Wavelet-Galerkin methods for ill-posed problems
Volker Dicken, P. MAASS · Journal of Inverse and Ill-Posed Problems · 1996
Projection methods based on wavelet functions combine optimal convergence rates with algorithmic efficiency. The proofs in this paper utilize the approximation properties of wavelets and results from the general theory of regularization methods. Moreover, adaptive strategies can be incorporated still leading to optimal convergence rates for the resulting algorithms. The so-called wavelet-vaguelette decompositions enable the realization of especially fast algorithms for certain operators. 1. INTRODUCTION A projection method for solving the inverse problem Af = g (1.1) where A : X ! Y denotes an operator between Hilbert spaces X and Y , and g 2 Y is the given data is defined by the sequences of subspaces X h ae X h 0 ae X, h 0 ! h, and Y h ae Y growing in size with the step width h. In a realistic setting we are only given noisy data g " with a bounded error kg \\Gamma g " k Y ! ". An approximate solution f h 2 X h is then computed from the requirement hAf h j yi Y = hg " j y...