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. 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 four parameters, 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 and coincide with the rates which are usual for homogeneous linear inverse problems. 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 wavelet-vaguelette decomposition or 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. We show that convergence rates of the hybrid estimator lie within a logarithmic factor of the optimal minimax convergence rates. The theory presented in the paper is supplemented by examples of deconvolution with a spatially inhomogeneous kernel, deconvolution in the presence of locally extreme noise or extremely inhomogeneous design. The first two problems are examined via a limited simulation study which demonstrates advantages of the hybrid estimator when the degree of spatial inhomogeneity is high. In addition, we apply the technique to recovery of a convolution signal transmitted via amplitude modulation.