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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
\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
\jour Problems Inform. Transmission
\yr 1999
\vol 35
\issue 2
\pages 136--149

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    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
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    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
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    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
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