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We consider linear inverse problems in a nonparametric statistical framework.Both the signal and the operator are unknown and subject to error measurements.We establish minimax rates of convergence under squared error loss when theoperator admits a blockwise singular value decomposition (blockwise SVD) andthe smoothness of the signal is measured in a Sobolev sense. We construct anonlinear procedure adapting simultaneously to the unknown smoothness of boththe signal and the operator and achieving the optimal rate of convergence towithin logarithmic terms. When the noise level in the operator is dominant, bytaking full advantage of the blockwise SVD property, we demonstrate that theblock SVD procedure overperforms classical methods based on Galerkin projectionor nonlinear wavelet thresholding. We subsequently apply our abstract frameworkto the specific case of blind deconvolution on the torus and on the sphere.

Blockwise SVD with error in the operator and application to blind deconvolution