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Liberating the Dimension for Function Approximation and Integration

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 23)

Abstract

We discuss recent results on the complexity and tractability of problems dealing with -variate functions. Such problems, especially path integrals, arise in many areas including mathematical finance, quantum physics and chemistry, and stochastic differential equations. It is possible to replace the -variate problem by one that has only d variables since the difference between the two problems diminishes with d approaching infinity. Therefore, one could use algorithms obtained in the Information-Based Complexity study, where problems with arbitrarily large but fixed d have been analyzed. However, to get the optimal results, the choice of a specific value of d should be a part of an efficient algorithm. This is why the approach discussed in the present paper is called liberating the dimension. Such a choice should depend on the cost of sampling d-variate functions and on the error demand \(\epsilon \). Actually, as recently observed for a specific class of problems, optimal algorithms are from a family of changing dimension algorithms which approximate -variate functions by a combination of special functions, each depending on a different set of variables. Moreover, each such set contains no more than \(d(\epsilon ) = \mathcal{O}(\ln (1/\epsilon )/\ln (\ln (1/\epsilon )))\) variables. This is why the new algorithms have the total cost polynomial in \(1/\epsilon \) even if the cost of sampling a d-variate function is exponential in d.

Notes

Acknowledgements

I would like to thank Henryk Woźniakowski for valuable comments and suggestions to this paper.

References

  1. 1.
    Caflisch, R. E., Morokoff, M., Owen, A. B.: Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension. J. Computational Finance 1, 27–46 (1997)Google Scholar
  2. 2.
    Creutzig, J., Dereich, S., Müller-Gronbach, T., Ritter, K.: Infinite-dimensional quadrature and approximation of distributions. Found. Comput. Math. 9, 391–429 (2009)Google Scholar
  3. 3.
    Das, A.: Field Theory: A Path Integral Approach. Lecture Notes in Physics, Vol. 52, World Scientific, Singapore, 1993Google Scholar
  4. 4.
    DeWitt-Morette, C. (editor): Special Issue on Functional Integration. J. Math. Physics 36, (1995)Google Scholar
  5. 5.
    Duffie, D.: Dynamic Asset Pricing Theory. Princeton University, Princeton, NJ, 1992Google Scholar
  6. 6.
    Egorov, R. P., Sobolevsky, P. I., Yanovich, L. A.: Functional Integrals: Approximate Evaluation and Applications. Kluver Academic, Dordrecht, 1993Google Scholar
  7. 7.
    Feynman, R. P., Hibbs, A. R.: Quantum Mechanics and Path-Integrals. McGraw-Hill, New York, 1965Google Scholar
  8. 8.
    Gnewuch, M.: Infinite-dimensional integration on weighted Hilbert spaces. Math. Comput. 81, 2175–2205 (2012)Google Scholar
  9. 9.
    Hickernell, F. J., Müller-Gronbach, T., Niu, B., Ritter, K.: Multi-level Monte Carlo algorithms for infinite-dimensional integration on \({\mathbb{R}}^{\mathbb{N}}\). J. Complexity 26, 229–254 (2010)Google Scholar
  10. 10.
    Hickernell, F. J., Wang, X.: The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension. Math. Comp. 71, 1641–1661 (2002)Google Scholar
  11. 11.
    Hinrichs, A., Novak, E., Vybiral, J.: Linear information versus function evaluations for L 2-approximation, J. Complexity 153, 97–107 (2008)Google Scholar
  12. 12.
    Hull, J.: Option, Futures, and Other Derivative Securities. 2nd ed., Prentice Hall, Engelwood Cliffs. NJ, 1993Google Scholar
  13. 13.
    Khandekar, D. C., Lawande, S. V., Bhagwat, K. V.: Path-Integral Methods and their Applications. World Scientific, Singapore, 1993Google Scholar
  14. 14.
    Kleinert, H.: Path Integrals in Quantum Mechanics, Statistics and Polymer Physics. World Scientific, Singapore, 1990Google Scholar
  15. 15.
    Kuo, F. Y., Sloan, I. H., Wasilkowski, G. W., Woźniakowski, H.: On decompositions of multivariate functions. Math. Comp. 79 953-966 (2010), DOI: 0.1090/S0025-5718-09-02319-9Google Scholar
  16. 16.
    Kuo, F. Y., Sloan, I. H., Wasilkowski, G. W., Woźniakowski, H.: Liberating the dimension. J. Complexity 26, 422–454 (2010)Google Scholar
  17. 17.
    Merton, R.: Continuous–Time Finance, Basil Blackwell, Oxford, 1990Google Scholar
  18. 18.
    Niu, B., Hickernell, F. J.: Monte Carlo simulation of stochastic integrals when the cost function evaluation is dimension dependent. In: Ecuyer, P. L., Owen, A. B. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2008, pp. 545-569, Springer (2008)Google Scholar
  19. 19.
    Niu, B., Hickernell, F. J., Müller-Gronbach, T., Ritter, K.: Deterministic multi-level algorithms for infinite-dimensional integration on \({\mathbb{R}}^{\mathbb{N}}\). Submitted (2010)Google Scholar
  20. 20.
    Novak, E.: Optimal linear randomized methods for linear operators in Hilbert spaces, J. Complexity 8, 22–36, (1992)Google Scholar
  21. 21.
    Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems, European Mathematical Society, Zürich (2008)Google Scholar
  22. 22.
    Novak, E., Woźniakowski, H.: On the power of function values for the approximation problem in various settings. Submitted (2010)Google Scholar
  23. 23.
    Plaskota, L., Wasilkowski, G. W.: Tractability of infinite-dimensional integration in the worst case and randomized settings. J. Complexity 27, 505–518 (2011)Google Scholar
  24. 24.
    Plaskota, L., Wasilkowski, G. W., Woźniakowski, H.: A new algorithm and worst case complexity for Feynman-Kac path integration. J. Computational Physics 164, 335–353 (2000)Google Scholar
  25. 25.
    Smolyak, S. A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Acad. Nauk SSSR 4, 240-243 (1963)Google Scholar
  26. 26.
    Traub, J. F., Wasilkowski, G. W., Woźniakowski, H.: Information-Based Complexity, Academic Press, New York (1988)Google Scholar
  27. 27.
    Wang, X., Fang, K. -T.: Effective dimensions and quasi-Monte Carlo integration. J. Complexity 19, 101-124 (2003)Google Scholar
  28. 28.
    Wang, X., Sloan, I. H.: Why are high-dimensional finance problems often of low effective dimension? SIAM J. Sci. Comput. 27, 159-183 (2005)Google Scholar
  29. 29.
    Wasilkowski, G. W.: Randomization for continuous problems, J. Complexity 5, 195–218 (1989)Google Scholar
  30. 30.
    Wasilkowski, G. W., Woźniakowski, H.: Explicit cost bounds for multivariate tensor product problems. J. Complexity 11, 1-56 (1995)Google Scholar
  31. 31.
    Wasilkowski, G. W., Woźniakowski, H.: On tractability of path integration, J. Math. Physics 37, 2071-2088 (1996)Google Scholar
  32. 32.
    Wasilkowski, G. W., Woźniakowski, H.: The power of standard information for multivariate approximation in the randomized setting, Mathematics of Computation 76, 965–988 (2007)Google Scholar
  33. 33.
    Wasilkowski, G. W., Woźniakowski, H.: Liberating the dimension for function approximation. J. Complexity 27, 86–110 (2011)Google Scholar
  34. 34.
    Wasilkowski, G. W., Woźniakowski, H.: Liberating the dimension for function approximation: standard information. J. Complexity 27, 417–440 (2011)Google Scholar
  35. 35.
    Wasilkowski, G. W.: Liberating the dimension for L 2 approximation. J. Complexity 28, 304–319 (2012)Google Scholar
  36. 36.
    Wiegel, F. W.: Path Integral Methods in Physics and Polymer Physics, World Scientific, Singapore (1986)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of KentuckyLexingtonUSA

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