Wasserstein gradient flow formulation of the time-fractional Fokker–Planck equation

Duong, Manh Hong; Jin, Bangti

doi:10.4310/cms.2020.v18.n7.a6

Cited by 6 publications

(2 citation statements)

References 33 publications

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“…, which appeared in (14). Note that by definition of the q-logarithmic function we have ln q C 0 (q, d)…”

Section: Penalization Of Barycenters For Q-gaussian Measuresmentioning

confidence: 99%

A GPM-based algorithm for solving regularized Wasserstein barycenter problems in some spaces of probability measures

Kum¹,

Duong²,

Lim³

et al. 2020

Preprint

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In this paper we study the penalization of barycenters in the Wasserstein space for ϕ-exponential distributions. We obtain an explicit characterization of the barycenter in terms of the variances of the measures generalizing existing results for Gaussian measures. We then develop a gradient projection method for the computation of the barycenter establishing a Lipstchitz continuity for the gradient function. We also numerically show the influence of parameters and stability of the algorithm under small perturbation of data.PENALIZATION OF BARYCENTERS FOR ϕ-EXPONENTIAL DISTRIBUTIONS Wasserstein barycenters and optimal transports have been explored [27,18]. Several computational methods for the computation of the barycenter have been developed [12,2,21,28].Recently Wasserstein barycenters has found many applications in statistics, image processing and machine learning [29,23,31]. We refer the reader to the mentioned papers and references therein for a more detailed account of the topic.The case γ > 0 has been studied in the recent paper [8] where the existence, uniqueness and stability of a minimizer, which is called the penalized barycenter, has been established. The regularization parameter γ was proved to provide smooth barycenters especially when the input probability measures are irregular which is useful for data analysis [7,30]. In addition, the penalized barycenter problem also resembles the discretization formulation of Wasserstein gradient flows for dissipative evolution equations [17,3,10] and the fractional heat equation [14] at a given time step where {µ i } represent discretized solutions at the previous steps and γ is proportional to the time-step parameter.Gaussian measures play an important role in the study of Wasserstein barycenter problem since in this case an useful characterization of the barycenter exists [1,6] which gives rise to efficient computational algorithms such as the fixed point approach [2] and the gradient projection method [21]. Our aim in this paper is to seek for a large class of probability measures so that the penalized barycenter can be explicitly characterized and computed similarly to the case of Gaussian measures. We will study the penalization problem (1) for an important classes of probability measures, namely ϕ-exponential measures, where the entropy functional is the Tsallis entropy functional respectively. The class of ϕ-exponential measures significantly enlarges that of Gaussian measures and containing also q-Gaussian measures as special cases, cf. Section 1.3 below. To state our main results, we briefly recall the definition of ϕ-exponential measures; more detailed will be given in Section 2.1.3. ϕ-exponential distributions. Let ϕ be an increasing, positive, continuous function on (0, ∞), the ϕ-logarithmic is defined by [33] ln ϕ (t) := t 1

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“…, which appeared in (14). Note that by definition of the q-logarithmic function we have ln q C 0 (q, d)…”

Section: Penalization Of Barycenters For Q-gaussian Measuresmentioning

confidence: 99%

A GPM-based algorithm for solving regularized Wasserstein barycenter problems in some spaces of probability measures

Kum¹,

Duong²,

Lim³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Some of their time-fractional counterparts have been investigated in the literature, e.g., the time-fractional gradient flows of type Allen-Cahn [20], Cahn-Hilliard [21], Keller-Segel [22], and Fokker-Planck [23] type. Up to now there is no unified theory for time-fractional gradient flows, and it is not yet known whether the dissipation of energy is fulfilled, see also the discussions in [24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy

Fritz¹,

Khristenko²,

Wohlmuth³

2021

Preprint

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Time-fractional partial differential equations are nonlocal in time and show an innate memory effect. In this work, we propose an augmented energy functional which includes the history of the solution. Further, we prove the equivalence of a time-fractional gradient flow problem to an integer-order one based on our new energy. This equivalence guarantees the dissipating character of the augmented energy. The state function of the integerorder gradient flow acts on an extended domain similar to the Caffarelli-Silvestre extension for the fractional Laplacian. Additionally, we apply a numerical scheme for solving time-fractional gradient flows, which is based on kernel compressing methods. We illustrate the behavior of the original and augmented energy in the case of the Ginzburg-Landau energy functional.

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On nonlocal Fokker–Planck equations with nonlinear force fields and perturbations

Ke,

Quyet,

Thanh

2024

Monatsh Math

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Wasserstein gradient flow formulation of the time-fractional Fokker–Planck equation

Cited by 6 publications

References 33 publications

A GPM-based algorithm for solving regularized Wasserstein barycenter problems in some spaces of probability measures

A GPM-based algorithm for solving regularized Wasserstein barycenter problems in some spaces of probability measures

Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy

On nonlocal Fokker–Planck equations with nonlinear force fields and perturbations

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