Model reduction of controlled Fokker–Planck and Liouville–von Neumann equations

Benner, Peter; Breiten, Tobias; Hartmann, Carsten; Schmidt, Burkhard

doi:10.3934/jcd.2020001

Cited by 2 publications

(3 citation statements)

References 55 publications

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“…Its form is suggested by (5) and (10). A justification of the differentiability of V p and a formula for its derivative, used in the above expression, can be found in [9,Lemma 7].…”

Section: The Value Function Is Continuously Differentiable Onmentioning

confidence: 99%

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Numerical study of polynomial feedback laws for a bilinear control problem

Breiten¹,

Kunisch²,

Pfeiffer³

2018

Mathematical Control &Amp; Related Fields

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An infinite-dimensional bilinear optimal control problem with infinite-time horizon is considered. The associated value function can be expanded in a Taylor series around the equilibrium, the Taylor series involving multilinear forms which are uniquely characterized by generalized Lyapunov equations. A numerical method for solving these equations is proposed. It is based on a generalization of the balanced truncation model reduction method and some techniques of tensor calculus, in order to attenuate the curse of dimensionality. Polynomial feedback laws are derived from the Taylor expansion and are numerically investigated for a control problem of the Fokker-Planck equation. Their efficiency is demonstrated for initial values which are sufficiently close to the equilibrium.

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“…Its form is suggested by (5) and (10). A justification of the differentiability of V p and a formula for its derivative, used in the above expression, can be found in [9,Lemma 7].…”

Section: The Value Function Is Continuously Differentiable Onmentioning

confidence: 99%

“…[6]. It has already been used in the context of the Fokker-Planck equation in [5]. Let us briefly summarize it.…”

Section: Model Reductionmentioning

confidence: 99%

Numerical study of polynomial feedback laws for a bilinear control problem

Breiten¹,

Kunisch²,

Pfeiffer³

2018

Mathematical Control &Amp; Related Fields

Self Cite

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“…Related work on model reduction of controlled multiscale diffusions using duality arguments and Fleming's technique of logarithmic transformations (see [17,Ch. VI]) has been carried by one of the authors [7,22,20,21]. Despite recent progress on the theoretical foundations of G-(F)BSDE and G-Brownian motion, there have been relatively few practically oriented works in the context of uncertainty quantification (see e.g.…”

Section: Introductionmentioning

confidence: 99%

Model reduction and uncertainty quantification of multiscale diffusions with parameter uncertainties using nonlinear expectations

Bouanani,

Hartmann,

Kebiri

2021

Preprint

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In this paper we study model reduction of linear and bilinear quadratic stochastic control problems with parameter uncertainties. Specifically, we consider slow-fast systems with unknown diffusion coefficient and study the convergence of the slow process in the limit of infinite scale separation. The aim of our work is two-fold: Firstly, we want to propose a general framework for averaging and homogenisation of multiscale systems with parametric uncertainties in the drift or in the diffusion coefficient. Secondly, we want to use this framework to quantify the uncertainty in the reduced system by deriving a limit equation that represents a worst-case scenario for any given (possibly path-dependent) quantity of interest. We do so by reformulating the slow-fast system as an optimal control problem in which the unknown parameter plays the role of a control variable that can take values in a closed bounded set. For systems with unknown diffusion coefficient, the underlying stochastic control problem admits an interpretation in terms of a stochastic differential equation driven by a G-Brownian motion. We prove convergence of the slow process with respect to the nonlinear expectation on the probability space induced by the G-Brownian motion. The idea here is to formulate the nonlinear dynamic programming equation of the underlying control problem as a forward-backward stochastic differential equation in the G-Brownian motion framework (in brief: G-FBSDE), for which convergence can be proved by standard means. We illustrate the theoretical findings with two simple numerical examples, exploiting the connection between fully nonlinear dynamic programming equations and second-order BSDE (2BSDE): a linear quadratic Gaussian regulator problem and a bilinear multiplicative triad that is a standard benchmark system in turbulence and climate modelling.

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Model reduction of controlled Fokker–Planck and Liouville–von Neumann equations

Cited by 2 publications

References 55 publications

Numerical study of polynomial feedback laws for a bilinear control problem

Numerical study of polynomial feedback laws for a bilinear control problem

Model reduction and uncertainty quantification of multiscale diffusions with parameter uncertainties using nonlinear expectations

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