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Towards an adaptive solver for high-dimensional PDE problems on clusters of multicore processors
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.
2012 (English)Licentiate 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.

Place, publisher, year, edition, pages
Uppsala University, 2012.
Series
Information technology licentiate theses: Licentiate theses from the Department of Information Technology, ISSN 1404-5117 ; 2012-003
National Category
Computer Science Computational Mathematics
Research subject
Scientific Computing
Identifiers
URN: urn:nbn:se:uu:diva-169259OAI: oai:DiVA.org:uu-169259DiVA: diva2:505808
Supervisors
Projects
eSSENCE
Available from: 2012-03-09 Created: 2012-02-25 Last updated: 2017-08-31Bibliographically approved
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, 417-424 p.Conference paper, Published paper (Refereed)
Place, publisher, year, edition, pages
Berlin: Springer-Verlag, 2010
National Category
Computer Science Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-132927 (URN)10.1007/978-3-642-11795-4_44 (DOI)978-3-642-11794-7 (ISBN)
Projects
eSSENCEUPMARC
Available from: 2010-10-29 Created: 2010-10-29 Last updated: 2014-11-21Bibliographically 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, 368-378 p.Conference paper, Published paper (Refereed)
Place, publisher, year, edition, pages
Berlin: Springer-Verlag, 2012
Series
Lecture Notes in Computer Science, 7134
National Category
Computer Science 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: 2014-11-21Bibliographically 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 Science
Identifiers
urn:nbn:se:uu:diva-169257 (URN)
Projects
eSSENCE
Available from: 2012-01-22 Created: 2012-02-25 Last updated: 2014-11-21Bibliographically approved

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Citation style
  • apa
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