Advertisement

An Implementation Framework for Solving High-Dimensional PDEs on Massively Parallel Computers

  • Magnus GustafssonEmail author
  • Sverker Holmgren
Conference paper

Abstract

Accurate solution of time-dependent, high-dimensional PDEs requires massive-scale parallel computing. In this paper, we describe an implementation framework for block-decomposed structured grids and discuss techniques for optimization on clusters where the nodes have one or more multicore processors. We use a two-level parallelization scheme with message passing between nodes and multithreading within each node, and argue that this is the best compromise for memory efficiency. We present some examples where the time-dependent Schrödinger equation is solved.

Keywords

Hamiltonian Matrix Ghost Cell Multicore Processor Implementation Framework Lanczos Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Datta, K., Kamil, S., Williams, S., Oliker, L., Shalf, J., Yelick, K.: Optimization and Performance Modeling of Stencil Computations on Modern Microprocessors, SIAM Rev., 51, 129–159 (2009)zbMATHCrossRefGoogle Scholar
  2. 2.
    Fornberg, B.: Calculation of Weights in Finite Difference Formulas, SIAM Rev., 40, 685–691 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Gustafson, J.L.: Reevaluating Amdahl’s law, Commun. ACM, 31, 532–533 (1988)CrossRefGoogle Scholar
  4. 4.
    Gustafsson, M.: A PDE Solver Framework Optimized for Clusters of Multi-core Processors, Master’s thesis UPTEC F09 004, School of Engineering, Uppsala University (2009)Google Scholar
  5. 5.
    Kim, S.K., Chronopoulos, A.T.: A Class of Lanczos-like Algorithms Implemented on Parallel Computers, Parallel Comput., 17, 763–778 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Lubich, C.: Integrators for Quantum Dynamics: A Numerical Analyst’s Brief Review, In: Grotendorst, J., Marx, D., Muramatsu, A. (eds.) Quantum Simulations of Complex Many-Body Systems: From Theory to Algorithms, vol 10, pp. 459–466. John von Neumann Institute for Computing, Jülich (2002)Google Scholar
  7. 7.
    Moler, C., van Loan, C.: Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later, SIAM Rev., 45, 3–49 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    van der Vorst, H.A.: Iterative Krylov Methods for Large Linear Systems, Cambridge University Press, London (2003)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Department of Information TechnologyUppsala UniversityUppsalaSweden

Personalised recommendations