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Probl. Peredachi Inf., 1999, Volume 35, Issue 2, Pages 51–66 (Mi ppi442)  

This article is cited in 16 scientific papers (total in 16 papers)

Methods of Signal Processing

A Statistical Approach to Some Inverse Problems for Partial Differential Equations

G. K. Golubev, R. Z. Khas'minskii


Abstract: Some inverse problems for the Laplace equation and heat-conduction equation are considered. The solutions of these equations are assumed to be observed under the Gaussian white noise of low intensity. The problem consists of the renewal of the unknown smooth boundary conditions or initial conditions from the solution observed against a noise background. It is shown that the minimax estimates of the second order are linear for low spectral density of the noise.

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English version:
Problems of Information Transmission, 1999, 35:2, 136–149

Bibliographic databases:
UDC: 621.391.1:519.27
Received: 15.09.1998

Citation: G. K. Golubev, R. Z. Khas'minskii, “A Statistical Approach to Some Inverse Problems for Partial Differential Equations”, Probl. Peredachi Inf., 35:2 (1999), 51–66; Problems Inform. Transmission, 35:2 (1999), 136–149

Citation in format AMSBIB
\Bibitem{GolKha99}
\by G.~K.~Golubev, R.~Z.~Khas'minskii
\paper A~Statistical Approach to Some Inverse Problems for Partial Differential Equations
\jour Probl. Peredachi Inf.
\yr 1999
\vol 35
\issue 2
\pages 51--66
\mathnet{http://mi.mathnet.ru/ppi442}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1728907}
\zmath{https://zbmath.org/?q=an:0947.35174}
\transl
\jour Problems Inform. Transmission
\yr 1999
\vol 35
\issue 2
\pages 136--149


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Kerkyacharian G., Picard D., “Thresholding Algorithms, Maxisets and Well-Concentrated Bases”, Test, 9:2 (2000), 283–328  crossref  mathscinet  isi
    2. Tsybakov A., “On the Best Rate of Adaptive Estimation in Some Inverse Problems”, Comptes Rendus Acad. Sci. Ser. I-Math., 330:9 (2000), 835–840  crossref  mathscinet  zmath  adsnasa  isi
    3. Cavalier, L, “Oracle inequalities for inverse problems”, Annals of Statistics, 30:3 (2002), 843  crossref  mathscinet  zmath  isi
    4. Cavalier L. Tsybakov A., “Sharp Adaptation for Inverse Problems with Random Noise”, Probab. Theory Relat. Field, 123:3 (2002), 323–354  crossref  mathscinet  zmath  isi
    5. Theory Probab. Appl., 48:3 (2004), 426–446  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. Lototsky S.V., “Optimal filtering of stochastic parabolic equations”, Recent Developments in Stochastic Analysis and Related Topics, 2004, 330–353  crossref  mathscinet  zmath  isi
    7. Johnstone I., Raimondo M., “Periodic Boxcar Deconvolution and Diophantine Approximation”, Ann. Stat., 32:5 (2004), 1781–1804  crossref  mathscinet  zmath  isi
    8. Golubev Y., “The Principle of Penalized Empirical Risk in Severely Ill-Posed Problems”, Probab. Theory Relat. Field, 130:1 (2004), 18–38  crossref  mathscinet  zmath  isi
    9. Cavalier L., “Nonparametric Statistical Inverse Problems”, Inverse Probl., 24:3 (2008), 034004  crossref  mathscinet  zmath  adsnasa  isi  elib
    10. Petsa, A, “Minimax convergence rates under the L-p-risk in the functional deconvolution model”, Statistics & Probability Letters, 79:13 (2009), 1568  crossref  mathscinet  zmath  isi
    11. Pensky, M, “FUNCTIONAL DECONVOLUTION IN A PERIODIC SETTING: UNIFORM CASE”, Annals of Statistics, 37:1 (2009), 73  crossref  mathscinet  zmath  isi
    12. Pensky M., Sapatinas T., “On Convergence Rates Equivalency and Sampling Strategies in Functional Deconvolution Models”, Ann Statist, 38:3 (2010), 1793–1844  crossref  mathscinet  zmath  isi
    13. Ingster Yu.I., Sapatinas T., Suslina I.A., “Minimax Signal Detection in Ill-Posed Inverse Problems”, Ann. Stat., 40:3 (2012), 1524–1549  crossref  mathscinet  zmath  isi  elib
    14. Knapik B.T., Van der Vaart A.W., Van Zanten J.H., “Bayesian Recovery of the Initial Condition for the Heat Equation”, Commun. Stat.-Theory Methods, 42:7, SI (2013), 1294–1313  crossref  mathscinet  zmath  isi
    15. Benhaddou R., Kulik R., Pensky M., Sapatinas T., “Multichannel Deconvolution With Long-Range Dependence: a Minimax Study”, J. Stat. Plan. Infer., 148 (2014), 1–19  crossref  mathscinet  zmath  isi
    16. Hohmann D., Holzmann H., “Weighted Angle Radon Transform: Convergence Rates and Efficient Estimation”, Stat. Sin., 26:1 (2016), 157–175  crossref  isi
  • Проблемы передачи информации Problems of Information Transmission
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