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Numerical evaluation of the Communication-Avoiding Lanczos algorithm
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.
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)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.

Place, publisher, year, edition, pages
2012.
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-001
National Category
Computational Mathematics Computer Sciences
Identifiers
URN: urn:nbn:se:uu:diva-169257OAI: oai:DiVA.org:uu-169257DiVA, id: diva2:505799
Projects
eSSENCEAvailable from: 2012-01-22 Created: 2012-02-25 Last updated: 2024-05-30Bibliographically approved
In thesis
1. Towards an adaptive solver for high-dimensional PDE problems on clusters of multicore processors
Open this publication in new window or tab >>Towards an adaptive solver for high-dimensional PDE problems on clusters of multicore processors
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 Sciences Computational Mathematics
Research subject
Scientific Computing
Identifiers
urn:nbn:se:uu:diva-169259 (URN)
Supervisors
Projects
eSSENCEUPMARC
Available from: 2012-03-09 Created: 2012-02-25 Last updated: 2019-02-25Bibliographically approved
2. Adaptive Solvers for High-Dimensional PDE Problems on Clusters of Multicore Processors
Open this publication in new window or tab >>Adaptive Solvers for High-Dimensional PDE Problems on Clusters of Multicore Processors
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
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:nbn:se:uu:diva-234984 (URN)978-91-554-9095-9 (ISBN)
Public defence
2014-12-12, Room 2446, Polacksbacken, Lägerhyddsvägen 2, Uppsala, 10:15 (English)
Opponent
Supervisors
Projects
eSSENCEUPMARC
Available from: 2014-11-21 Created: 2014-10-27 Last updated: 2019-02-25Bibliographically approved

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Gustafsson, MagnusHolmgren, Sverker

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