Spatially Inhomogeneous Linear Inverse Problems with Possible Singularities

• Activity Number: 4
• Type: Invited
• Date/Time: Sunday, August 4, 2013 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Marianna Pensky*+
• Companies: University of Central Florida
• Keywords: Linear inverse problems ; inhomogeneous ; minimax convergence rates ; singularity

Abstract

We introduce the concept of a spatially inhomogeneous linear inverse problem which is emerging due to a variety of practical applications. The special feature of the problem is that the degree of ill-posedness depends not only on the scale but also on location. Estimators obtained in the paper are based either on wavelet-vaguelette decomposition or on a hybrid of wavelet-vaguelette decomposition and Galerkin method (if vaguelettes in the neighborhood of the singularity point have infinite norms). 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 (1990) method. Subsequently, this resolution level is used as the lowest resolution level for the nonlinear wavelet estimator. The theory presented in the paper is supplemented by examples and a simulation study which demonstrates advantages of the hybrid estimator when the degree of spatial inhomogeneity is high. Applications of the theory to DCE-CT imaging is discussed.