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  • 1.
    Grandin, Magnus
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    Adaptive Solvers for High-Dimensional PDE Problems on Clusters of Multicore Processors2014Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Accurate numerical solution of time-dependent, high-dimensional partial differential equations (PDEs) usually requires efficient numerical techniques and massive-scale parallel computing. In this thesis, we implement and evaluate discretization schemes suited for PDEs of higher dimensionality, focusing on high order of accuracy and low computational cost.

    Spatial discretization is particularly challenging in higher dimensions. The memory requirements for uniform grids quickly grow out of reach even on large-scale parallel computers. We utilize high-order discretization schemes and implement adaptive mesh refinement on structured hyperrectangular domains in order to reduce the required number of grid points and computational work. We allow for anisotropic (non-uniform) refinement by recursive bisection and show how to construct, manage and load balance such grids efficiently. In our numerical examples, we use finite difference schemes to discretize the PDEs. In the adaptive case we show how a stable discretization can be constructed using SBP-SAT operators. However, our adaptive mesh framework is general and other methods of discretization are viable.

    For integration in time, we implement exponential integrators based on the Lanczos/Arnoldi iterative schemes for eigenvalue approximations. Using adaptive time stepping and a truncated Magnus expansion, we attain high levels of accuracy in the solution at low computational cost. We further investigate alternative implementations of the Lanczos algorithm with reduced communication costs.

    As an example application problem, we have considered the time-dependent Schrödinger equation (TDSE). We present solvers and results for the solution of the TDSE on equidistant as well as adaptively refined Cartesian grids.

    List of papers
    1. An implementation framework for solving high-dimensional PDEs on massively parallel computers
    Open this publication in new window or tab >>An implementation framework for solving high-dimensional PDEs on massively parallel computers
    2010 (English)In: Numerical Mathematics and Advanced Applications: 2009, Berlin: Springer-Verlag , 2010, p. 417-424Conference paper, Published paper (Refereed)
    Place, publisher, year, edition, pages
    Berlin: Springer-Verlag, 2010
    National Category
    Computer Sciences Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-132927 (URN)10.1007/978-3-642-11795-4_44 (DOI)000395207900044 ()978-3-642-11794-7 (ISBN)
    Projects
    eSSENCEUPMARC
    Available from: 2010-10-29 Created: 2010-10-29 Last updated: 2018-06-16Bibliographically approved
    2. Communication-efficient algorithms for numerical quantum dynamics
    Open this publication in new window or tab >>Communication-efficient algorithms for numerical quantum dynamics
    2012 (English)In: Applied Parallel and Scientific Computing: Part II, Berlin: Springer-Verlag , 2012, p. 368-378Conference paper, Published paper (Refereed)
    Place, publisher, year, edition, pages
    Berlin: Springer-Verlag, 2012
    Series
    Lecture Notes in Computer Science ; 7134
    National Category
    Computer Sciences Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-135980 (URN)10.1007/978-3-642-28145-7_36 (DOI)000309716000036 ()978-3-642-28144-0 (ISBN)
    Conference
    PARA 2010: State of the Art in Scientific and Parallel Computing
    Projects
    eSSENCEUPMARC
    Available from: 2012-02-16 Created: 2010-12-09 Last updated: 2018-01-12Bibliographically approved
    3. Numerical evaluation of the Communication-Avoiding Lanczos algorithm
    Open this publication in new window or tab >>Numerical evaluation of the Communication-Avoiding Lanczos algorithm
    2012 (English)Report (Other academic)
    Series
    Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-001
    National Category
    Computational Mathematics Computer Sciences
    Identifiers
    urn:nbn:se:uu:diva-169257 (URN)
    Projects
    eSSENCE
    Available from: 2012-01-22 Created: 2012-02-25 Last updated: 2018-01-12Bibliographically approved
    4. Stable difference methods for block-oriented adaptive grids
    Open this publication in new window or tab >>Stable difference methods for block-oriented adaptive grids
    2015 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 65, p. 486-511Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-234977 (URN)10.1007/s10915-014-9969-z (DOI)000362911900003 ()
    Projects
    eSSENCE
    Available from: 2014-12-18 Created: 2014-10-27 Last updated: 2017-12-05Bibliographically approved
    5. Data structures and algorithms for high-dimensional structured adaptive mesh refinement
    Open this publication in new window or tab >>Data structures and algorithms for high-dimensional structured adaptive mesh refinement
    2014 (English)Report (Other academic)
    Series
    Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2014-019
    National Category
    Computer Sciences Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-234980 (URN)
    Projects
    eSSENCE
    Available from: 2014-10-30 Created: 2014-10-27 Last updated: 2018-01-11Bibliographically approved
    6. Parallel data structures and algorithms for high-dimensional structured adaptive mesh refinement
    Open this publication in new window or tab >>Parallel data structures and algorithms for high-dimensional structured adaptive mesh refinement
    2014 (English)Report (Other academic)
    Series
    Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2014-020
    National Category
    Computer Sciences Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-234981 (URN)
    Projects
    eSSENCEUPMARC
    Available from: 2014-10-31 Created: 2014-10-27 Last updated: 2018-01-11Bibliographically approved
    Download full text (pdf)
    fulltext
    Download (jpg)
    presentationsbild
  • 2.
    Grandin, Magnus
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    Data structures and algorithms for high-dimensional structured adaptive mesh refinement2015In: Advances in Engineering Software, ISSN 0965-9978, E-ISSN 1873-5339, Vol. 82, p. 75-86Article in journal (Refereed)
  • 3.
    Grandin, Magnus
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    Data structures and algorithms for high-dimensional structured adaptive mesh refinement2014Report (Other academic)
  • 4.
    Grandin, Magnus
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    Holmgren, Sverker
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    Parallel data structures and algorithms for high-dimensional structured adaptive mesh refinement2014Report (Other academic)
  • 5.
    Gustafsson, Magnus
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    Towards an adaptive solver for high-dimensional PDE problems on clusters of multicore processors2012Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    Accurate numerical simulation of time-dependent phenomena in many spatial dimensions is a challenging computational task apparent in a vast range of application areas, for instance quantum dynamics, financial mathematics, systems biology and plasma physics. Particularly problematic is that the number of unknowns in the governing equations (the number of grid points) grows exponentially with the number of spatial dimensions introduced, often referred to as the curse of dimensionality. This limits the range of problems that we can solve, since the computational effort and requirements on memory storage directly depend on the number of unknowns for which to solve the equations.

    In order to push the limit of tractable problems, we are developing an implementation framework, HAParaNDA, for high-dimensional PDE-problems. By using high-order accurate schemes and adaptive mesh refinement (AMR) in space, we aim at reducing the number of grid points used in the discretization, thereby enabling the solution of larger and higher-dimensional problems. Within the framework, we use structured grids for spatial discretization and a block-decomposition of the spatial domain for parallelization and load balancing. For integration in time, we use exponential integration, although the framework allows the flexibility of other integrators to be implemented as well. Exponential integrators using the Lanzcos or the Arnoldi algorithm has proven a succesful and efficient approach for large problems. Using a truncation of the Magnus expansion, we can attain high levels of accuracy in the solution.

    As an example application, we have implemented a solver for the time-dependent Schrödinger equation using this framework. We provide scaling results for small and medium sized clusters of multicore nodes, and show that the solver fulfills the expected rate of convergence.

    List of papers
    1. An implementation framework for solving high-dimensional PDEs on massively parallel computers
    Open this publication in new window or tab >>An implementation framework for solving high-dimensional PDEs on massively parallel computers
    2010 (English)In: Numerical Mathematics and Advanced Applications: 2009, Berlin: Springer-Verlag , 2010, p. 417-424Conference paper, Published paper (Refereed)
    Place, publisher, year, edition, pages
    Berlin: Springer-Verlag, 2010
    National Category
    Computer Sciences Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-132927 (URN)10.1007/978-3-642-11795-4_44 (DOI)000395207900044 ()978-3-642-11794-7 (ISBN)
    Projects
    eSSENCEUPMARC
    Available from: 2010-10-29 Created: 2010-10-29 Last updated: 2018-06-16Bibliographically approved
    2. Communication-efficient algorithms for numerical quantum dynamics
    Open this publication in new window or tab >>Communication-efficient algorithms for numerical quantum dynamics
    2012 (English)In: Applied Parallel and Scientific Computing: Part II, Berlin: Springer-Verlag , 2012, p. 368-378Conference paper, Published paper (Refereed)
    Place, publisher, year, edition, pages
    Berlin: Springer-Verlag, 2012
    Series
    Lecture Notes in Computer Science ; 7134
    National Category
    Computer Sciences Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-135980 (URN)10.1007/978-3-642-28145-7_36 (DOI)000309716000036 ()978-3-642-28144-0 (ISBN)
    Conference
    PARA 2010: State of the Art in Scientific and Parallel Computing
    Projects
    eSSENCEUPMARC
    Available from: 2012-02-16 Created: 2010-12-09 Last updated: 2018-01-12Bibliographically approved
    3. Stable difference methods for block-structured adaptive grids
    Open this publication in new window or tab >>Stable difference methods for block-structured adaptive grids
    2011 (English)Report (Other academic)
    Series
    Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2011-022
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-159854 (URN)
    Projects
    eSSENCE
    Available from: 2011-10-11 Created: 2011-10-11 Last updated: 2013-11-29Bibliographically approved
    4. Numerical evaluation of the Communication-Avoiding Lanczos algorithm
    Open this publication in new window or tab >>Numerical evaluation of the Communication-Avoiding Lanczos algorithm
    2012 (English)Report (Other academic)
    Series
    Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-001
    National Category
    Computational Mathematics Computer Sciences
    Identifiers
    urn:nbn:se:uu:diva-169257 (URN)
    Projects
    eSSENCE
    Available from: 2012-01-22 Created: 2012-02-25 Last updated: 2018-01-12Bibliographically approved
    Download full text (pdf)
    fulltext
  • 6.
    Gustafsson, Magnus
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    Demmel, James
    Holmgren, Sverker
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    Numerical evaluation of the Communication-Avoiding Lanczos algorithm2012Report (Other academic)
  • 7.
    Gustafsson, Magnus
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    Holmgren, Sverker
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    An implementation framework for solving high-dimensional PDEs on massively parallel computers2010In: Numerical Mathematics and Advanced Applications: 2009, Berlin: Springer-Verlag , 2010, p. 417-424Conference paper (Refereed)
  • 8.
    Gustafsson, Magnus
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    Holmgren, Sverker
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    Efficient implementation of a high-dimensional PDE-solver on multicore processors2009In: Proc. 2nd Swedish Workshop on Multi-Core Computing, Uppsala, Sweden: Department of Information Technology, Uppsala University , 2009, p. 64-66Conference paper (Other academic)
  • 9.
    Gustafsson, Magnus
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    Kormann, Katharina
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    Holmgren, Sverker
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    Communication-efficient algorithms for numerical quantum dynamics2012In: Applied Parallel and Scientific Computing: Part II, Berlin: Springer-Verlag , 2012, p. 368-378Conference paper (Refereed)
  • 10.
    Gustafsson, Magnus
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    Nissen, Anna
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Kormann, Katharina
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    Stable difference methods for block-structured adaptive grids2011Report (Other academic)
  • 11. Nissen, Anna
    et al.
    Kormann, Katharina
    Grandin, Magnus
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    Virta, Kristoffer
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Stable difference methods for block-oriented adaptive grids2015In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 65, p. 486-511Article in journal (Refereed)
1 - 11 of 11
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  • ieee
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