Skip to main content

Linear problems: Definitions and a classical example

  • 2705 Accesses

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

Keywords

  • Linear Problem
  • Random Function
  • Quadrature Formula
  • Gaussian Measure
  • Brownian Bridge

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

1.5. Notes and References

  • Gihman, I. I., and Skorohod, A. V. (1974), The theory of stochastic processes I, Springer-Verlag, Berlin.

    CrossRef  MATH  Google Scholar 

  • Adler, R. J. (1981), The geometry of random fields, Wiley, New York.

    MATH  Google Scholar 

  • Vakhania, N. N., Tarieladze, V. I., and Chobanyan, S. A. (1987), Probability distributions on Banach spaces, Reidel, Dordrecht.

    CrossRef  Google Scholar 

  • Lifshits, M. A. (1995), Gaussian random functions, Kluwer, Dordrecht.

    CrossRef  MATH  Google Scholar 

2.8. Notes and References

  • Suldin, A. V. (1959), Wiener measure and its applications to approximation methods I (in Russian), Izv. Vyssh. Ucheb. Zaved. Mat. 13, 145–158.

    MathSciNet  Google Scholar 

  • Suldin, A. V. (1960), Wiener measure and its applications to approximation methods II (in Russian), Izv. Vyssh. Ucheb. Zaved. Mat. 18, 165–179.

    MathSciNet  Google Scholar 

  • Sarma, V. L. N. (1968), Eberlein measure and mechanical quadrature formulae. I: Basic Theory, Math. Comp. 22, 607–616.

    MathSciNet  MATH  Google Scholar 

  • Sarma, V. L. N., and Stroud, A. H. (1969), Eberlein measure and mechanical quadrature formulae. II: Numerical Results, Math. Comp. 23, 781–784.

    MathSciNet  MATH  Google Scholar 

  • Stroud, A. H. (1971), Approximate calculation of multiple integrals, Prentice Hall, Englewood Cliff.

    MATH  Google Scholar 

  • Larkin, F. M. (1972), Gaussian measure in Hilbert space and application in numerical analysis, Rocky Mountain J. Math. 2, 379–421.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Sacks, J., and Ylvisaker, D. (1966), Designs for regression with correlated errors, Ann. Math. Statist. 37, 68–89.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Sacks, J., and Ylvisaker, D. (1968), Designs for regression problems with correlated errors; many parameters, Ann. Math. Statist. 39, 49–69.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Sacks, J., and Ylvisaker, D. (1970a), Design for regression problems with correlated errors III, Ann. Math. Statist. 41, 2057–2074.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Sacks, J., and Ylvisaker, D. (1970b), Statistical design and integral approximation, in: Proc. 12th Bienn. Semin. Can. Math. Congr., R. Pyke, ed., pp. 115–136, Can. Math. Soc., Montreal.

    Google Scholar 

  • Cambanis, S. (1985), Sampling designs for time series, in: Time series in the time domain, Handbook of Statistics, Vol. 5, E. J. Hannan, P. R. Krishnaiah, and M. M. Rao, eds., pp. 337–362, North-Holland, Amsterdam.

    Google Scholar 

  • Traub, J. F., Wasilkowski, G. W., and Woźniakowski, H. (1988), Information-based complexity, Academic Press, New York.

    MATH  Google Scholar 

  • Micchelli, C. A., and Rivlin, T. J. (1985), Lectures on optimal recovery, in: Numerical analysis Lancaster 1984, P. R. Turner, ed., pp. 21–93, Lect. Notes in Math. 1129, Springer-Verlag, Berlin.

    CrossRef  Google Scholar 

  • Novak, E. (1988), Deterministic and stochastic error bounds in numerical analysis, Lect. Notes in Math. 1349, Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Kadane, J. B., and Wasilkowski, G. W. (1985), Average case ε-complexity in computer science — a Bayesian view, in: Bayesian statistics, J. M. Bernardo, ed., pp. 361–374, Elsevier, North-Holland.

    Google Scholar 

  • Diaconis, P. (1988), Bayesian numerical analysis, in: Statistical decision theory and related topics IV, Vol. 1, S. S. Gupta and J. O. Berger, eds., pp. 163–175, Springer-Verlag, New York.

    CrossRef  Google Scholar 

  • Richter, M. (1992), Approximation of Gaussian random elements and statistics, Teubner Verlagsgesellschaft, Stuttgart.

    MATH  Google Scholar 

  • Sard, A. (1949), Best approximate integration formulas, best approximation formulas, Amer. J. Math. 71, 80–91.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Kiefer, J. (1957), Optimum sequential search and approximation methods under regularity assumptions, J. Soc. Indust. Appl. Math. 5, 105–136.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Golomb, M., and Weinberger, H. F. (1959), Optimal approximation and error bounds, in: On numerical approximation, R. E. Langer, ed., pp. 117–190, Univ. of Wisconsin Press, Madison.

    Google Scholar 

  • Braß, H. (1977), Quadraturverfahren, Vandenhoek & Ruprecht, Göttingen.

    MATH  Google Scholar 

  • Levin, M., and Girshovich, J. (1979), Optimal quadrature formulas, Teubner Verlagsgesellschaft, Leipzig.

    MATH  Google Scholar 

  • Engels, H. (1980), Numerical quadrature and cubature, Academic Press, London.

    MATH  Google Scholar 

  • Sobolev, S. L. (1992), Cubature formulas and modern analysis, Gordon and Breach, Philadelphia.

    MATH  Google Scholar 

  • Tikhomirov, V. M. (1976), Some problems in approximation theory (in Russian), Moscow State Univ., Moscow.

    Google Scholar 

  • Tikhomirov, V. M. (1990), Approximation theory, in: Analysis II, R. V. Gamkrelidze, ed., Encyclopaedia of mathematical sciences, Vol. 14, Springer-Verlag, Berlin.

    CrossRef  Google Scholar 

  • Korneichuk, N. P. (1991), Exact constants in approximation theory, Cambridge Univ. Press, Cambridge.

    CrossRef  Google Scholar 

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

    MATH  Google Scholar 

  • Micchelli, C. A., and Rivlin, T. J. (1977), A survey of optimal recovery, in: Optimal estimation in approxmation theory, C. A. Micchelli and T. J. Rivlin, eds., pp. 1–54, Plenum, New York.

    CrossRef  Google Scholar 

  • Micchelli, C. A., and Rivlin, T. J. (1985), Lectures on optimal recovery, in: Numerical analysis Lancaster 1984, P. R. Turner, ed., pp. 21–93, Lect. Notes in Math. 1129, Springer-Verlag, Berlin.

    CrossRef  Google Scholar 

  • Traub, J. F., Wasilkowski, G. W., and Woźniakowski, H. (1983), Information, uncertainty, complexity, Addison-Wesley, Reading.

    MATH  Google Scholar 

  • Traub, J. F., Wasilkowski, G. W., and Woźniakowski, H. (1988), Information-based complexity, Academic Press, New York.

    MATH  Google Scholar 

  • Novak, E. (1988), Deterministic and stochastic error bounds in numerical analysis, Lect. Notes in Math. 1349, Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Plaskota, L. (1996), Noisy information and computational complexity, Cambridge Univ. Press, Cambridge.

    CrossRef  MATH  Google Scholar 

  • Kon, M. A., Ritter, K., and Werschulz, A. G. (1991), On the average case solvability of ill-posed problems, J. Complexity 7, 220–224.

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Wasilkowski, G. W. (1993), Integration and approximation of multivariate functions: average case complexity with isotropic Wiener measure, Bull. Amer. Math. Soc. (N. S.) 28, 308–314. Full version (1994), J. Approx. Theory 77, 212–227.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Christensen, R. (1991), Linear models for multivariate, time series, and spatial data, Springer-Verlag, New York.

    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 

  • Christensen, R. (1991), Linear models for multivariate, time series, and spatial data, Springer-Verlag, New York.

    CrossRef  MATH  Google Scholar 

  • Cressie, N. A. C. (1993), Statistics for spatial data, Wiley, New York.

    MATH  Google Scholar 

  • Hjort, N. L., and Omre, H. (1994), Topics in spatial statistics, Scand. J. Statist. 21, 289–357.

    MathSciNet  MATH  Google Scholar 

  • Sacks, J., Welch, W. J., Mitchell, T. J., and Wynn, H. P. (1989), Design and analysis of computer experiments, Statist. Sci. 4, 409–435.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Currin, C., Mitchell, T., Morris, M., and Ylvisaker, D. (1991), Bayesian prediction of deterministic functions, with applications to the design and analysis of computer experiments, J. Amer. Statist. Assoc. 86, 953–963.

    CrossRef  MathSciNet  Google Scholar 

  • Koehler, J. R., and Owen, A. B. (1996), Computer experiments, in: Design and analysis of experiments, Handbook of Statistics, Vol. 13, S. Gosh and C. R. Rao, eds., pp. 261–308, North Holland, Amsterdam.

    CrossRef  Google Scholar 

3.8. Notes and References

  • Suldin, A. V. (1959), Wiener measure and its applications to approximation methods I (in Russian), Izv. Vyssh. Ucheb. Zaved. Mat. 13, 145–158.

    MathSciNet  Google Scholar 

  • Samaniego, F. J. (1976), The optimal sampling design for estimating the integral of a process with stationary independent increments, IEEE Trans. Inform. Theory IT-22, 375–376.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Cressie, N. A. C. (1978), Estimation of the integral of a stochastic process, Bull. Austr. Math. Soc. 18, 83–93.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Yanovich, L. A. (1988), On the best quadrature formulas in the spaces of random process trajectories (in Russian), Dokl. Akad. Nauk BSSR 32, 9–12.

    MathSciNet  MATH  Google Scholar 

  • Lee, D. (1986), Approximation of linear operators on a Wiener space, Rocky Mountain J. Math. 16, 641–659.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Abt, M. (1992), Some exact optimal designs for linear covariance functions in one dimension, Commun. Statist.-Theory Meth. 21, 2059–2069.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Ritter, K. (1990), Approximation and optimization on the Wiener space, J. Complexity 6, 337–364.

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2000 Springer-Verlag

About this chapter

Cite this chapter

Ritter, K. (2000). Linear problems: Definitions and a classical example. In: Ritter, K. (eds) Average-Case Analysis of Numerical Problems. Lecture Notes in Mathematics, vol 1733. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0103936

Download citation

  • DOI: https://doi.org/10.1007/BFb0103936

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67449-8

  • Online ISBN: 978-3-540-45592-9

  • eBook Packages: Springer Book Archive