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Ålund, Oskar
Publications (3 of 3) Show all publications
Ålund, O., Iaccarino, G. & Nordström, J. (2020). Learning to Differentiate. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>Learning to Differentiate
2020 (English)Report (Other academic)
Abstract [en]

Artificial neural networks together with associated computational libraries provide a powerful framework for constructing both classification and regression algorithms. In this paper we use neural networks to design linear and non-linear discrete differential operators. We show that neural network based operators can be used to construct stable discretizations of initial boundary-value problems by ensuring that the operators satisfy a discrete analogue of integration-byparts known as summation-by-parts. Furthermore we demonstrate the benefits of building the summation-by-parts property into the network by weight restriction, rather than enforcing it through a regularizer. We conclude that, if possible, known structural elements of an operation are best implemented as innate—rather than learned—properties of the network. The strategy developed in this work also opens the door for constructing stable differential operators on general meshes.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2020. p. 27
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2020:1
Keywords
Neural network, discrete differential operators, stability, initial boundary value problem, summation-by-parts, regularization, weight restriction, general mesh
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-163120 (URN)LiTH-MAT-R--2020/01--SE (ISRN)
Available from: 2020-01-14 Created: 2020-01-14 Last updated: 2020-01-14Bibliographically approved
Ålund, O. & Nordström, J. (2019). Encapsulated high order difference operators on curvilinear non-conforming grids. Journal of Computational Physics, 385, 209-224
Open this publication in new window or tab >>Encapsulated high order difference operators on curvilinear non-conforming grids
2019 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 385, p. 209-224Article in journal (Refereed) Published
Abstract [en]

Constructing stable difference schemes on complex geometries is an arduous task. Even fairly simple partial differential equations end up very convoluted in their discretized form, making them difficult to implement and manage. Spatial discretizations using so called summation-by-parts operators have mitigated this issue to some extent, particularly on rectangular domains, making it possible to formulate stable discretizations in a compact and understandable manner. However, the simplicity of these formulations is lost for curvilinear grids, where the standard procedure is to transform the grid to a rectangular one, and change the structure of the original equation. In this paper we reinterpret the grid transformation as a transformation of the summation-by-parts operators. This results in operators acting directly on the curvilinear grid. Together with previous developments in the field of nonconforming grid couplings we can formulate simple, implementable, and provably stable schemes on general nonconforming curvilinear grids. The theory is applicable to methods on summation-by-parts form, including finite differences, discontinuous Galerkin spectral element, finite volume, and flux reconstruction methods. Time dependent advection–diffusion simulations corroborate the theoretical development.

Keywords
Non-conforming grids, Curvilinear mappings, Weak interface couplings, Summation-by-parts, Stability, Energy method
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-154938 (URN)10.1016/j.jcp.2019.02.007 (DOI)000460889200011 ()
Available from: 2019-03-06 Created: 2019-03-06 Last updated: 2019-04-01
Ålund, O. & Nordström, J. (2018). A stable, high order accurate and efficient hybrid method for flow calculations in complex geometries. In: 2018 AIAA Aerospace Sciences Meeting, AIAA SciTech Forum, (AIAA 2018-1096): . Paper presented at 56th AIAA Aerospace Sciences Meeting 2018, Kissimmee, Florida, USA, 8-12 January 2018 (pp. 1-9). American Institute of Aeronautics and Astronautics (210059)
Open this publication in new window or tab >>A stable, high order accurate and efficient hybrid method for flow calculations in complex geometries
2018 (English)In: 2018 AIAA Aerospace Sciences Meeting, AIAA SciTech Forum, (AIAA 2018-1096), American Institute of Aeronautics and Astronautics, 2018, no 210059, p. 1-9Conference paper, Published paper (Refereed)
Abstract [en]

The suitability of a discretization method is highly dependent on the shape of the domain. Finite difference schemes are typically efficient, but struggle with complex geometry, while finite element methods are expensive but well suited for complex geometries. In this paper we propose a provably stable hybrid method for a 2D advection–diffusion problem, using a class of inner product compatible projection operators to couple the non-conforming grids that arise due to varying the discretization method throughout the domain.

Place, publisher, year, edition, pages
American Institute of Aeronautics and Astronautics, 2018
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-154082 (URN)10.2514/6.2018-1096 (DOI)2-s2.0-85044403973 (Scopus ID)9781624105241 (ISBN)9781510857032 (ISBN)
Conference
56th AIAA Aerospace Sciences Meeting 2018, Kissimmee, Florida, USA, 8-12 January 2018
Available from: 2019-01-28 Created: 2019-01-28 Last updated: 2019-01-28Bibliographically approved
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