# On the Convergence Theorem for the Regularized Functional Matching Pursuit (RFMP) Algorithm

## TL;DR

This paper revises and extends the convergence theorem for the RFMP algorithm, a regularization method for linear inverse problems, including infinite-dimensional spaces and non-injective/non-surjective operators.

## Contribution

It reformulates and proves the convergence theorem for RFMP, extending it to infinite-dimensional Hilbert spaces and addressing non-injective and non-surjective operators.

## Key findings

- Reformulated the convergence theorem for RFMP.
- Extended RFMP to infinite-dimensional Hilbert spaces.
- Addressed convergence in non-injective and non-surjective cases.

## 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 [Fischer2011] and [FischerMichel2012], it was shown that the iteration converges in the unregularized case to an exact solution. In [Michel2015] and [MichelTelschow2016], 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 (ROFMP) algorithm in [MichelTelschow2016]. Moreover, the convergence proof in [Michel2015] 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.