Spatially inhomogeneous linear inverse problems with possible singularities

The objective of the present paper is to introduce the concept of a spatially inhomogeneous linear inverse problem which, to the best of the author's knowledge, has never been considered previously in statistical framework. The special feature of the problem is that the degree of ill-posedness depends not only on the scale but also on location. In this case, the rates of convergence are determined by the interaction of the smoothness and spatial homogeneity of the unknown function f and degrees of ill-posedness and spatial inhomogeneity of operator Q. An interesting property here is that, if operator Q is weakly inhomogeneous, then the rates of convergence are not influenced by spatial inhomogeneity of operator Q. On the other hand, if operator Q is moderately or strongly inhomogeneous, convergence rates are significantly affected by the degree of spatial inhomogeneity. Estimators obtained in the paper are based either on a hybrid of wavelet-vaguelette decomposition and Galerkin method. The hybrid estimator is a combination of a linear part in the vicinity of the singularity point and the nonlinear block thresholding wavelet estimator elsewhere. To attain adaptivity, an optimal resolution level for the linear, singularity affected, portion of the estimator is obtained using Lepskii method. Subsequently, this resolution level is used as the lowest resolution level for the nonlinear wavelet estimator. We show that convergence rates of the hybrid estimator lie within a logarithmic factor of the optimal minimax convergence rates. The theory is supplemented by examples of deconvolution based on irregularly spaced observations or in the presence of a heteroscedastic noise. The latter problem is also examined via a limited simulation study which demonstrates advantages of the hybrid estimator when the degree of spatial inhomogeneity is high.