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Integration and approximation of multivariate functions

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1733)

Keywords

  • Tensor Product
  • Minimal Error
  • Sparse Grid
  • Cubature Formula
  • Multivariate Function

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2.8. Notes and References

  • Ylvisaker, D. (1975), Designs on random fields, in: A survey of statistical design and linear models, J. Srivastava, ed., pp. 593–607, North-Holland, Amsterdam.

    Google Scholar 

  • Wittwer, Gisela (1978), Über asymptotisch optimale Versuchsplanung im Sinne von Sacks-Ylvisaker, Math. Operationsforsch. u. Statist. 9, 61–71.

    MathSciNet  MATH  Google Scholar 

  • Micchelli, C. A., and Wahba, G. (1981), Design problems for optimal surface interpolation, in: Approximation theory and applications, Z. Ziegler, ed., pp. 329–347, Academic Press, New York.

    Google Scholar 

  • Papageorgiou, A., and Wasilkowski, G. W. (1990), On the average complexity of multivariate problems, J. Complexity 6, 1–23.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Woźniakowski, H. (1991), Average case complexity of multivariate integration, Bull. Amer. Math. Soc. (N. S.) 24, 185–194.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Aronszajn, N. (1950), Theory of reproducing kernels, Trans. Amer. Math. Soc. 68, 337–404.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Delvos, F.-J., and Schempp, W. (1989), Boolean methods in interpolation and approximation, Pitman Research Notes in Mathematics Series 230, Longman, Essex.

    MATH  Google Scholar 

  • Smolyak, S. A. (1963), Quadrature and interpolation formulas for tensor products of certain classes of functions, Soviet Math. Dokl. 4, 240–243.

    MATH  Google Scholar 

  • Novak, E., Ritter, K., and Woźniakowski, H. (1995), Average case optimality of a hybrid secant-bisection method, Math. Comp. 64, 1517–1539.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Gordon, W. J. (1971), Blending function methods of bivariate and multivariate interpolation and approximation, SIAM J. Numer. Anal. 8, 158–177.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Wahba, G. (1978b), Interpolating surfaces: high order convergence rates and their associated designs, with applications to X-ray image reconstruction, Tech. Rep. No. 523, Dept. of Statistics, Univ. of Wisconsin, Madison.

    Google Scholar 

  • Delvos, F.-J., and Schempp, W. (1989), Boolean methods in interpolation and approximation, Pitman Research Notes in Mathematics Series 230, Longman, Essex.

    MATH  Google Scholar 

  • Delvos, F.-J. (1990), Boolean methods for double integration, Math. Comp. 55, 683–692.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Ritter, K. (1996a), Asymptotic optimality of regular sequence designs, Ann. Statist. 24, 2081–2096.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Novak, E., and Ritter, K. (1999), Simple cubature formulas with high polynomial exactness, Constr. Approx. 15, 499–522.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Genz, A. C. (1986), Fully symmetric interpolatory rules for multiple integrals, SIAM J. Numer. Anal. 23, 1273–1283.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Pereverzev, S. V. (1986), On optimization of approximate methods of solving integral equations, Soviet Math. Dokl. 33, 347–351.

    MATH  Google Scholar 

  • Pereverzev, S. V. (1996), Optimization of methods for approximate solution of operator equations, Nova Science, New York.

    MATH  Google Scholar 

  • Zenger, Ch. (1991), Sparse grids, in: Parallel algorithms for partial differential equations, W. Hackbusch, ed., pp. 241–251. Vieweg, Braunschweig.

    Google Scholar 

  • Griebel, M., Schneider, M., and Zenger, Ch. (1992), A combination technique for the solution of sparse grid problems, in: Iterative methods in linear algebra, R. Beauwens and P. de Groen, eds., pp. 263–281, Elsevier, North-Holland.

    Google Scholar 

  • Werschulz, A. G. (1996), The complexity of the Poisson problem for spaces of bounded mixed derivatives, in: The mathematics of numerical analysis, J. Renegar, M. Shub, S. Smale, eds., pp. 895–914, Lect. in Appl. Math. 32, AMS, Providence.

    Google Scholar 

  • Bungartz, H.-J., and Griebel, M. (1999), A note on the complexity of solving Poisson's equation for spaces of bounded mixed derivatives, J. Complexity 15, 167–199.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Ritter, K. (1996b), Almost optimal differentiation using noisy data, J. Approx. Theory bf 86, 293–309.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Temlyakov, V. N. (1994), Approximation of periodic functions, Nova Science, New York.

    MATH  Google Scholar 

  • Novak, E., Ritter, K., and Steinbauer, A. (1998), A multiscale method for the evaluation of Wiener integrals, in: Approximation Theory IX, Vol. 2, C. K. Chui and L. L. Schumaker, eds., pp. 251–258, Vanderbilt Univ. Press, Nashville.

    Google Scholar 

  • Wasilkowski, G. W., and Woźniakowski, H. (1999), Weighted tensor product algorithms for linear multivariate problems, J. Complexity 15, 402–447.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Steinbauer, A. (1999), Quadrature formulas for the Wiener measure, J. Complexity 15, 476–498.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Wasilkowski, G. W., and Woźniakowski, H. (1999), Weighted tensor product algorithms for linear multivariate problems, J. Complexity 15, 402–447.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Novak, E., and Ritter, K. (1993), Some complexity results for zero finding for univariate functions, J. Complexity 9, 15–40.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Papageorgiou, A., and Wasilkowski, G. W. (1990), On the average complexity of multivariate problems, J. Complexity 6, 1–23.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Novak, E., Ritter, K., and Woźniakowski, H. (1995), Average case optimality of a hybrid secant-bisection method, Math. Comp. 64, 1517–1539.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Woźniakowski, H. (1991), Average case complexity of multivariate integration, Bull. Amer. Math. Soc. (N. S.) 24, 185–194.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Papageorgiou, A., and Wasilkowski, G. W. (1990), On the average complexity of multivariate problems, J. Complexity 6, 1–23.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Woźniakowski, H. (1992), Average case complexity of linear multivariate problems, Part 1: Theory, Part 2: Applications, J. Complexity 8, 337–372, 373–392.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Woźniakowski, H. (1991), Average case complexity of multivariate integration, Bull. Amer. Math. Soc. (N. S.) 24, 185–194.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Roth, K. (1954), On irregularities of distribution, Mathematika 1, 73–79.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Roth, K. (1980), On irregularities of distribution, IV, Acta Arith. 37, 67–75.

    MathSciNet  MATH  Google Scholar 

  • Frank, K., and Heinrich, S. (1996), Computing discrepancies of Smolyak quadrature rules, J. Complexity 12, 287–314.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Niederreiter, H. (1992), Random number generation and quasi-Monte Carlo methods, CBSM-NSF Regional Conf. Ser. Appl. Math. 63, SIAM, Philadelphia.

    CrossRef  MATH  Google Scholar 

  • Drmota, M., and Tichy, R. F. (1997), Sequences, discrepancies, and applications, Lect. Notes in Math. 1651, Springer-Verlag, Berlin.

    MATH  Google Scholar 

3.1. Notes and References

  • Bellman, R. (1961), Adaptive Control Processes: a Guided Tour, Princeton University, Princeton.

    CrossRef  MATH  Google Scholar 

  • Woźniakowski, H. (1992), Average case complexity of linear multivariate problems, Part 1: Theory, Part 2: Applications, J. Complexity 8, 337–372, 373–392.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Woźniakowski, H. (1994a), Tractability and strong tractability of linear multivariate problems, J. Complexity 10, 96–128.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Woźniakowski, H. (1992), Average case complexity of linear multivariate problems, Part 1: Theory, Part 2: Applications, J. Complexity 8, 337–372, 373–392.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Woźniakowski, H. (1994a), Tractability and strong tractability of linear multivariate problems, J. Complexity 10, 96–128.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Woźniakowski, H. (1994b), Tractability and strong tractability of multivariate tensor product problems, J. Computing Inform. 4, 1–19.

    MATH  Google Scholar 

  • Wasilkowski, G. W., and Woźniakowski, H. (1995), Explicit cost bounds of algorithms for multivariate tensor product problems, J. Complexity 11, 1–56.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Wasilkowski, G. W., and Woźniakowski, H. (1997), The exponent of discrepancy is at most 1.4778, Math. Comp. 66, 1125–1132.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Sloan, I. H., and Woźniakowski, H. (1998), When are quasi-Monte Carlo algorithms efficient for high dimensional integrals, J. Complexity 14, 1–33.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Wasilkowski, G. W., and Woźniakowski, H. (1999), Weighted tensor product algorithms for linear multivariate problems, J. Complexity 15, 402–447.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Wasilkowski, G. W., and Woźniakowski, H. (1999), Weighted tensor product algorithms for linear multivariate problems, J. Complexity 15, 402–447.

    CrossRef  MathSciNet  MATH  Google Scholar 

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Ritter, K. (2000). Integration and approximation of multivariate functions. In: Ritter, K. (eds) Average-Case Analysis of Numerical Problems. Lecture Notes in Mathematics, vol 1733. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0103940

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  • DOI: https://doi.org/10.1007/BFb0103940

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