Geodesic
Convergence rates for regularization with sparsity constraints
Electronic transactions on numerical analysis, Tome 37 (2010), pp. 87-104
Tikhonov regularization with p-powers of the weighted $\ell p$ norms as penalties, with p $\in $(1, 2), have been employed recently in reconstruction of sparse solutions of ill-posed inverse problems. This paper shows convergence rates for such a regularization with respect to the norm of the weighted spaces by assuming that the solutions satisfy a certain smoothness (source) condition. The meaning of the latter is analyzed in some detail. Moreover, converse results are established: Linear convergence rates for the residual, together with convergence of the approximations to the solution, can be achieved only if the solution satisfies a source condition. Further insights for the particular case of a convolution equation are provided by analyzing the equation both theoretically and numerically.
Classification : 47A52, 65J20
Keywords: ill-posed problem, regularization, Bregman distance, sparsity 
@article{ETNA_2010__37__a21,
     author = {Ramlau,  Ronny and Resmerita,  Elena},
     title = {Convergence rates for regularization with sparsity constraints},
     journal = {Electronic transactions on numerical analysis},
     pages = {87--104},
     year = {2010},
     volume = {37},
     zbl = {1206.47016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2010__37__a21/}
}
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%A Resmerita,  Elena
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%J Electronic transactions on numerical analysis
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Ramlau,  Ronny; Resmerita,  Elena. Convergence rates for regularization with sparsity constraints. Electronic transactions on numerical analysis, Tome 37 (2010), pp. 87-104. http://geodesic.mathdoc.fr/item/ETNA_2010__37__a21/