An MBO scheme for minimizing the graph Ohta-Kawasaki functional
We study a graph based version of the Ohta-Kawasaki functional, which was originally introduced in a continuum setting to model pattern formation in diblock copolymer melts and has been studied extensively as a paradigmatic example of a variational model for pattern formation. Graph based problems i...
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| Format: | Article |
| Language: | English English |
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Springer
2018
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| Online Access: | https://eprints.nottingham.ac.uk/51873/ |
| _version_ | 1848798593726545920 |
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| author | van Gennip, Yves |
| author_facet | van Gennip, Yves |
| author_sort | van Gennip, Yves |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | We study a graph based version of the Ohta-Kawasaki functional, which was originally introduced in a continuum setting to model pattern formation in diblock copolymer melts and has been studied extensively as a paradigmatic example of a variational model for pattern formation. Graph based problems inspired by partial differential equations (PDEs) and varational methods have been the subject of many recent papers in the mathematical literature, because of their applications in areas such as image processing and data classification. This paper extends the area of PDE inspired graph based problems to pattern forming models, while continuing in the tradition of recent papers in the field.
We introduce a mass conserving Merriman-Bence-Osher (MBO) scheme for minimizing the graph Ohta-Kawasaki functional with a mass constraint. We present three main results: (1) the Lyapunov functionals associated with this MBO scheme Γ-converge to the Ohta-Kawasaki functional (which includes the standard graph based MBO scheme and total variation as a special case); (2) there is a class of graphs on which the Ohta-Kawasaki MBO scheme corresponds to a standard MBO scheme on a transformed graph and for which generalized comparison principles hold; (3) this MBO scheme allows for the numerical computation of (approximate) minimizers of the graph Ohta-Kawasaki functional with a mass constraint. |
| first_indexed | 2025-11-14T20:22:14Z |
| format | Article |
| id | nottingham-51873 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English English |
| last_indexed | 2025-11-14T20:22:14Z |
| publishDate | 2018 |
| publisher | Springer |
| recordtype | eprints |
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| spelling | nottingham-518732018-06-12T04:57:21Z https://eprints.nottingham.ac.uk/51873/ An MBO scheme for minimizing the graph Ohta-Kawasaki functional van Gennip, Yves We study a graph based version of the Ohta-Kawasaki functional, which was originally introduced in a continuum setting to model pattern formation in diblock copolymer melts and has been studied extensively as a paradigmatic example of a variational model for pattern formation. Graph based problems inspired by partial differential equations (PDEs) and varational methods have been the subject of many recent papers in the mathematical literature, because of their applications in areas such as image processing and data classification. This paper extends the area of PDE inspired graph based problems to pattern forming models, while continuing in the tradition of recent papers in the field. We introduce a mass conserving Merriman-Bence-Osher (MBO) scheme for minimizing the graph Ohta-Kawasaki functional with a mass constraint. We present three main results: (1) the Lyapunov functionals associated with this MBO scheme Γ-converge to the Ohta-Kawasaki functional (which includes the standard graph based MBO scheme and total variation as a special case); (2) there is a class of graphs on which the Ohta-Kawasaki MBO scheme corresponds to a standard MBO scheme on a transformed graph and for which generalized comparison principles hold; (3) this MBO scheme allows for the numerical computation of (approximate) minimizers of the graph Ohta-Kawasaki functional with a mass constraint. Springer 2018-06-01 Article PeerReviewed application/pdf en cc_by https://eprints.nottingham.ac.uk/51873/31/MBO%2010.1007_s00332-018-9468-8.pdf application/pdf en cc_by https://eprints.nottingham.ac.uk/51873/2/graphOK_suppl.pdf van Gennip, Yves (2018) An MBO scheme for minimizing the graph Ohta-Kawasaki functional. Journal of Nonlinear Science . ISSN 1432-1467 https://link.springer.com/article/10.1007/s00332-018-9468-8 doi:10.1007/s00332-018-9468-8 doi:10.1007/s00332-018-9468-8 |
| spellingShingle | van Gennip, Yves An MBO scheme for minimizing the graph Ohta-Kawasaki functional |
| title | An MBO scheme for minimizing the graph Ohta-Kawasaki
functional |
| title_full | An MBO scheme for minimizing the graph Ohta-Kawasaki
functional |
| title_fullStr | An MBO scheme for minimizing the graph Ohta-Kawasaki
functional |
| title_full_unstemmed | An MBO scheme for minimizing the graph Ohta-Kawasaki
functional |
| title_short | An MBO scheme for minimizing the graph Ohta-Kawasaki
functional |
| title_sort | mbo scheme for minimizing the graph ohta-kawasaki
functional |
| url | https://eprints.nottingham.ac.uk/51873/ https://eprints.nottingham.ac.uk/51873/ https://eprints.nottingham.ac.uk/51873/ |