Kinetic and Moment Models for Cell Motion in Fiber Structures

Borsche, Raul; Klar, Axel; Schneider, Florian

doi:10.1007/978-3-030-20297-2_1

Cited by 3 publications

(3 citation statements)

References 57 publications

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“…Accordingly, the term 'particle' is used here in a fairly general sense and may refer not only to classical particles like gas molecules but also to biological entities such as cells and animals (e.g. [25,49,69,78,79,81,118,130]). However, in this thesis we will mostly restrict ourselves to the radiative transfer equation [39,162] in three dimensions, where the particles are photons and the collision operator of the form (1.3.3) describes scattering of photons in a medium.…”

Section: Radiative Transfermentioning

confidence: 99%

First-order continuous- and discontinuous-Galerkin moment models for a linear kinetic equation: Model derivation and realizability theory

Schneider

Leibner

2020

Journal of Computational Physics

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Moment models are a class of specialized approximate models for kinetic transport equations. These models transform the kinetic equation to a system of equations for weighted velocity averages of the solution, called moments, thereby removing the velocity dependency. The properties of the resulting models depend on the chosen weight functions for the moments and on the approach used to close the equations. Closing the system by specifying a linear ansatz function results in linear models that are comparatively easy to solve but may show non-physical behaviour. Minimum-entropy moment closures, on the other hand, conserve many of the fundamental physical properties but require the solution of a non-linear optimization problem for every cell in the space-time grid. In addition, these models are only well-defined if the moments can be kept within a particular subset of the real coordinate space, the so-called realizable set, which is particularly challenging for higher-order numerical solvers.Schließlich beschäftigen wir uns mit der Möglichkeit einer zusätzlichen Modellreduktion für parameterabhängige Momentenmodelle. Hierzu nutzen wir die Reduzierte-Basis-Methode und berechnen mit der Hauptkomponentenanalyse ("Proper Orthogonal Decomposition", POD) ein reduziertes Modell. Wir stellen die hierarchische approximative POD (HAPOD) vor, ein universelles und einfach zu implementierendes Verfahren um eine approximative POD zu berechnen. Die HAPOD nutzt eine fast beliebig an die verfügbaren Ressourcen anpassbare Baumstruktur, so dass jeder Rechenknoten nur die POD einer relativ kleinen Teilmenge der Eingangsvektoren berechnen muss. Wir führen eine ausführliche theoretische Analyse der HAPOD durch und zeigen die Anwendbarkeit auf lineare Momentenmodelle sowie eine gegenüber der POD deutlich verbesserte Effizienz.

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Section: Radiative Transfermentioning

confidence: 99%

First-order continuous- and discontinuous-Galerkin moment models for a linear kinetic equation: Model derivation and realizability theory

Schneider

Leibner

2020

Journal of Computational Physics

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“…The Boltzmann equation or similar kinetic equations proved to be applicable not only to classical gases but also to electron transport in solids and plasmas, neutron transport in nuclear reactors, photon transport in superfluids and radiative transfer, among others [13,35,[39][40][41]. More recently, kinetic equations were also derived in the context of biological modelling, e.g., for studying cell movement or wolf migration [10,27,30].…”

Section: Introductionmentioning

confidence: 99%

A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations

Leibner

Ohlberger

2021

ESAIM: M2AN

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In this contribution we derive and analyze a new numerical method for kinetic equations based on a variable transformation of the moment approximation. Classical minimum-entropy moment closures are a class of reduced models for kinetic equations that conserve many of the fundamental physical properties of solutions. However, their practical use is limited by their high computational cost, as an optimization problem has to be solved for every cell in the space-time grid. In addition, implementation of numerical solvers for these models is hampered by the fact that the optimization problems are only well-defined if the moment vectors stay within the realizable set. For the same reason, further reducing these models by, e.g., reduced-basis methods is not a simple task. Our new method overcomes these disadvantages of classical approaches. The transformation is performed on the semi-discretized level which makes them applicable to a wide range of kinetic schemes and replaces the nonlinear optimization problems by inversion of the positive-definite Hessian matrix. As a result, the new scheme gets rid of the realizability-related problems. Moreover, a discrete entropy law can be enforced by modifying the time stepping scheme. Our numerical experiments demonstrate that our new method is often several times faster than the standard optimization-based scheme.

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“…The Boltzmann equation or similar kinetic equations proved to be applicable not only to classical gases but also to electron transport in solids and plasmas, neutron transport in nuclear reactors, photon transport in superfluids and radiative transfer, among others [9,26,[28][29][30]. More recently, kinetic equations were also derived in the context of biological modelling, e.g., for studying cell movement or wolf migration [7,18,21].…”

Section: Introductionmentioning

confidence: 99%

A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations

Leibner¹,

Ohlberger²

2020

Preprint

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In this contribution we derive and analyze a new numerical method for kinetic equations based on a coordinate transformation of the moment approximation. Classical minimum-entropy moment (M N ) closures are a class of reduced models for kinetic equations that conserve many of the fundamental physical properties of the solutions. However, their practical use is limited by their high-computational cost, as an optimization problem has to be solved for every cell in the space-time grid. In addition, implementation of numerical solvers for these models is hampered by the fact that the optimization problems are only well-defined if the moment vectors stay within the realizable set. For the same reason, further reducing these models by, e.g., reduced-basis methods is not a simple task. Our new method overcomes these disadvantages of classical approaches. The coordinate transformation is performed on the semi-discretized level which makes them applicable to a wide range of kinetic schemes and replaces the nonlinear optimization problems by inversion of the positive-definite Hessian matrix. As a result, the new scheme gets rid of the realizability-related problems. Moreover, our numerical experiments demonstrate that our new method is often several times faster than the standard optimization-based scheme.

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Kinetic and Moment Models for Cell Motion in Fiber Structures

Cited by 3 publications

References 57 publications

First-order continuous- and discontinuous-Galerkin moment models for a linear kinetic equation: Model derivation and realizability theory

First-order continuous- and discontinuous-Galerkin moment models for a linear kinetic equation: Model derivation and realizability theory

A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations

A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations

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