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Computing and Visualization in Science

, Volume 17, Issue 2, pp 67–78 | Cite as

Parallel tensor sampling in the hierarchical Tucker format

  • Lars GrasedyckEmail author
  • Ronald Kriemann
  • Christian Löbbert
  • Arne Nägel
  • Gabriel Wittum
  • Konstantinos Xylouris
Article

Abstract

We consider the problem of uncertainty quantification for extreme scale parameter dependent problems where an underlying low rank property of the parameter dependency is assumed. For this type of dependency the hierarchical Tucker format offers a suitable framework to approximate a given output function of the solutions of the parameter dependent problem from a number of samples that is linear in the number of parameters. In particular we can a posteriori compute the mean, variance or other interesting statistical quantities of interest. In the extreme scale setting it is already assumed that the underlying fixed-parameter problem is distributed and solved for in parallel. We provide in addition a parallel evaluation scheme for the sampling phase that allows us on the one hand to combine several solves and on the other hand parallelise the sampling.

Keywords

Parallel sampling UQ Hierarchical tucker 

References

  1. 1.
    Ballani, J., Grasedyck, L., Kluge, M.: Black box approximation of tensors in hierarchical Tucker format. Linear Algeb. Appl. 438, 639–657 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ballani, J., Grasedyck, L.: Hierarchical tensor approximation of output quantities of parameter-dependent PDEs, is accepted for SIAM J. UQ, preprint 385, RWTH Aachen. www.igpm.rwth-aachen.de/forschung/preprints/385 (2014)
  3. 3.
    Bebendorf, M.: Approximation of boundary element matrices. Numer. Math. 86, 565–589 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Caflisch, R.: Monte Carlo and quasi-Monte Carlo methods. Acta Numer. 7, 1–49 (1998)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Giles, M.: Multilevel Monte Carlo methods. Springer, Berlin (2013)CrossRefGoogle Scholar
  6. 6.
    Goreinov SA, Tyrtyshnikov EE (2001) The maximal-volume concept in approximation of low-rank matrices, Contemp. Math. 280Google Scholar
  7. 7.
    Goreinov, S.A., Tyrtyshnikov, E.E., Zamarashkin, N.L.: A theory of pseudoskeleton approximations. Linear Algeb. Appl. 261, 1–22 (1997)Google Scholar
  8. 8.
    Goreinov, S., Zamarashkin, N., Tyrtyshnikov, E.: Pseudo-skeleton approximations by matrices of maximal volume. Math. Notes 62, 515–519 (1997)Google Scholar
  9. 9.
    Grasedyck, L.: Hierarchical singular value decomposition of tensors. SIAM J. Matrix Anal. Appl. 31, 2029–2054 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hackbusch, W., Kühn, S.: A new scheme for the tensor representation. J. Fourier Anal. Appl. 15, 706–722 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Heisig, M., Lieckfeldt, R., Wittum, G., Mazurkevich, G., Lee, G.: Non steady-state descriptions of drug permeation through stratum corneum. I. The biphasic brick-and-mortar model. Pharm. Res. 13, 421–426 (1996)CrossRefGoogle Scholar
  13. 13.
    Nägel, A., Hansen, S., Neumann, D., Lehr, C.-M., Schaefer, U.F., Wittum, G., Heisig, M.: In-silico model of skin penetration based on experimentally determined input parameters. Part II: Mathematical modelling of in-vitro diffusion experiments. Identification of critical input parameters. Eur. J. Pharm. Biopharm. 68, 368–379 (2008)CrossRefGoogle Scholar
  14. 14.
    Schwab, C., Gittelson, C.: Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. Acta Num. 20, 291–467 (2011)Google Scholar
  15. 15.
    Vogel, A., Reiter, S., Rupp, M., Nägel, A., Wittum, G.: UG 4—a novel flexible software system for simulating pde based models on high performance computers. Comp. Vis. Sci. 16, 165–179 (2013)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Lars Grasedyck
    • 1
    Email author
  • Ronald Kriemann
    • 2
  • Christian Löbbert
    • 1
  • Arne Nägel
    • 3
  • Gabriel Wittum
    • 3
  • Konstantinos Xylouris
    • 3
  1. 1.IGPM, RWTH-AachenAachenGermany
  2. 2.MPI MIS LeipzigLeipzigGermany
  3. 3.G-CSC, Univ. FrankfurtFrankfurt am MainGermany

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