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Statistical approach to some ill-posed problems for linear partial differential equations
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  • Published: February 1999

Statistical approach to some ill-posed problems for linear partial differential equations

  • Pao-Liu Chow1,
  • Ildar A. Ibragimov2 &
  • Rafail Z. Khasminskii1 

Probability Theory and Related Fields volume 113, pages 421–441 (1999)Cite this article

Abstract

For linear partial differential equations, some inverse source problems are treated statistically based on nonparametric estimation ideas. By observing the solution in a small Gaussian white noise, the kernel type of estimators is used to estimate the unknown source function and its partial derivatives.. It is proved that such estimators are consistent as the noise intensity tends to zero. Depending on the principal part of the differential operator, the optimal asymptotic rate of convergence is ascertained within a wide class of risk functions in a minimax sense.

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Authors and Affiliations

  1. Department of Mathematics, Wayne State University, Detroit, MI 48202, USA. e-mail: rafail@math.wagne.edu, , , , , , US

    Pao-Liu Chow & Rafail Z. Khasminskii

  2. St. Peterburg Branch of Steklov Mathematical Institute, 27 Fontanka, St. Peterburg 191011, Russia , , , , , , RU

    Ildar A. Ibragimov

Authors
  1. Pao-Liu Chow
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  2. Ildar A. Ibragimov
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  3. Rafail Z. Khasminskii
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Received: 5 May 1997 / Revised version: 18 June 1998

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Chow, PL., Ibragimov, I. & Khasminskii, R. Statistical approach to some ill-posed problems for linear partial differential equations . Probab. Theory Relat. Fields 113, 421–441 (1999). https://doi.org/10.1007/s004400050212

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  • Issue Date: February 1999

  • DOI: https://doi.org/10.1007/s004400050212

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  • Mathematics Subject Classification (1991): 60G, 62G, 35R
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