Compressive Algorithms--Adaptive Solutions of PDEs and Variational Problems
Proceedings of the 13th IMA International Conference on Mathematics of Surfaces XIII
A proximal iteration for deconvolving Poisson noisy images using sparse representations
IEEE Transactions on Image Processing
Neuroelectric current localization from combined EEG/MEG data
Proceedings of the 7th international conference on Curves and Surfaces
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We introduce and analyze 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. In a first step, we simply combine a thresholding algorithm on the data with a Galerkin inversion on a fixed linear space. In a second step, a more elaborate method performs the inversion by an adaptive procedure in which a smaller space adapted to the solution is iteratively constructed; this leads to a significant reduction of the computational cost.