Spatially Adaptive Refinement

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

Abstract

While sparse grids allow one to tackle problems in higher dimensionalities than possible for standard grid-based discretizations, real-world applications often come along with requirements or restrictions which enforce problem-dependent adaptations of the standard sparse grid technique. Consider, for example, interpolations where the function values at grid points are obtained via time-consuming numerical simulations. Then, only very few grid points can be spent; classical convergence might be out of reach. Another hurdle is that real-world problems often do not meet the smoothness requirements of the sparse grid method. Thus, the standard approach has to be fine-tuned to the problem at hand, especially in higher-dimensional settings. Therefore, a suitable choice of basis functions can be required, as well as criteria for problem-adapted refinement. Fortunately, and in contrast to full grids, the hierarchical basis formulation of the direct sparse grid approach conveniently provides a reasonable criterion for spatially adaptive refinement practically for free. This can serve as a starting point to develop suitable modifications. We show several problems stemming from different fields of application and demonstrate modifications of the standard sparse grid approach. They enable one to cope with the properties and requirements of the corresponding problem and can serve as examples for similar challenges.

Keywords

Basis Function Grid Point Mean Square Error Monte Carlo Sparse Grid 
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.
    J. K. Adelman-McCarthy et al. The fifth data release of the Sloan Digital Sky Survey. ApJS, 172:634–644, 2007.CrossRefGoogle Scholar
  2. 2.
    D. M. Allen. The relationship between variable selection and data augmentation and a method for prediction. Technometrics, 16(1):125–127, 1974.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    J. Benk, H.-J. Bungartz, A.-E. Nagy, and S. Schraufstetter. An option pricing framework based on theta-calculus and sparse grids. In Progress in Industrial Mathematics at ECMI 2010, 2010.Google Scholar
  4. 4.
    H.-J. Bungartz and M. Griebel. Sparse grids. Acta Numerica, 13:147–269, 2004.MathSciNetCrossRefGoogle Scholar
  5. 5.
    H.-J. Bungartz, A. Heinecke, D. Pflüger, and S. Schraufstetter. Option pricing with a direct adaptive sparse grid approach. Journal of Computational and Applied Mathematics, 2011.Google Scholar
  6. 6.
    H.-J. Bungartz, D. Pflüger, and S. Zimmer. Adaptive sparse grid techniques for data mining. In H.G. Bock, E. Kostina, X.P. Hoang, and R. Rannacher, editors, Modelling, Simulation and Optimization of Complex Processes, Proceedings of the High Performance Scientific Computing 2006, Hanoi, Vietnam, pages 121–130. Springer, 2008.Google Scholar
  7. 7.
    J. F. Carrière. Valuation of early-exercise price of options using simulation and nonparametric regression. Insurance: Mathematics and Economics, 19:19–30, 1996.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    I. Csabai et al. The application of photometric redshifts to the SDSS Early Data Release. Astron. J., 125:580–592, 2003.CrossRefGoogle Scholar
  9. 9.
    T. Evgeniou, M. Pontil, and T. Poggio. Regularization networks and support vector machines. In Advances in Computational Mathematics, pages 1–50. MIT Press, 2000.Google Scholar
  10. 10.
    J. H. Friedman. Multivariate adaptive regression splines. Annals of Statistics, 19, 1991.Google Scholar
  11. 11.
    B. Ganapathysubramanian and N. Zabaras. Sparse grid collocation schemes for stochastic natural convection problems. J. Comput. Phys., 225(1):652–685, 2007.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    J. Garcke. Regression with the optimised combination technique. In ICML ’06: Proceedings of the 23rd international conference on Machine learning, pages 321–328, New York, NY, USA, 2006. ACM Press.Google Scholar
  13. 13.
    J. Garcke. A dimension adaptive sparse grid combination technique for machine learning. In Wayne Read, Jay 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
  14. 14.
    J. Garcke, M. Griebel, and M. Thess. Data mining with sparse grids. Computing, 67(3):225–253, 2001.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    J. Garcke and M. Hegland. Fitting multidimensional data using gradient penalties and the sparse grid combination technique. Computing, 84(1–2):1–25, 2009.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    M. Hegland. Adaptive sparse grids. In K. Burrage and Roger B. Sidje, editors, Proc. of 10th Computational Techniques and Applications Conference CTAC-2001, volume 44, pages C335–C353, 2003.Google Scholar
  17. 17.
    A. Heinecke and D. Pflüger. Multi- and many-core data mining with adaptive sparse grids. In Proceedings of the 8th ACM International Conference on Computing Frontiers, pages 29:1–29:10, New York, USA, May 2011. ACM Press.Google Scholar
  18. 18.
    M. Holtz. Sparse Grid Quadrature in High Dimensions with Applications in Finance and Insurance. Dissertation, Institut für Numerische Simulation, Universität Bonn, 2008.Google Scholar
  19. 19.
    A. Klimke, R. Nunes, and B. Wohlmuth. Fuzzy arithmetic based on dimension-adaptive sparse grids: a case study of a large-scale finite element model under uncertain parameters. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems, 14:561–577, 2006.CrossRefGoogle Scholar
  20. 20.
    D. Pflüger. Data Mining mit Dünnen Gittern. Diplomarbeit, IPVS, Universität Stuttgart, 2005.Google Scholar
  21. 21.
    D. Pflüger. Spatially Adaptive Sparse Grids for High-Dimensional Problems. Verlag Dr. Hut, München, 2010.Google Scholar
  22. 22.
    C. Reisinger and G. Wittum. Efficient hierarchical approximation of high-dimensional option pricing problems. SIAM J. Scientific Computing, 29(1):440–458, 2007.MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    B. D. Ripley and N. L. Hjort. Pattern Recognition and Neural Networks. Cambridge University Press, New York, NY, USA, 1995.Google Scholar
  24. 24.
    A. A. Suchkov, R. J. Hanisch, and B. Margon. A census of object types and redshift estimates in the SDSS photometric catalog from a trained decision-tree classifier. Astron. J., 130:2439–2452, 2005.CrossRefGoogle Scholar
  25. 25.
    T. von Petersdorff and C. Schwab. Sparse finite element methods for operator equations with stochastic data. Appl. Math., 51(2):145–180, 2006.MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Y. Wadadekar. Estimating photometric redshifts using support vector machines. Publications of the Astronomical Society of the Pacific, 117:79, 2005.CrossRefGoogle Scholar
  27. 27.
    G. Widmer, R. Hiptmair, and C. Schwab. Sparse adaptive finite elements for radiative transfer. Journal of Computational Physics, 227:6071–6105, 2008.MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    C. Zenger. Sparse grids. In Wolfgang Hackbusch, editor, Parallel Algorithms for Partial Differential Equations, volume 31 of Notes on Numerical Fluid Mechanics, pages 241–251. Vieweg, 1991.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute for Parallel and Distributed SystemsUniversity of StuttgartStuttgartGermany

Personalised recommendations