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The Regularized Weak Functional Matching Pursuit for linear inverse problems

Max Kontak ORCID logo and Volker Michel ORCID logo

Abstract

In this work, we present the so-called Regularized Weak Functional Matching Pursuit (RWFMP) algorithm, which is a weak greedy algorithm for linear ill-posed inverse problems. In comparison to the Regularized Functional Matching Pursuit (RFMP), on which it is based, the RWFMP possesses an improved theoretical analysis including the guaranteed existence of the iterates, the convergence of the algorithm for inverse problems in infinite-dimensional Hilbert spaces, and a convergence rate, which is also valid for the particular case of the RFMP. Another improvement is the cancellation of the previously required and difficult to verify semi-frame condition. Furthermore, we provide an a-priori parameter choice rule for the RWFMP, which yields a convergent regularization. Finally, we will give a numerical example, which shows that the “weak” approach is also beneficial from the computational point of view. By applying an improved search strategy in the algorithm, which is motivated by the weak approach, we can save up to 90  of computation time in comparison to the RFMP, whereas the accuracy of the solution does not change as much.

Funding statement: The authors gratefully acknowledge the financial support by the School of Science and Technology of the University of Siegen, Germany

Acknowledgements

We would like to thank the anonymous reviewers for their valuable comments, which helped to improve the paper.

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Received: 2018-02-27
Revised: 2018-07-13
Accepted: 2018-08-27
Published Online: 2018-10-16
Published in Print: 2019-06-01

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