Estimation non-paramétrique dans les problèmes inverses à opérateur bruité

Thomas Vareschi 1
1 Statistiques
LPMA - Laboratoire de Probabilités et Modèles Aléatoires
Abstract : This thesis focuses on the impact of the imprecision on a linear operator when the latter is at stake in an inverse problem. The usual framework of an inverse problem involves the recovery of an input signal, when one observes its response through a linear operator (the output signal) . The output signal is usually observed with an additive random Gaussian noise, and we suppose that the operator is observed with an additive Gaussian noise as well, independent of the former, the error amplitudes being potentially different. We will study more precisely the case of kernel operators, when the kernel is subject to observation noise. This covers the case of periodic Fourier convolution, Laplace/Volterra convolution or spherical convolution. In each of those preceding cases, we develop statistical procedures of estimation, which rely on the adequate treatment of the Galerkin matrix involved when discretizing the inverse problem. More precisely, we study the quadratic risk in the case where the latter matrix is diagonal, block-diagonal or lower triangular Toeplitz. In each case we put into evidence new rates of convergence with an explicit dependency on the two noise amplitudes (noise contaminating the output signal or the kernel) and we prove them to be minimax. Finally, we focus on the specific case of spherical deconvolution and show how spherical needlets (or second generation wavelets) allow us to design a procedure which controls the risk measured in Lp norm.
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Submitted on : Tuesday, March 11, 2014 - 2:58:30 PM
Last modification on : Sunday, March 31, 2019 - 1:37:40 AM
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  • HAL Id : tel-00957985, version 1

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Thomas Vareschi. Estimation non-paramétrique dans les problèmes inverses à opérateur bruité. Statistiques [math.ST]. Université Paris-Diderot - Paris VII, 2013. Français. ⟨tel-00957985⟩

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