Adaptive wavelet Galerkin methods for linear inverse problems

Albert Cohen, Marc Hoffmann () and Markus Reiß

No 2002,50, SFB 373 Discussion Papers from Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes

Abstract: We introduce and analyse numerical methods for the treatment of inverse problems, based on an adaptive wavelet Galerkin discretization. These methods combine the theoretical advantages of the wavelet-vaguelette decomposition (WVD) in terms of optimally adapting to the unknown smoothness of the solution, together with the numerical simplicity of Galerkin methods. Two strategies are proposed: the first one simply combines a thresholding algorithm on the data with a Galerkin inversion on a fixed liner space, while the second one performs the inversion through an adaptive procedure in which a smaller space adapted to the solution is iteratively constructed. For both methods, we recover the same minimax rates achieved by WVD for various function classes modeling the solution.

Date: 2002
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