Well posedness and maximum entropy approximation for the dynamics of quantitative traits

Handle URI:
http://hdl.handle.net/10754/626135
Title:
Well posedness and maximum entropy approximation for the dynamics of quantitative traits
Authors:
Boďová, Katarína; Haskovec, Jan ( 0000-0003-3464-304X ) ; Markowich, Peter A. ( 0000-0002-3704-1821 )
Abstract:
We study the Fokker–Planck equation derived in the large system limit of the Markovian process describing the dynamics of quantitative traits. The Fokker–Planck equation is posed on a bounded domain and its transport and diffusion coefficients vanish on the domain’s boundary. We first argue that, despite this degeneracy, the standard no-flux boundary condition is valid. We derive the weak formulation of the problem and prove the existence and uniqueness of its solutions by constructing the corresponding contraction semigroup on a suitable function space. Then, we prove that for the parameter regime with high enough mutation rate the problem exhibits a positive spectral gap, which implies exponential convergence to equilibrium.Next, we provide a simple derivation of the so-called Dynamic Maximum Entropy (DynMaxEnt) method for approximation of observables (moments) of the Fokker–Planck solution, which can be interpreted as a nonlinear Galerkin approximation. The limited applicability of the DynMaxEnt method inspires us to introduce its modified version that is valid for the whole range of admissible parameters. Finally, we present several numerical experiments to demonstrate the performance of both the original and modified DynMaxEnt methods. We observe that in the parameter regimes where both methods are valid, the modified one exhibits slightly better approximation properties compared to the original one.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Katarína Boďová, Jan Haskovec, Peter Markowich, Well posedness and maximum entropy approximation for the dynamics of quantitative traits, Physica D: Nonlinear Phenomena, Available online 6 November 2017, ISSN 0167-2789, https://doi.org/10.1016/j.physd.2017.10.015. (https://www.sciencedirect.com/science/article/pii/S0167278917302282)
Publisher:
Elsevier BV
Journal:
Physica D: Nonlinear Phenomena
KAUST Grant Number:
1000000193
Issue Date:
6-Nov-2017
DOI:
10.1016/j.physd.2017.10.015
ARXIV:
arXiv:1704.08757
Type:
Article
ISSN:
0167-2789
Sponsors:
We thank Nicholas Barton (IST Austria) for his useful comments and suggestions. JH and PM are funded by KAUST baseline funds and grant no. 1000000193.
Additional Links:
http://www.sciencedirect.com/science/article/pii/S0167278917302282
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorBoďová, Katarínaen
dc.contributor.authorHaskovec, Janen
dc.contributor.authorMarkowich, Peter A.en
dc.date.accessioned2017-11-09T06:33:07Z-
dc.date.available2017-11-09T06:33:07Z-
dc.date.issued2017-11-06en
dc.identifier.citationKatarína Boďová, Jan Haskovec, Peter Markowich, Well posedness and maximum entropy approximation for the dynamics of quantitative traits, Physica D: Nonlinear Phenomena, Available online 6 November 2017, ISSN 0167-2789, https://doi.org/10.1016/j.physd.2017.10.015. (https://www.sciencedirect.com/science/article/pii/S0167278917302282)-
dc.identifier.issn0167-2789en
dc.identifier.doi10.1016/j.physd.2017.10.015en
dc.identifier.urihttp://hdl.handle.net/10754/626135-
dc.description.abstractWe study the Fokker–Planck equation derived in the large system limit of the Markovian process describing the dynamics of quantitative traits. The Fokker–Planck equation is posed on a bounded domain and its transport and diffusion coefficients vanish on the domain’s boundary. We first argue that, despite this degeneracy, the standard no-flux boundary condition is valid. We derive the weak formulation of the problem and prove the existence and uniqueness of its solutions by constructing the corresponding contraction semigroup on a suitable function space. Then, we prove that for the parameter regime with high enough mutation rate the problem exhibits a positive spectral gap, which implies exponential convergence to equilibrium.Next, we provide a simple derivation of the so-called Dynamic Maximum Entropy (DynMaxEnt) method for approximation of observables (moments) of the Fokker–Planck solution, which can be interpreted as a nonlinear Galerkin approximation. The limited applicability of the DynMaxEnt method inspires us to introduce its modified version that is valid for the whole range of admissible parameters. Finally, we present several numerical experiments to demonstrate the performance of both the original and modified DynMaxEnt methods. We observe that in the parameter regimes where both methods are valid, the modified one exhibits slightly better approximation properties compared to the original one.en
dc.description.sponsorshipWe thank Nicholas Barton (IST Austria) for his useful comments and suggestions. JH and PM are funded by KAUST baseline funds and grant no. 1000000193.en
dc.publisherElsevier BVen
dc.relation.urlhttp://www.sciencedirect.com/science/article/pii/S0167278917302282en
dc.rightsNOTICE: this is the author’s version of a work that was accepted for publication in Physica D: Nonlinear Phenomena. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Physica D: Nonlinear Phenomena, [, , (2017-11-06)] DOI: 10.1016/j.physd.2017.10.015 . © 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/en
dc.titleWell posedness and maximum entropy approximation for the dynamics of quantitative traitsen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalPhysica D: Nonlinear Phenomenaen
dc.eprint.versionPost-printen
dc.contributor.institutionInstitute of Science and Technology Austria (IST Austria), Klosterneuburg A-3400, Austriaen
dc.identifier.arxividarXiv:1704.08757-
kaust.authorHaskovec, Janen
kaust.authorMarkowich, Peter A.en
kaust.grant.number1000000193en
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