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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. , p. 34
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1199
Keywords [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, id: diva2:758606
Public defence
2014-12-12, Room 2446, Polacksbacken, Lägerhyddsvägen 2, Uppsala, 10:15 (English)
Opponent
Supervisors
Projects
eSSENCEUPMARCAvailable from: 2014-11-21 Created: 2014-10-27 Last updated: 2019-02-25Bibliographically 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, 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)
Abstract [en]

The Lanczos algorithm is widely used for solving large sparse symmetric eigenvalue problems when only a few eigenvalues from the spectrum are needed. Due to sparse matrix-vector multiplications and frequent synchronization, the algorithm is communication intensive leading to poor performance on parallel computers and modern cache-based processors. The Communication-Avoiding Lanczos algorithm [Hoemmen; 2010] attempts to improve performance by taking the equivalence of s steps of the original algorithm at a time. The scheme is equivalent to the original algorithm in exact arithmetic but as the value of s grows larger, numerical roundoff errors are expected to have a greater impact. In this paper, we investigate the numerical properties of the Communication-Avoiding Lanczos (CA-Lanczos) algorithm and how well it works in practical computations. Apart from the algorithm itself, we have implemented techniques that are commonly used with the Lanczos algorithm to improve its numerical performance, such as semi-orthogonal schemes and restarting. We present results that show that CA-Lanczos is often as accurate as the original algorithm. In many cases, if the parameters of the s-step basis are chosen appropriately, the numerical behaviour of CA-Lanczos is close to the standard algorithm even though it is somewhat more sensitive to loosing mutual orthogonality among the basis vectors.

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: 2024-05-30Bibliographically 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: 2024-05-29Bibliographically 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: 2024-05-29Bibliographically approved

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Grandin, Magnus

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Citation style
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