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Adaptive Solvers 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.
2014 (English)Doctoral 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.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2014. , 34 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1199
Keyword [en]
adaptive mesh refinement, anisotropic refinement, exponential integrators, Lanczos' algorithm, hybrid parallelization, time-dependent Schrödinger equation
National Category
Computational Mathematics
Research subject
Scientific Computing
Identifiers
URN: urn:nbn:se:uu:diva-234984ISBN: 978-91-554-9095-9 (print)OAI: oai:DiVA.org:uu-234984DiVA: diva2:758606
Public defence
2014-12-12, Room 2446, Polacksbacken, Lägerhyddsvägen 2, Uppsala, 10:15 (English)
Opponent
Supervisors
Projects
eSSENCE
Available from: 2014-11-21 Created: 2014-10-27 Last updated: 2015-02-03Bibliographically 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. 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
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, 486-511 p.Article 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 Science Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-234980 (URN)
Projects
eSSENCE
Available from: 2014-10-30 Created: 2014-10-27 Last updated: 2014-11-21Bibliographically 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 Science Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-234981 (URN)
Projects
eSSENCEUPMARC
Available from: 2014-10-31 Created: 2014-10-27 Last updated: 2014-11-21Bibliographically approved

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