On the convergence theorem for the regularized functional matching pursuit (RFMP) algorithm

Abstract

The RFMP is an iterative regularization method for a class of linear inverse problems. It has proved to be applicable to problems which occur, for example, in the geosciences. In the early publications (Fischer in Sparse Regularization of a Joint Inversion of Gravitational Data and NormalMode Anomalies, 2011; Fischer and Michel in Inverse Probl 28:065012, 2012), it was shown that the iteration converges in the unregularized case to an exact solution. In Michel (in: Freeden, Nashed, Sonar (eds) Handbook of geomathematics, 2nd edn, Springer, Berlin, pp 2121–2147, 2015) and Michel and Telschow (Int J Geomath 5:195–224, 2014), it was later shown (for two different scenarios) that the iteration also converges in the regularized case, where the limit of the iteration is the solution of the Tikhonov-regularized normal equation. However, the condition of these convergence proofs cannot be satisfied and, therefore, has to be weakened, as it was pointed out for the convergence theorem of the related iterated regularized orthogonal functional matching pursuit algorithm in Michel and Telschow (SIAM J Numer Anal 54:262–287, 2016). Moreover, the convergence proof in Michel (2015) contained a minor error. For these reasons, we reformulate here the convergence theorem for the RFMP and its proof. We also use this opportunity to extend the algorithm for an arbitrary infinite-dimensional separable Hilbert space setting. In addition, we particularly elaborate the cases of non-injective and non-surjective operators.