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Neural network enhanced computations on coarse grids
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering. Department of Mathematics and Applied Mathematics, University of Johannesburg, South Africa.ORCID iD: 0000-0002-7972-6183
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0001-9797-3834
2021 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 425, article id 109821Article in journal (Refereed) Published
Abstract [en]

Unresolved gradients produce numerical oscillations and inaccurate results. The most straightforward solution to such a problem is to increase the resolution of the computational grid. However, this is often prohibitively expensive and may lead to ecessive execution times. By training a neural network to predict the shape of the solution, we show that it is possible to reduce numerical oscillations and increase both accuracy and efficiency. Data from the neural network prediction is imposed using multiple penalty terms inside the domain.

Place, publisher, year, edition, pages
Elsevier, 2021. Vol. 425, article id 109821
Keywords [en]
Boundary layer, Numerical oscillations, Neural network, Summation-by-parts, Penalty terms, Coarse grids
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-170823DOI: 10.1016/j.jcp.2020.109821ISI: 000630256300003OAI: oai:DiVA.org:liu-170823DiVA, id: diva2:1478983
Available from: 2020-10-23 Created: 2020-10-23 Last updated: 2021-12-28
In thesis
1. Applications of summation-by-parts operators
Open this publication in new window or tab >>Applications of summation-by-parts operators
2020 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Numerical solvers of initial boundary value problems will exhibit instabilities and loss of accuracy unless carefully designed. The key property that leads to convergence is stability, which this thesis primarily deals with. By employing discrete differential operators satisfying a so called summation-by-parts property, it is possible to prove stability in a systematic manner by mimicking the continuous analysis if the energy has a bound. The articles included in the thesis all aim to solve the problem of ensuring stability of a numerical scheme in some context. This includes a domain decomposition procedure, a non-conforming grid coupling procedure, an application in high energy physics, and two methods at the intersection of machine learning and summation-by-parts theory.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2020. p. 32
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 2106
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-171230 (URN)10.3384/diss.diva-171230 (DOI)9789179297534 (ISBN)
Public defence
2021-01-22, Ada Lovelace, B-Building, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2020-11-11 Created: 2020-11-11 Last updated: 2021-12-28Bibliographically approved

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Nordström, JanÅlund, Oskar

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