Pattern formation of a nonlocal, anisotropic interaction model

Handle URI:
http://hdl.handle.net/10754/626979
Title:
Pattern formation of a nonlocal, anisotropic interaction model
Authors:
Burger, Martin; Düring, Bertram; Kreusser, Lisa Maria; Markowich, Peter A. ( 0000-0002-3704-1821 ) ; Schönlieb, Carola-Bibiane
Abstract:
We consider a class of interacting particle models with anisotropic, repulsive–attractive interaction forces whose orientations depend on an underlying tensor field. An example of this class of models is the so-called Kücken–Champod model describing the formation of fingerprint patterns. This class of models can be regarded as a generalization of a gradient flow of a nonlocal interaction potential which has a local repulsion and a long-range attraction structure. In contrast to isotropic interaction models the anisotropic forces in our class of models cannot be derived from a potential. The underlying tensor field introduces an anisotropy leading to complex patterns which do not occur in isotropic models. This anisotropy is characterized by one parameter in the model. We study the variation of this parameter, describing the transition between the isotropic and the anisotropic model, analytically and numerically. We analyze the equilibria of the corresponding mean-field partial differential equation and investigate pattern formation numerically in two dimensions by studying the dependence of the parameters in the model on the resulting patterns.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Applied Mathematics and Computational Science Program; Computer Science Program
Citation:
Burger M, Düring B, Kreusser LM, Markowich PA, Schönlieb C-B (2018) Pattern formation of a nonlocal, anisotropic interaction model. Mathematical Models and Methods in Applied Sciences 28: 409–451. Available: http://dx.doi.org/10.1142/S0218202518500112.
Publisher:
World Scientific Pub Co Pte Lt
Journal:
Mathematical Models and Methods in Applied Sciences
Issue Date:
24-Nov-2017
DOI:
10.1142/S0218202518500112
Type:
Article
ISSN:
0218-2025; 1793-6314
Sponsors:
M.B. acknowledges support by ERC via Grant EU FP 7—ERC Consolidator Grant 615216 LifeInverse and by the German Science Foundation DFG via EXC 1003 Cells in Motion Cluster of Excellence, Munster, Germany. B.D. has been supported by the Leverhulme Trust Research Project Grant “Novel discretizations for higherorder nonlinear PDE” (RPG-2015-69). L.M.K. was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) Grant EP/L016516/1. C.-B.S. acknowledges support from Leverhulme Trust Project on breaking the non-convexity barrier, EPSRC Grant No. EP/M00483X/1, the EPSRC Center No. EP/N014588/1 and the Cantab Capital Institute for the Mathematics of Information. The authors would like to thank Carsten Gottschlich and Stephan Huckemann for introducing them to the Kucken–Champod model and for very useful discussions on the dynamics required for simulating fingerprints. The authors are grateful to the referees for their thorough review and valuable remarks that helped to improve the paper.
Additional Links:
http://www.worldscientific.com/doi/abs/10.1142/S0218202518500112
Appears in Collections:
Articles; Applied Mathematics and Computational Science Program; Computer Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorBurger, Martinen
dc.contributor.authorDüring, Bertramen
dc.contributor.authorKreusser, Lisa Mariaen
dc.contributor.authorMarkowich, Peter A.en
dc.contributor.authorSchönlieb, Carola-Bibianeen
dc.date.accessioned2018-02-01T07:25:00Z-
dc.date.available2018-02-01T07:25:00Z-
dc.date.issued2017-11-24en
dc.identifier.citationBurger M, Düring B, Kreusser LM, Markowich PA, Schönlieb C-B (2018) Pattern formation of a nonlocal, anisotropic interaction model. Mathematical Models and Methods in Applied Sciences 28: 409–451. Available: http://dx.doi.org/10.1142/S0218202518500112.en
dc.identifier.issn0218-2025en
dc.identifier.issn1793-6314en
dc.identifier.doi10.1142/S0218202518500112en
dc.identifier.urihttp://hdl.handle.net/10754/626979-
dc.description.abstractWe consider a class of interacting particle models with anisotropic, repulsive–attractive interaction forces whose orientations depend on an underlying tensor field. An example of this class of models is the so-called Kücken–Champod model describing the formation of fingerprint patterns. This class of models can be regarded as a generalization of a gradient flow of a nonlocal interaction potential which has a local repulsion and a long-range attraction structure. In contrast to isotropic interaction models the anisotropic forces in our class of models cannot be derived from a potential. The underlying tensor field introduces an anisotropy leading to complex patterns which do not occur in isotropic models. This anisotropy is characterized by one parameter in the model. We study the variation of this parameter, describing the transition between the isotropic and the anisotropic model, analytically and numerically. We analyze the equilibria of the corresponding mean-field partial differential equation and investigate pattern formation numerically in two dimensions by studying the dependence of the parameters in the model on the resulting patterns.en
dc.description.sponsorshipM.B. acknowledges support by ERC via Grant EU FP 7—ERC Consolidator Grant 615216 LifeInverse and by the German Science Foundation DFG via EXC 1003 Cells in Motion Cluster of Excellence, Munster, Germany. B.D. has been supported by the Leverhulme Trust Research Project Grant “Novel discretizations for higherorder nonlinear PDE” (RPG-2015-69). L.M.K. was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) Grant EP/L016516/1. C.-B.S. acknowledges support from Leverhulme Trust Project on breaking the non-convexity barrier, EPSRC Grant No. EP/M00483X/1, the EPSRC Center No. EP/N014588/1 and the Cantab Capital Institute for the Mathematics of Information. The authors would like to thank Carsten Gottschlich and Stephan Huckemann for introducing them to the Kucken–Champod model and for very useful discussions on the dynamics required for simulating fingerprints. The authors are grateful to the referees for their thorough review and valuable remarks that helped to improve the paper.en
dc.publisherWorld Scientific Pub Co Pte Lten
dc.relation.urlhttp://www.worldscientific.com/doi/abs/10.1142/S0218202518500112en
dc.subjectdynamical systemsen
dc.subjectNonlocal interactionsen
dc.subjectpattern formationen
dc.titlePattern formation of a nonlocal, anisotropic interaction modelen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.contributor.departmentComputer Science Programen
dc.identifier.journalMathematical Models and Methods in Applied Sciencesen
dc.contributor.institutionInstitute for Computational and Applied Mathematics, University of Münster, Einsteinstr. 62, 48149 Münster, Germanyen
dc.contributor.institutionDepartment of Mathematics, University of Sussex, Pevensey II, Brighton BN1 9QH, UKen
dc.contributor.institutionDepartment of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UKen
kaust.authorMarkowich, Peter A.en
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