Show simple item record

dc.contributor.authorGomes, Diogo A.
dc.contributor.authorYang, Xianjin
dc.date.accessioned2020-10-18T08:48:42Z
dc.date.available2020-03-25T13:55:34Z
dc.date.available2020-10-18T08:48:42Z
dc.date.issued2020-05-12
dc.date.submitted2019-08-09
dc.identifier.citationGomes, D. A., & Yang, X. (2020). The Hessian Riemannian flow and Newton’s method for effective Hamiltonians and Mather measures. ESAIM: Mathematical Modelling and Numerical Analysis, 54(6), 1883–1915. doi:10.1051/m2an/2020036
dc.identifier.issn1290-3841
dc.identifier.issn0764-583X
dc.identifier.doi10.1051/m2an/2020036
dc.identifier.urihttp://hdl.handle.net/10754/662306
dc.description.abstractEffective Hamiltonians arise in several problems, including homogenization of Hamilton-Jacobi equations, nonlinear control systems, Hamiltonian dynamics, and Aubry-Mather theory. In Aubry-Mather theory, related objects, Mather measures, are also of great importance. Here, we combine ideas from mean-field games with the Hessian Riemannian flow to compute effective Hamiltonians and Mather measures simultaneously. We prove the convergence of the Hessian Riemannian flow in the continuous setting. For the discrete case, we give both the existence and the convergence of the Hessian Riemannian flow. In addition, we explore a variant of Newton's method that greatly improves the performance of the Hessian Riemannian flow. In our numerical experiments, we see that our algorithms preserve the non-negativity of Mather measures and are more stable than related methods in problems that are close to singular. Furthermore, our method also provides a way to approximate stationary MFGs.
dc.description.sponsorshipThe authors were supported by King Abdullah University of Science and Technology (KAUST) baseline funds and KAUST OSR-CRG2017-3452.
dc.publisherEDP Sciences
dc.relation.urlhttps://www.esaim-m2an.org/10.1051/m2an/2020036
dc.rightsArchived with thanks to ESAIM: Mathematical Modelling and Numerical Analysis
dc.titleThe Hessian Riemannian flow and Newton's method for effective Hamiltonians and Mather measures
dc.typeArticle
dc.contributor.departmentApplied Mathematics and Computational Science Program
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
dc.identifier.journalESAIM: Mathematical Modelling and Numerical Analysis
dc.eprint.versionPost-print
dc.identifier.volume54
dc.identifier.issue6
dc.identifier.pages1883-1915
dc.identifier.arxivid1810.03483
kaust.personGomes, Diogo A.
kaust.personYang, Xianjin
kaust.grant.numberOSR-CRG2017-3452
dc.date.accepted2020-05-08
dc.versionv1
dc.identifier.eid2-s2.0-85092317338
refterms.dateFOA2020-03-25T13:58:15Z
kaust.acknowledged.supportUnitOSR
dc.date.posted2018-10-08


Files in this item

Thumbnail
Name:
HomogenizationHamiltonian.pdf
Size:
4.287Mb
Format:
PDF
Description:
Accepted Manuscript

This item appears in the following Collection(s)

Show simple item record

VersionItemEditorDateSummary

*Selected version