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  • 1.
    Nordström, Jan
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering. Department of Mathematics and Applied Mathematics, University of Johannesburg, South Africa.
    Ålund, Oskar
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Neural network enhanced computations on coarse grids2021In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 425, article id 109821Article in journal (Refereed)
    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.

    Download full text (pdf)
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  • 2.
    Wahlsten, Markus
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Ålund, Oskar
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, Faculty of Science & Engineering. Department of Mathematics and Applied Mathematics, University of Johannesburg, Johannesburg, South Africa.
    An efficient hybrid method for uncertainty quantification2022In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 62, p. 607-629Article in journal (Refereed)
    Abstract [en]

    A technique for coupling an intrusive and non-intrusive uncertainty quantification method is proposed. The intrusive approach uses a combination of polynomial chaos and stochastic Galerkin projection. The non-intrusive method uses numerical integration by combining quadrature rules and the probability density functions of the prescribed uncertainties. A stable coupling procedure between the two methods at an interface is constructed. The efficiency of the hybrid method is exemplified using hyperbolic systems of equations, and verified by numerical experiments.

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  • 3. Order onlineBuy this publication >>
    Ålund, Oskar
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Applications of summation-by-parts operators2020Doctoral 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.

    List of papers
    1. A Stable Domain Decomposition Technique for Advection–Diffusion Problems
    Open this publication in new window or tab >>A Stable Domain Decomposition Technique for Advection–Diffusion Problems
    2018 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 77, no 2, p. 755-774Article in journal (Refereed) Published
    Abstract [en]

    The use of implicit methods for numerical time integration typically generates very large systems of equations, often too large to fit in memory. To address this it is necessary to investigate ways to reduce the sizes of the involved linear systems. We describe a domain decomposition approach for the advection–diffusion equation, based on the Summation-by-Parts technique in both time and space. The domain is partitioned into non-overlapping subdomains. A linear system consisting only of interface components is isolated by solving independent subdomain-sized problems. The full solution is then computed in terms of the interface components. The Summation-by-Parts technique provides a solid theoretical framework in which we can mimic the continuous energy method, allowing us to prove both stability and invertibility of the scheme. In a numerical study we show that single-domain implementations of Summation-by-Parts based time integration can be improved upon significantly. Using our proposed method we are able to compute solutions for grid resolutions that cannot be handled efficiently using a single-domain formulation. An order of magnitude speed-up is observed, both compared to a single-domain formulation and to explicit Runge–Kutta time integration.

    Keywords
    Domain decomposition, Partial differential equations, Summation-by-Parts, Finite difference methods, Stability
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-147768 (URN)10.1007/s10915-018-0722-x (DOI)000446594600003 ()
    Available from: 2018-05-14 Created: 2018-05-14 Last updated: 2021-12-28
    2. Encapsulated high order difference operators on curvilinear non-conforming grids
    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: 2021-12-28
    3. Trace preserving quantum dynamics using a novel reparametrization-neutral summation-by-parts difference operator
    Open this publication in new window or tab >>Trace preserving quantum dynamics using a novel reparametrization-neutral summation-by-parts difference operator
    Show others...
    2021 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 425, article id 109917Article in journal (Refereed) Published
    Abstract [en]

    We develop a novel numerical scheme for the simulation of dissipative quantum dynamics, following from two-body Lindblad master equations. It exactly preserves the trace of the density matrix and shows only mild deviations from hermiticity and positivity, which are the defining properties of the continuum Lindblad dynamics. The central ingredient is a new spatial difference operator, which not only fulfills the summation by parts (SBP) property, but also implements a continuum reparametrization property. Using the time evolution of a heavy-quark anti-quark bound state in a hot thermal medium as an explicit example, we show how the reparametrization neutral summation-by-parts (RN-SBP) operator enables an accurate simulation of the full dissipative dynamics of this open quantum system.

    Place, publisher, year, edition, pages
    Elsevier, 2021
    Keywords
    Time integration, Initial boundary value problems, Dissipative systems, Open quantum systems, Summation-by-parts operators, Mimetic operator
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:liu:diva-171869 (URN)10.1016/j.jcp.2020.109917 (DOI)000598924000002 ()
    Funder
    Swedish Research Council, 2018-05084_VRSwedish e‐Science Research Center, ABL in SESSI
    Note

    Funding agencies: The work of Y.A. is supported by JSPS KAKENHI Grant Number JP18K13538. O.Å., F.L. and J.N. acknowledge funding from the Swedish Research Council (Stockholm) under grant number 2018-05084_VR and from the Swedish e-Science Research Center (SeRC) through project ABL in SESSI. A.R. acknowledges discussions with M. Riesch and gladly acknowledges support by the Research Council of Norway under the FRIPRO Young Research Talent grant 286883. This work has utilized computing resources provided by UNINETT Sigma2 -the National Infrastructure for High Performance Computing and Data Storage in Norway under project NN9578K-QCDrtX “Real-time dynamics of nuclear matter under extreme conditions”.

    Available from: 2020-12-10 Created: 2020-12-10 Last updated: 2021-12-28
    4. Neural network enhanced computations on coarse grids
    Open this publication in new window or tab >>Neural network enhanced computations on coarse grids
    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
    Keywords
    Boundary layer, Numerical oscillations, Neural network, Summation-by-parts, Penalty terms, Coarse grids
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:liu:diva-170823 (URN)10.1016/j.jcp.2020.109821 (DOI)000630256300003 ()
    Available from: 2020-10-23 Created: 2020-10-23 Last updated: 2021-12-28
    5. Learning to differentiate
    Open this publication in new window or tab >>Learning to differentiate
    2021 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 424, article id 109873Article in journal (Refereed) Published
    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-by-parts known as summation-by-parts. Our neural network approach with linear activation functions is compared and contrasted with a more traditional linear algebra approach. An application to overlapping grids is explored. The strategy developed in this work opens the door for constructing stable differential operators on general meshes.

    Place, publisher, year, edition, pages
    Elsevier, 2021
    Keywords
    Neural networks, Discrete differential operators, Stability, Summation-by-parts, Overlapping grids
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:liu:diva-170279 (URN)10.1016/j.jcp.2020.109873 (DOI)000588203600029 ()
    Available from: 2020-10-07 Created: 2020-10-07 Last updated: 2021-12-28Bibliographically approved
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  • 4.
    Ålund, Oskar
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Akamatsu, Yukinao
    Department of Physics, Osaka University, Toyonaka, Japan.
    Laurén, Fredrik
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Miura, Takahiro
    Department of Physics, Osaka University, Toyonaka, Japan.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering. Department of Mathematics and Applied Mathematics, University of Johannesburg, South Africa.
    Rothkopf, Alexander
    Faculty of Science and Technology, University of Stavanger, Norway.
    Trace preserving quantum dynamics using a novel reparametrization-neutral summation-by-parts difference operator2021In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 425, article id 109917Article in journal (Refereed)
    Abstract [en]

    We develop a novel numerical scheme for the simulation of dissipative quantum dynamics, following from two-body Lindblad master equations. It exactly preserves the trace of the density matrix and shows only mild deviations from hermiticity and positivity, which are the defining properties of the continuum Lindblad dynamics. The central ingredient is a new spatial difference operator, which not only fulfills the summation by parts (SBP) property, but also implements a continuum reparametrization property. Using the time evolution of a heavy-quark anti-quark bound state in a hot thermal medium as an explicit example, we show how the reparametrization neutral summation-by-parts (RN-SBP) operator enables an accurate simulation of the full dissipative dynamics of this open quantum system.

    Download full text (pdf)
    fulltext
  • 5.
    Ålund, Oskar
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Iaccarino, Gianluca
    Department of Mechanical Engineering and Institute for Computational Mathematical Engineering, Stanford University, Stanford, California, USA.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Learning to Differentiate2020Report (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.

    Download full text (pdf)
    Learning to Differentiate
  • 6.
    Ålund, Oskar
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Iaccarino, Gianluca
    Stanford University, Stanford, United States of America.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering. University of Johannesburg, South Africa.
    Learning to differentiate2021In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 424, article id 109873Article in journal (Refereed)
    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-by-parts known as summation-by-parts. Our neural network approach with linear activation functions is compared and contrasted with a more traditional linear algebra approach. An application to overlapping grids is explored. The strategy developed in this work opens the door for constructing stable differential operators on general meshes.

    Download full text (pdf)
    fulltext
  • 7.
    Ålund, Oskar
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    A provably stable, non-iterative domain decomposition technique for the advection-diffusion equation2016Report (Other academic)
    Abstract [en]

    We describe an efficient, non-iterative domain decomposition approach for the onedimensional advection–diffusion equation based on the Summation-by-Parts technique in both time and space. A fully discrete multidomain analogue of the continuous equation is formulated and a linear system consisting only of the solution components involved in the coupling between the subdomain interfaces is isolated. Once the coupling system is solved, the full solution is found by computing linear combinations of known vectors, weighted by the coupling components. Both stability and invertibility of the discrete scheme is proved using standard Summation-by-Parts procedures.

    In a numerical study we show that perfunctory implementations of monodomain Summation-by-Parts based time integration can be improved upon significantly. Using our proposed method we are able to reduce execution time and memory footprint by up to 80% and 95% respectively. Similar improvements in execution time is shown also when compared against explicit Runge–Kutta time integration.

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    Provably Stable, Non-iterative Domain Decomposition Technique for the Advection-Diffusion Equation
  • 8.
    Ålund, Oskar
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    A Stable Domain Decomposition Technique for Advection–Diffusion Problems2018In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 77, no 2, p. 755-774Article in journal (Refereed)
    Abstract [en]

    The use of implicit methods for numerical time integration typically generates very large systems of equations, often too large to fit in memory. To address this it is necessary to investigate ways to reduce the sizes of the involved linear systems. We describe a domain decomposition approach for the advection–diffusion equation, based on the Summation-by-Parts technique in both time and space. The domain is partitioned into non-overlapping subdomains. A linear system consisting only of interface components is isolated by solving independent subdomain-sized problems. The full solution is then computed in terms of the interface components. The Summation-by-Parts technique provides a solid theoretical framework in which we can mimic the continuous energy method, allowing us to prove both stability and invertibility of the scheme. In a numerical study we show that single-domain implementations of Summation-by-Parts based time integration can be improved upon significantly. Using our proposed method we are able to compute solutions for grid resolutions that cannot be handled efficiently using a single-domain formulation. An order of magnitude speed-up is observed, both compared to a single-domain formulation and to explicit Runge–Kutta time integration.

    Download full text (pdf)
    fulltext
  • 9.
    Ålund, Oskar
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    A stable, high order accurate and efficient hybrid method for flow calculations in complex geometries2018In: 2018 AIAA Aerospace Sciences Meeting, AIAA SciTech Forum, (AIAA 2018-1096), American Institute of Aeronautics and Astronautics, 2018, no 210059, p. 1-9Conference 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.

    Download full text (pdf)
    A stable, high order accurate and efficient hybrid method for flow calculations in complex geometries
  • 10.
    Ålund, Oskar
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Encapsulated high order difference operators on curvilinear non-conforming grids2019In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 385, p. 209-224Article in journal (Refereed)
    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.

    Download full text (pdf)
    fulltext
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