Efficient Analysis of High Dimensional Data in Tensor Formats

  • Mike Espig
  • Wolfgang Hackbusch
  • Alexander Litvinenko
  • Hermann G. Matthies
  • Elmar Zander
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 88)

Abstract

In this article we introduce new methods for the analysis of high dimensional data in tensor formats, where the underling data come from the stochastic elliptic boundary value problem. After discretisation of the deterministic operator as well as the presented random fields via KLE and PCE, the obtained high dimensional operator can be approximated via sums of elementary tensors. This tensors representation can be effectively used for computing different values of interest, such as maximum norm, level sets and cumulative distribution function. The basic concept of the data analysis in high dimensions is discussed on tensors represented in the canonical format, however the approach can be easily used in other tensor formats. As an intermediate step we describe efficient iterative algorithms for computing the characteristic and sign functions as well as pointwise inverse in the canonical tensor format. Since during majority of algebraic operations as well as during iteration steps the representation rank grows up, we use lower-rank approximation and inexact recursive iteration schemes.

Keywords

Sparse Grid Tensor Format Stochastic Dimension Polynomial Chaos Expansion Tensor Space 
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.

References

  1. 1.
    S. Acharjee and N. Zabaras. A non-intrusive stochastic Galerkin approach for modeling uncertainty propagation in deformation processes. Computers & Structures, 85:244–254, 2007.Google Scholar
  2. 2.
    I. Babuška, R. Tempone, and G. E. Zouraris. Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal., 42(2):800–825, 2004.Google Scholar
  3. 3.
    I. Babuška, R. Tempone, and G. E. Zouraris. Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation. Comput. Methods Appl. Mech. Engrg., 194(12–16):1251–1294, 2005.Google Scholar
  4. 4.
    R. E. Caflisch. Monte Carlo and quasi-Monte Carlo methods. Acta Numerica, 7:1–49, 1998.Google Scholar
  5. 5.
    S. R. Chinnamsetty, M. Espig, B. N. Khoromskij, W. Hackbusch, and H. J. Flad. Tensor product approximation with optimal rank in quantum chemistry. The Journal of chemical physics, 127(8):084–110, 2007.Google Scholar
  6. 6.
    G. Christakos. Random Field Models in Earth Sciences. Academic Press, San Diego, CA, 1992.Google Scholar
  7. 7.
    P. G. Ciarlet. The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978.Google Scholar
  8. 8.
    O. G. Ernst, C. E. Powell, D. J. Silvester, and E. Ullmann. Efficient solvers for a linear stochastic Galerkin mixed formulation of diffusion problems with random data. SIAM J. Sci. Comput., 31(2):1424–1447, 2008/09.Google Scholar
  9. 9.
    O. G. Ernst and E. Ullmann. Stochastic Galerkin matrices. SIAM Journal on Matrix Analysis and Applications, 31(4):1848–1872, 2010.Google Scholar
  10. 10.
    M. Espig. Effiziente Bestapproximation mittels Summen von Elementartensoren in hohen Dimensionen. PhD thesis, Dissertation, Universität Leipzig, 2008.Google Scholar
  11. 11.
    M. Espig, L. Grasedyck, and W. Hackbusch. Black box low tensor rank approximation using fibre-crosses. Constructive approximation, 2009.Google Scholar
  12. 12.
    M. Espig and W. Hackbusch. A regularized newton method for the efficient approximation of tensors represented in the canonical tensor format. submitted Num. Math., 2011.Google Scholar
  13. 13.
    M. Espig, W. Hackbusch, T. Rohwedder, and R. Schneider. Variational calculus with sums of elementary tensors of fixed rank. paper submitted to: Numerische Mathematik, 2009.Google Scholar
  14. 14.
    P. Frauenfelder, Ch. Schwab, and R. A. Todor. Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Engrg., 194(2–5):205–228, 2005.Google Scholar
  15. 15.
    T. Gerstner and M. Griebel. Numerical integration using sparse grids. Numer. Algorithms, 18(3–4):209–232, 1998.Google Scholar
  16. 16.
    R. Ghanem. Ingredients for a general purpose stochastic finite elements implementation. Comput. Methods Appl. Mech. Engrg., 168(1–4):19–34, 1999.Google Scholar
  17. 17.
    R. Ghanem. Stochastic finite elements for heterogeneous media with multiple random non-Gaussian properties. Journal of Engineering Mechanics, 125:24–40, 1999.Google Scholar
  18. 18.
    R. G. Ghanem and R. M. Kruger. Numerical solution of spectral stochastic finite element systems. Computer Methods in Applied Mechanics and Engineering, 129(3):289–303, 1996.Google Scholar
  19. 19.
    G. H. Golub and C. F. Van Loan. Matrix Computations. Wiley-Interscience, New York, 1984.Google Scholar
  20. 20.
    L. Grasedyck. Theorie und Anwendungen Hierarchischer Matrizen. Doctoral thesis, Universität Kiel, 2001.Google Scholar
  21. 21.
    W. Hackbusch, B. Khoromskij, and E. Tyrtyshnikov. Approximate iterations for structured matrices. Numerische Mathematik, 109:365–383, 2008.Google Scholar
  22. 22.
    M. Jardak, C.-H. Su, and G. E. Karniadakis. Spectral polynomial chaos solutions of the stochastic advection equation. In Proceedings of the Fifth International Conference on Spectral and High Order Methods (ICOSAHOM-01) (Uppsala), volume 17, pages 319–338, 2002.Google Scholar
  23. 23.
    B. N. Khoromskij and A. Litvinenko. Data sparse computation of the Karhunen-Loève expansion. Numerical Analysis and Applied Mathematics: Intern. Conf. on Num. Analysis and Applied Mathematics, AIP Conf. Proc., 1048(1):311–314, 2008.Google Scholar
  24. 24.
    B. N. Khoromskij, A. Litvinenko, and H. G. Matthies. Application of hierarchical matrices for computing Karhunen-Loève expansion. Computing, 84(1–2):49–67, 2009.Google Scholar
  25. 25.
    A. Klimke. Sparse Grid Interpolation Toolbox – user’s guide. Technical Report IANS report 2007/017, University of Stuttgart, 2007.Google Scholar
  26. 26.
    A. Klimke and B. Wohlmuth. Algorithm 847: spinterp: Piecewise multilinear hierarchical sparse grid interpolation in MATLAB. ACM Transactions on Mathematical Software, 31(4), 2005.Google Scholar
  27. 27.
    P. Krée and Ch. Soize. Mathematics of random phenomena, volume 32 of Mathematics and its Applications. D. Reidel Publishing Co., Dordrecht, 1986. Random vibrations of mechanical structures, Translated from the French by Andrei Iacob, With a preface by Paul Germain.Google Scholar
  28. 28.
    O. P. Le Maître, H. N. Najm, R. G. Ghanem, and O. M. Knio. Multi-resolution analysis of Wiener-type uncertainty propagation schemes. J. Comput. Phys., 197(2):502–531, 2004.Google Scholar
  29. 29.
    A. Litvinenko and H. G. Matthies. Sparse data formats and efficient numerical methods for uncertainties quantification in numerical aerodynamics. Informatikbericht-Nr. 2010-01, http://www.digibib.tu-bs.de/?docid=00036490, Technische Universität Braunschweig, Braunschweig, 2010.
  30. 30.
    A. Litvinenko and H. G. Matthies. Sparse data formats and efficient numerical methods for uncertainties quantification in numerical aerodynamics. In Proceedings of the IV European Congress on Computational Mechanics, http://www.eccm2010.org/complet/fullpaper\_1036.pdf, Paris, France, 2010.
  31. 31.
    H. G. Matthies. Computational aspects of probability in non-linear mechanics. In A. Ibrahimbegović and B. Brank, editors, Engineering Structures under Extreme Conditions. Multi-physics and multi-scale computer models in non-linear analysis and optimal design of engineering structures under extreme conditions, volume 194 of NATO Science Series III: Computer and System Sciences. IOS Press, Amsterdam, 2005.Google Scholar
  32. 32.
    H. G. Matthies. Quantifying uncertainty: Modern computational representation of probability and applications. In A. Ibrahimbegović, editor, Extreme Man-Made and Natural Hazards in Dynamics of Structures, NATO-ARW series. Springer Verlag, Berlin, 2007.Google Scholar
  33. 33.
    H. G. Matthies. Uncertainty quantification with stochastic finite elements. 2007. Part 1. Fundamentals. Encyclopedia of Computational Mechanics, John Wiley and Sons, Ltd.Google Scholar
  34. 34.
    H. G. Matthies. Stochastic finite elements: Computational approaches to stochastic partial differential equations. Zeitschr. Ang. Math. Mech.(ZAMM), 88(11):849–873, 2008.Google Scholar
  35. 35.
    H. G. Matthies, Ch. E. Brenner, Ch. G. Bucher, and C. Guedes Soares. Uncertainties in probabilistic numerical analysis of structures and solids-stochastic finite elements. Structural Safety, 19(3):283–336, 1997.Google Scholar
  36. 36.
    H. G. Matthies and Ch. Bucher. Finite elements for stochastic media problems. Comput. Meth. Appl. Mech. Eng., 168(1–4):3–17, 1999.Google Scholar
  37. 37.
    H. G. Matthies and A. Keese. Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Engrg., 194(12–16):1295–1331, 2005.Google Scholar
  38. 38.
    H. G. Matthies and E. Zander. Solving stochastic systems with low-rank tensor compression. Linear Algebra and its Applications, 436:3819–3838, 2012.Google Scholar
  39. 39.
    F. Nobile, R. Tempone, and C. G. Webster. A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal., 46(5):2309–2345, 2008.Google Scholar
  40. 40.
    E. Novak and K. Ritter. The curse of dimension and a universal method for numerical integration. In Multivariate approximation and splines (Mannheim, 1996), volume 125 of Internat. Ser. Numer. Math., pages 177–187. Birkhäuser, Basel, 1997.Google Scholar
  41. 41.
    E. Novak and K. Ritter. Simple cubature formulas with high polynomial exactness. Constr. Approx., 15(4):499–522, 1999.Google Scholar
  42. 42.
    K. Petras. Fast calculation of coefficients in the Smolyak algorithm. Numer. Algorithms, 26(2):93–109, 2001.Google Scholar
  43. 43.
    L. J. Roman and M. Sarkis. Stochastic Galerkin method for elliptic SPDEs: a white noise approach. Discrete Contin. Dyn. Syst. Ser. B, 6(4):941–955 (electronic), 2006.Google Scholar
  44. 44.
    Ch. Schwab and C. J. Gittelson. Sparse tensor discretizations of high-dimensional parametric and stochastic pdes. Acta Numerica, 20:291–467, 2011.Google Scholar
  45. 45.
    S. A. Smoljak. Quadrature and interpolation formulae on tensor products of certain function classes. Dokl. Akad. Nauk SSSR, 148:1042–1045, 1963.Google Scholar
  46. 46.
    G. Strang and G. J. Fix. An Analysis of the Finite Element Method. Wellesley-Cambridge Press, Wellesley, MA, 1988.Google Scholar
  47. 47.
    E. Ullmann. A Kronecker product preconditioner for stochastic Galerkin finite element discretizations. SIAM Journal on Scientific Computing, 32(2):923–946, 2010.Google Scholar
  48. 48.
    D. Xiu and G. E. Karniadakis. Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Meth. Appl. Mech. Eng., 191:4927–4948, 2002.Google Scholar
  49. 49.
    X. Frank Xu. A multiscale stochastic finite element method on elliptic problems involving uncertainties. Comput. Methods Appl. Mech. Engrg., 196(25–28):2723–2736, 2007.Google Scholar
  50. 50.
    E. Zander. Stochastic Galerkin library. Technische Universität Braunschweig, http://github.com/ezander/sglib, 2008.
  51. 51.
    O. C. Zienkiewicz and R. L. Taylor. The Finite Element Method. Butterwort-Heinemann, Oxford, 5th ed., 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Mike Espig
    • 1
  • Wolfgang Hackbusch
    • 1
  • Alexander Litvinenko
    • 2
  • Hermann G. Matthies
    • 2
  • Elmar Zander
    • 2
  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.Technische Universität BraunschweigBraunschweigGermany

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