Sparse Grids in a Nutshell

Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 88)

Abstract

The technique of sparse grids allows to overcome the curse of dimensionality, which prevents the use of classical numerical discretization schemes in more than three or four dimensions, under suitable regularity assumptions. The approach is obtained from a multi-scale basis by a tensor product construction and subsequent truncation of the resulting multiresolution series expansion. This entry level article gives an introduction to sparse grids and the sparse grid combination technique.

References

  1. 1.
    S. Achatz. Higher order sparse grid methods for elliptic partial differential equations with variable coefficients. Computing, 71(1):1–15, 2003.Google Scholar
  2. 2.
    K. I. Babenko. Approximation of periodic functions of many variables by trigonometric polynomials. Dokl. Akad. Nauk SSSR, 132:247–250, 1960. Russian, Engl. Transl.: Soviet Math. Dokl. 1:513–516, 1960.Google Scholar
  3. 3.
    R. Balder. Adaptive Verfahren für elliptische und parabolische Differentialgleichungen auf dünnen Gittern. Dissertation, Technische Universität München, 1994.Google Scholar
  4. 4.
    G. Baszenski, F.-J. Delvos, and S. Jester. Blending approximations with sine functions. In D. Braess, editor, Numerical Methods in Approximation Theory IX, ISNM 105, pages 1–19. Birkhäuser, Basel, 1992.Google Scholar
  5. 5.
    H.-J. Bungartz. Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung. Dissertation, Institut für Informatik, Technische Universität München, 1992.Google Scholar
  6. 6.
    H.-J. Bungartz. Finite Elements of Higher Order on Sparse Grids. Habilitation, Institut für Informatik, Technische Universität München and Shaker Verlag, Aachen, 1998.Google Scholar
  7. 7.
    H.-J. Bungartz and M. Griebel. A note on the complexity of solving Poisson’s equation for spaces of bounded mixed derivatives. J. Complexity, 15:167–199, 1999. also as Report No 524, SFB 256, Univ. Bonn, 1997.Google Scholar
  8. 8.
    H.-J. Bungartz and M. Griebel. Sparse grids. Acta Numerica, 13:147–269, 2004.Google Scholar
  9. 9.
    H.-J. Bungartz, M. Griebel, D. Röschke, and C. Zenger. Pointwise convergence of the combination technique for the Laplace equation. East-West J. Numer. Math., 2:21–45, 1994.Google Scholar
  10. 10.
    H.-J. Bungartz, M. Griebel, and U. Rüde. Extrapolation, combination, and sparse grid techniques for elliptic boundary value problems. Comput. Methods Appl. Mech. Eng., 116:243–252, 1994. Also in C. Bernardi and Y. Maday, Editoren, International conference on spectral and high order methods, ICOSAHOM 92. Elsevier, 1992.Google Scholar
  11. 11.
    F.-J. Delvos. d-variate Boolean interpolation. J. Approx. Theory, 34:99–114, 1982.Google Scholar
  12. 12.
    F.-J. Delvos and W. Schempp. Boolean Methods in Interpolation and Approximation. Pitman Research Notes in Mathematics Series 230. Longman Scientific & Technical, Harlow, 1989.Google Scholar
  13. 13.
    G. Faber. Über stetige Funktionen. Mathematische Annalen, 66:81–94, 1909.Google Scholar
  14. 14.
    C. Feuersänger. Sparse Grid Methods for Higher Dimensional Approximation. Dissertation, Institut für Numerische Simulation, Universität Bonn, Sept. 2010.Google Scholar
  15. 15.
    J. Garcke. Maschinelles Lernen durch Funktionsrekonstruktion mit verallgemeinerten dünnen Gittern. Doktorarbeit, Institut für Numerische Simulation, Universität Bonn, 2004.Google Scholar
  16. 16.
    J. Garcke. Regression with the optimised combination technique. In W. Cohen and A. Moore, editors, Proceedings of the 23rd ICML ’06, pages 321–328, New York, NY, USA, 2006. ACM Press.Google Scholar
  17. 17.
    J. Garcke. A dimension adaptive sparse grid combination technique for machine learning. In W. Read, J. W. Larson, and A. J. Roberts, editors, Proceedings of the 13th Biennial Computational Techniques and Applications Conference, CTAC-2006, volume 48 of ANZIAM J., pages C725–C740, 2007.Google Scholar
  18. 18.
    J. Garcke. An optimised sparse grid combination technique for eigenproblems. PAMM, 7(1):1022301–1022302, 2007.Google Scholar
  19. 19.
    J. Garcke. A dimension adaptive combination technique using localised adaptation criteria. In H. G. Bock, X. P. Hoang, R. Rannacher, and J. P. Schlöder, editors, Modeling, Simulation and Optimization of Complex Processes, pages 115–125. Springer Berlin Heidelberg, 2012.Google Scholar
  20. 20.
    T. Gerstner and M. Griebel. Numerical Integration using Sparse Grids. Numer. Algorithms, 18:209–232, 1998.Google Scholar
  21. 21.
    T. Gerstner and M. Griebel. Dimension–Adaptive Tensor–Product Quadrature. Computing, 71(1):65–87, 2003.Google Scholar
  22. 22.
    M. Griebel. A parallelizable and vectorizable multi-level algorithm on sparse grids. In W. Hackbusch, editor, Parallel Algorithms for partial differential equations, Notes on Numerical Fluid Mechanics, volume 31, pages 94–100. Vieweg Verlag, Braunschweig, 1991. also as SFB Bericht, 342/20/90 A, Institut für Informatik, TU München, 1990.Google Scholar
  23. 23.
    M. Griebel. Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences. Computing, 61(2):151–179, 1998.Google Scholar
  24. 24.
    M. Griebel and S. Knapek. Optimized tensor-product approximation spaces. Constructive Approximation, 16(4):525–540, 2000.Google Scholar
  25. 25.
    M. Griebel, M. Schneider, and C. Zenger. A combination technique for the solution of sparse grid problems. In P. de Groen and R. Beauwens, editors, Iterative Methods in Linear Algebra, pages 263–281. IMACS, Elsevier, North Holland, 1992.Google Scholar
  26. 26.
    M. Hegland. Adaptive sparse grids. In K. Burrage and R. B. Sidje, editors, Proc. of 10th CTAC-2001, volume 44 of ANZIAM J., pages C335–C353, 2003.Google Scholar
  27. 27.
    M. Hegland, J. Garcke, and V. Challis. The combination technique and some generalisations. Linear Algebra and its Applications, 420(2–3):249–275, 2007.Google Scholar
  28. 28.
    R. Hochmuth. Wavelet Bases in numerical Analysis and Restrictal Nonlinear Approximation. Habilitation, Freie Universität Berlin, 1999.Google Scholar
  29. 29.
    R. Hochmuth, S. Knapek, and G. Zumbusch. Tensor products of Sobolev spaces and applications. Technical Report 685, SFB 256, Univ. Bonn, 2000.Google Scholar
  30. 30.
    S. Knapek. Approximation und Kompression mit Tensorprodukt-Multiskalenräumen. Doktorarbeit, Universität Bonn, April 2000.Google Scholar
  31. 31.
    C. B. Liem, T. Lü, and T. M. Shih. The Splitting Extrapolation Method. World Scientific, Singapore, 1995.Google Scholar
  32. 32.
    J. Noordmans and P. Hemker. Application of an adaptive sparse grid technique to a model singular perturbation problem. Computing, 65:357–378, 2000.Google Scholar
  33. 33.
    C. Pflaum and A. Zhou. Error analysis of the combination technique. Numer. Math., 84(2): 327–350, 1999.Google Scholar
  34. 34.
    D. Pflüger. Spatially Adaptive Sparse Grids for High-Dimensional Problems. Verlag Dr. Hut, München, Aug. 2010. Dissertation.Google Scholar
  35. 35.
    D. Pflüger, B. Peherstorfer, and H.-J. Bungartz. Spatially adaptive sparse grids for high-dimensional data-driven problems. Journal of Complexity, 26(5):508–522, 2010.Google Scholar
  36. 36.
    C. Reisinger. Numerische Methoden für hochdimensionale parabolische Gleichungen am Beispiel von Optionspreisaufgaben. PhD thesis, Ruprecht-Karls-Universität Heidelberg, 2004. in Vorbereitung.Google Scholar
  37. 37.
    S. A. Smolyak. Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR, 148:1042–1043, 1963. Russian, Engl. Transl.: Soviet Math. Dokl. 4:240–243, 1963.Google Scholar
  38. 38.
    V. N. Temlyakov. Approximation of functions with bounded mixed derivative. Proc. Steklov Inst. Math., 1, 1989.Google Scholar
  39. 39.
    V. N. Temlyakov. Approximation of Periodic Functions. Nova Science, New York, 1993.Google Scholar
  40. 40.
    V. N. Temlyakov. On approximate recovery of functions with bounded mixed derivative. J. Complexity, 9:41–59, 1993.Google Scholar
  41. 41.
    G. Wahba. Spline models for observational data, volume 59 of Series in Applied Mathematics. SIAM, Philadelphia, 1990.Google Scholar
  42. 42.
    H. Yserentant. On the multi-level splitting of finite element spaces. Numerische Mathematik, 49:379–412, 1986.Google Scholar
  43. 43.
    H. Yserentant. Hierarchical bases. In J. R. E. O’Malley et al., editors, Proc. ICIAM’91, Philadelphia, 1992. SIAM.Google Scholar
  44. 44.
    A. Zeiser. Fast Matrix-Vector Multiplication in the Sparse-Grid Galerkin Method. Journal of Scientific Computing, 47(3):328–346, Nov. 2010.Google Scholar
  45. 45.
    C. Zenger. Sparse grids. In W. Hackbusch, editor, Parallel Algorithms for Partial Differential Equations, Proceedings of the Sixth GAMM-Seminar, Kiel, 1990, volume 31 of Notes on Num. Fluid Mech., pages 241–251. Vieweg-Verlag, 1991.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institut für Numerische SimulationUniversität BonnBonnGermany

Personalised recommendations