Convergence rates for regularization with sparsity constraints
نویسندگان
چکیده
Tikhonov regularization with p-powers of the weighted `p norms as penalties, with p ∈ (1, 2), have been lately employed in reconstruction of sparse solutions of ill-posed inverse problems. This paper points out convergence rates for such a regularization with respect to the norm of the weighted spaces, by assuming that the solutions satisfy 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.
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