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A note on the minimization of a Tikhonov functional with \(\ell^1\)-penalty. (English) Zbl 1455.65086
This paper is concerned with the minimization of a Tikhonov functional with an \(\ell^1\) penalty, arising from sparse reconstruction. The authors propose a transformation with \(\ell^2\) penalty and a nonlinear operator. The novelty lies in the fact that the resulting functional is a twice differentiable functional that can now be minimized using efficient second order methods, e.g., Newton’s method. The authors provide a convergence analysis of the scheme and several numerical experiments.
65J22 Numerical solution to inverse problems in abstract spaces
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
47J06 Nonlinear ill-posed problems
Full Text: DOI
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