Lagrangian Numerical Approximations to One‐Dimensional Convolution‐Diffusion Equations

Gosse, Laurent; Toscani, Giuseppe

doi:10.1137/050628015

Cited by 52 publications

(45 citation statements)

References 33 publications

(93 reference statements)

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“…Moreover, these ideas have been used in [27,28] for numerical purposes. Explicit in time numerical schemes for the equations of the inverse distribution function are proposed keeping the contraction of the Wasserstein distance at the discrete level.…”

Section: One-dimensional Casementioning

confidence: 99%

Convergence of the Mass-Transport Steepest Descent Scheme for the Subcritical Patlak–Keller–Segel Model

Blanchet¹,

Cálvez²,

Carrillo³

2008

SIAM J. Numer. Anal.

187

292

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Abstract. Variational steepest descent approximation schemes for the modified Patlak-Keller-Segel equation with a logarithmic interaction kernel in any dimension are considered. We prove the convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of the Euclidean Wasserstein distance, associated to this equation for sub-critical masses. As a consequence, we recover the recent result about the global in time existence of weak-solutions to the modified Patlak-Keller-Segel equation for the logarithmic interaction kernel in any dimension in the sub-critical case. Moreover, we show how this method performs numerically in one dimension. In this particular case, this numerical scheme corresponds to a standard implicit Euler method for the pseudo-inverse of the cumulative distribution function. We demonstrate its capabilities to reproduce easily without the need of mesh-refinement the blow-up of solutions for super-critical masses.

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Section: One-dimensional Casementioning

confidence: 99%

Convergence of the Mass-Transport Steepest Descent Scheme for the Subcritical Patlak–Keller–Segel Model

Blanchet¹,

Cálvez²,

Carrillo³

2008

SIAM J. Numer. Anal.

187

292

View full text Add to dashboard Cite

show abstract

“…This scheme can be understood as an explicit firstorder version of the advection-diffusion operator introduced in the previous section. In this setting we can follow the lines of [15,14] and derive a bound on ∆t that makes the scheme (3.1) monotonicity preserving:…”

Section: Monotonicity Preservation Of the Diffusion-taxis Operatormentioning

confidence: 99%

A Hybrid Mass Transport Finite Element Method for Keller–Segel Type Systems

2019

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We propose a new splitting scheme for general reaction-taxis-diffusion systems in one spatial dimension capable to deal with simultaneous concentrated and diffusive regions as well as travelling waves and merging phenomena. The splitting scheme is based on a mass transport strategy for the cell density coupled with classical finite element approximations for the rest of the system. The built-in mass adaption of the scheme allows for an excellent performance even with respect to dedicated mesh-adapted AMR schemes in original variables.

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“…A major bottleneck of this approach is the computation of W 2 2 (ρ, ρ k ), which is an infinite dimensional minimization problem. Hence, early works that use (3) avoid direct computing W 2 2 (ρ, ρ k ) either by linearization [11,42] or by diffeomorphisms [10,23,24], which lead to methods that lose some inherited properties in (3) or are limited by complicated geometry and structure. Only recent progress in computing W 2 has enabled the direct application of (3) [6,27,32,56].…”

Section: Introductionmentioning

confidence: 99%

Fisher information regularization schemes for Wasserstein gradient flows

Wang

2020

Journal of Computational Physics

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We propose a variational scheme for computing Wasserstein gradient flows. The scheme builds upon the Jordan-Kinderlehrer-Otto framework with the Benamou-Brenier's dynamic formulation of the quadratic Wasserstein metric and adds a regularization by the Fisher information. This regularization can be derived in terms of energy splitting and is closely related to the Schrödinger bridge problem. It improves the convexity of the variational problem and automatically preserves the non-negativity of the solution. As a result, it allows us to apply sequential quadratic programming to solve the sub-optimization problem. We further save the computational cost by showing that no additional time interpolation is needed in the underlying dynamic formulation of the Wasserstein-2 metric, and therefore, the dimension of the problem is vastly reduced. Several numerical examples, including porous media equation, nonlinear Fokker-Planck equation, aggregation diffusion equation, and Derrida-Lebowitz-Speer-Spohn equation, are provided. These examples demonstrate the simplicity and stableness of the proposed scheme.As written, equation (1) possesses two immediate properties: it preserves the nonnegativity of the solution and conserves total mass. Therefore, in what follows, we will always consider nonnegative initial data with mass one, so that the solution is in the set of probability measures on Ω, P(Ω). The third property of (1) is the dissipation of the energy, which can be seen as follows. Given an energy E : P(Ω) → R ∪ {+∞}, we may formally define its gradient with respect to the quadratic Wasserstein metric W 2 as ∇ W 2 E(ρ) = −∇ · (ρ∇δE) ,

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Lagrangian Numerical Approximations to One‐Dimensional Convolution‐Diffusion Equations

Cited by 52 publications

References 33 publications

Convergence of the Mass-Transport Steepest Descent Scheme for the Subcritical Patlak–Keller–Segel Model

Convergence of the Mass-Transport Steepest Descent Scheme for the Subcritical Patlak–Keller–Segel Model

A Hybrid Mass Transport Finite Element Method for Keller–Segel Type Systems

Fisher information regularization schemes for Wasserstein gradient flows

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