Conservative‐dissipative approximation schemes for a generalized Kramers equation

Duong, Manh Hong; Peletier, MA Mark; Zimmer, Johannes

doi:10.1002/mma.2994

Cited by 23 publications

(40 citation statements)

References 30 publications

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“…Gradient flows and large-deviation principles As mentioned in the introduction, this approach using the duality formulation of the rate functionals is motivated by our recent results on the connection between generalised gradient flows and large-deviation principles [2,3,24,26,27,52]. We want to discuss here how the two overlap but are not the same.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Variational approach to coarse-graining of generalized gradient flows

et al. 2017

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In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (a) a natural interaction between the duality structure and the coarse-graining, (b) application to systems with nondissipative effects, and (c) application to coarse-graining of approximate solutions which solve the equation only to some error. As examples, we use this technique to solve three limit problems, the overdamped limit of the Vlasov-Fokker-Planck equation and the smallnoise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom. Mathematics Subject Classification

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Section: Conclusion and Discussionmentioning

confidence: 99%

Variational approach to coarse-graining of generalized gradient flows

et al. 2017

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“…where the vector b n .h/ and the two matrices B n .h/ and A n .h/ are given explicitly in (20), (24) and (19), respectively.…”

Section: Theorem 12mentioning

confidence: 99%

“…When

n = 2, C_{2, h} (x_{0}, x_{1}; y_{0}, y_{1}) = \frac{1}{h} ([], | y_{1} - y_{0} |^{2} + 12 {(||, \frac{x_{1} - x_{0}}{h} - \frac{y_{1} + y_{0}}{2})}^{2})

and

{scriptW}_{2}

is the minimal acceleration cost function. This cost function has been used to construct variational formulation for the Kramers equation (Equation previously with additional terms coming from external and frictional forces) showing that the Kramer equation is a (generalised) gradient flow of the Boltzmann entropy with respect to the Monge–Kantorovich transport cost

{scriptW}_{2}

. In addition,

{scriptW}_{2}

has also been used in constructing variational schemes for other evolution equations such as the system of isentropic Euler equations and the compressible Euler equations .…”

Section: Introductionmentioning

confidence: 99%

Analysis of the mean squared derivative cost function

Duong

Tran

2017

Math Methods in App Sciences

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In this paper, we investigate the mean squared derivative cost functions that arise in various applications such as in motor control, biometrics and optimal transport theory. We provide qualitative properties, explicit analytical formulas and computational algorithms for the cost functions. We also perform numerical simulations to illustrate the analytical results. In addition, as a by-product of our analysis, we obtain an explicit formula for the inverse of a Wronskian matrix that is of independent interest in linear algebra and differential equations theory.Comment: 28 page

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“…Many equations are now proven to belong to this class [2,[11][12][13][14]20,21,23,24]. Recently attempts have been made to extend this theory to discrete settings [22,25] and to systems that also contain conservative behavior [10,15,16]. The second approach is to understand why the Wasserstein metric and the combination of it with the entropy functional appear in the JKOformulation.…”

Section: The Porous Medium Equationmentioning

confidence: 99%

Asymptotic equivalence of the discrete variational functional and a rate-large-deviation-like functional in the Wasserstein gradient flow of the porous medium equation

Duong

2015

Asymptotic Analysis

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In this paper, we study the Wasserstein gradient flow structure of the porous medium equation restricted to q-Gaussians. The JKO-formulation of the porous medium equation gives a variational functional K h , which is the sum of the (scaled-) Wasserstein distance and the internal energy, for a time step h. We prove that, for the case of q-Gaussians on the real line, K h is asymptotically equivalent, in the sense of Γ -convergence as h tends to zero, to a rate-large-deviation-like functional. The result explains why the Wasserstein metric as well as the combination of it with the internal energy play an important role.1 E 1 is the Boltzmann-Shannon entropy. 0921-7134/15/$35.00 © 2015 -IOS Press and the authors. All rights reserved 86 M.H. Duong / Asymptotic equivalence of the discrete variational functional with respect to the Wasserstein distance W 2 on the space of probability measures with finite second moment P 2 (R d ),where Γ (μ, ν) is the set of all probability measures on R d × R d with marginals μ and ν. This statement can be understood in a variety of different ways. In [27], Otto shows this from a differential geometry point of view by considering (P 2 (R d ), W 2 ) as an infinite dimensional Riemannian manifold. However, for the purpose of this paper, the most useful way is that the solution t → ρ(t, x) can be approximated by the JKO-discretization scheme as follows [17,31].

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Conservative‐dissipative approximation schemes for a generalized Kramers equation

Cited by 23 publications

References 30 publications

Variational approach to coarse-graining of generalized gradient flows

Variational approach to coarse-graining of generalized gradient flows

Analysis of the mean squared derivative cost function

Asymptotic equivalence of the discrete variational functional and a rate-large-deviation-like functional in the Wasserstein gradient flow of the porous medium equation

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