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Fractional-order regularization and wavelet approximation to the inverse estimation problem for random fields

Author

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  • Ruiz-Medina, M. D.
  • Angulo, J. M.
  • Anh, V. V.

Abstract

The least-squares linear inverse estimation problem for random fields is studied in a fractional generalized framework. First, the second-order regularity properties of the random fields involved in this problem are analysed in terms of the fractional Sobolev norms. Second, the incorporation of prior information in the form of a fractional stochastic model, with covariance operator bicontinuous with respect to a certain fractional Sobolev norm, leads to a regularization of this problem. Third, a multiresolution approximation to the class of linear inverse problems considered is obtained from a wavelet-based orthogonal expansion of the input and output random models. The least-squares linear estimate of the input random field is then computed using these orthogonal wavelet decompositions. The results are applied to solving two important cases of linear inverse problems defined in terms of fractional integral operators.

Suggested Citation

  • Ruiz-Medina, M. D. & Angulo, J. M. & Anh, V. V., 2003. "Fractional-order regularization and wavelet approximation to the inverse estimation problem for random fields," Journal of Multivariate Analysis, Elsevier, vol. 85(1), pages 192-216, April.
    • Handle: RePEc:eee:jmvana:v:85:y:2003:i:1:p:192-216
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    References listed on IDEAS

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    1. Angulo, J. M. & Ruiz-Medina, M. D., 1997. "On the orthogonal representation of generalized random fields," Statistics & Probability Letters, Elsevier, vol. 31(3), pages 145-153, January.
    2. Ruiz-Medina, M. D. & Anh, V. V. & Angulo, J. M., 2001. "Stochastic fractional-order differential models with fractal boundary conditions," Statistics & Probability Letters, Elsevier, vol. 54(1), pages 47-60, August.
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    Cited by:

    1. Mekoth, Chitra & George, Santhosh & Jidesh, P., 2021. "Fractional Tikhonov regularization method in Hilbert scales," Applied Mathematics and Computation, Elsevier, vol. 392(C).
    2. Rosaura Fernández-Pascual & María Ruiz-Medina & Jose Angulo, 2004. "Wavelet-based functional reconstruction and extrapolation of fractional random fields," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 13(2), pages 417-444, December.
    3. M. D. Ruiz-Medina & V. V. Anh & R. M. Espejo & J. M. Angulo & M. P. Frías, 2015. "Least-Squares Estimation of Multifractional Random Fields in a Hilbert-Valued Context," Journal of Optimization Theory and Applications, Springer, vol. 167(3), pages 888-911, December.
    4. Fernández-Pascual, Rosaura & Ruiz-Medina, María D. & Angulo, José M., 2006. "Estimation of intrinsic processes affected by additive fractal noise," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1361-1381, July.
    5. Beran, Jan & Ghosh, Sucharita & Schell, Dieter, 2009. "On least squares estimation for long-memory lattice processes," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2178-2194, November.

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