order truncation of such series then represents an approximation to the exact MZ-PDF equation.…”

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confidence: 99%

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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: Time-convolutionless Form Of the Mori-zwanzig Equationmentioning

confidence: 99%

Section: Effective Propagatorsmentioning

confidence: 99%

Section: Effective Propagatorsmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Mori-Zwanzig Approach to Uncertainty Quantification

Venturi¹,

Cho²,

Karniadakis³

2016

Handbook of Uncertainty Quantification

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“…The second choice (A 35), introduced by Terwiel [56] (see also [39,55]), yields a solution in the form (A 33) if we select…”

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confidence: 99%

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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