The problem of the minimization of least squares functionals with $\ell^1$penalties is considered in an infinite dimensional Hilbert space setting. Whilethere are several algorithms available in the finite dimensional setting thereare only a few of them which come with a proper convergence analysis in theinfinite dimensional setting. In this work we provide an algorithm from a classwhich have not been considered for $\ell^1$ minimization before, namely aproximal-point method in combination with a projection step. We show that thisidea gives a simple and easy to implement algorithm. We present experimentswhich indicate that the algorithm may perform better than other algorithms ifwe employ them without any special tricks. Hence, we may conclude that theprojection proximal-point idea is a promising idea in the context of$\ell^1$-minimization.
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