Asymptotic preserving time‐discretization of optimal control problems for the Goldstein–Taylor model

Albi, Giacomo; Herty, Michaël; Jörres, Christian; Pareschi, Lorenzo

doi:10.1002/num.21877

Cited by 5 publications

(11 citation statements)

References 39 publications

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“…This will allow us to obtain not only explicit decay rates for each Fourier mode, but also an "optimal Lyapunov functional" for such given mode, with which we where the defectiveness appears (hence the circle). From that point onwards the decay rate decreases, and is of order O 1 σ will then be able to construct a non-modal entropy functional in terms of a pseudo-differential operator as defined in (4). As was mentioned in §2, this will give us intuition to the large time behaviour of the equation in several cases even when σ (x) is not constant.…”

Section: Constant Relaxation Functionmentioning

confidence: 88%

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Large Time Convergence of the Non-homogeneous Goldstein-Taylor Equation

et al. 2021

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The Goldstein-Taylor equations can be thought of as a simplified version of a BGK system, where the velocity variable is constricted to a discrete set of values. It is intimately related to turbulent fluid motion and the telegrapher’s equation. A detailed understanding of the large time behaviour of the solutions to these equations has been mostly achieved in the case where the relaxation function, measuring the intensity of the relaxation towards equally distributed velocity densities, is constant. The goal of the presented work is to provide a general method to tackle the question of convergence to equilibrium when the relaxation function is not constant, and to do so as quantitatively as possible. In contrast to the usual modal decomposition of the equations, which is natural when the relaxation function is constant, we define a new Lyapunov functional of pseudodifferential nature, one that is motivated by the modal analysis in the constant case, that is able to deal with full spatial dependency of the relaxation function. The approach we develop is robust enough that one can apply it to multi-velocity Goldstein-Taylor models, and achieve explicit rates of convergence. The convergence rate we find, however, is not optimal, as we show by comparing our result to those found in [8].

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Section: Constant Relaxation Functionmentioning

confidence: 88%

“…Various AP-schemes for this model in the stiff relaxation regime (i.e. for ) were constructed and analyzed in [ 4 , 16 , 17 ]. Since the large time convergence of solutions to ( 1 ) towards its unique steady state is also the topic of this work, we shall review the related literature in more detail:…”

Section: Introductionmentioning

confidence: 99%

Large Time Convergence of the Non-homogeneous Goldstein-Taylor Equation

et al. 2021

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“…Under the natural physical assumption of symmetry in the velocities (i.e. n i =1 v i = 0) and the expectation that the solutions will converge towards a state that is equally distributed in v and constant in x 4 , we find one potential multi-velocity extension of the Goldstein-Taylor model on T × (0, ∞):…”

Section: Convergence To Equilibrium In a 3−velocity Goldstein-taylor ...mentioning

confidence: 99%

“…Various AP-schemes for this model in the stiff relaxation regime (i.e. for σ → ∞) were constructed and analyzed in [17,16,4]. Since the large time convergence of solutions to (1.1) towards its unique steady state is also the topic of this work, we shall review the related literature in more detail:…”

Section: Introductionmentioning

confidence: 99%

Large time convergence of the non-homogeneous Goldstein-Taylor Equation

Arnold,

Einav,

Signorello

et al. 2020

Preprint

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The Goldstein-Taylor equations can be thought of as a simplified version of a BGK system, where the velocity variable is constricted to a discrete set of values. It is intimately related to turbulent fluid motion and the telegrapher's equation. A detailed understanding of the large time behaviour of the solutions to these equations has been mostly achieved in the case where the relaxation function, measuring the intensity of the relaxation towards equally distributed velocity densities, is constant. The goal of the presented work is to provide a general method to tackle the question of convergence to equilibrium when the relaxation function is not constant, and to do so as quantitatively as possible. In contrast to the usual modal decomposition of the equations, which is natural when the relaxation function is constant, we define a new Lyapunov functional of pseudodifferential nature, one that is motivated by the modal analysis in the constant case, that is able to deal with full spatial dependency of the relaxation function. The approach we develop is robust enough that one can apply it to multi-velocity Goldstein-Taylor models, and achieve explicit rates of convergence. The convergence rate we find, however, is not optimal, as we show by comparing our result to the that found in [8]. KEYWORDS. BGK equation, hypocoercivity, large time behaviour, exponential decay, Lyapunov functional MSC. 82C40 (Kinetic theory of gases in time-dependent statistical mechanics), 35B40 (Asymptotic behavior of solutions to PDEs), 35Q82 (PDEs in connection with statistical mechanics), 35S05 (Pseudodifferential operators as generalizations of partial differential operators)

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“…However, direct applications of standard numerical schemes to the adjoint differential systems of the optimal control problem may lead to order reduction problems [19,33]. Besides classical applications to ODEs these problems gained interest recently in PDEs, in particular in the field of hyperbolic and kinetic equations [1,2,24,29].…”

Section: Introductionmentioning

confidence: 99%

Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems

Albi

Herty

Pareschi

2019

Applied Mathematics and Computation

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We are interested in high-order linear multistep schemes for time discretization of adjoint equations arising within optimal control problems. First we consider optimal control problems for ordinary differential equations and show loss of accuracy for Adams-Moulton and Adams-Bashford methods, whereas BDF methods preserve high-order accuracy. Subsequently we extend these results to semi-lagrangian discretizations of hyperbolic relaxation systems. Computational results illustrate theoretical findings.

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Asymptotic preserving time‐discretization of optimal control problems for the Goldstein–Taylor model

Cited by 5 publications

References 39 publications

Large Time Convergence of the Non-homogeneous Goldstein-Taylor Equation

Large Time Convergence of the Non-homogeneous Goldstein-Taylor Equation

Large time convergence of the non-homogeneous Goldstein-Taylor Equation

Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems

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