Optimal rates of linear convergence of the averaged alternating modified reflections method for two subspaces
The averaged alternating modified reflections (AAMR) method is a projection algorithm for finding the closest point in the intersection of convex sets to any arbitrary point in a Hilbert space. This method can be seen as an adequate modification of the Douglas–Rachford method that yields a solution to the best approximation problem. In this paper, we consider the particular case of two subspaces in a Euclidean space. We obtain the rate of linear convergence of the AAMR method in terms of the Friedrichs angle between the subspaces and the parameters defining the scheme, by studying the linear convergence rates of the powers of matrices. We further optimize the value of these parameters in order to get the minimal convergence rate, which turns out to be better than the one of other projection methods. Finally, we provide some numerical experiments that demonstrate the theoretical results.
KeywordsBest approximation problem Linear convergence Averaged alternating modified reflections method Linear subspaces Friedrichs angle
Mathematics Subject Classification (2010)65F10 65K05 65F15 15A08 47H09 90C25 41A25
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The authors are thankful to the anonymous referees for their very careful reading and constructive suggestions. We would also like to thank one of the referees of  for suggesting us this interesting project.
This work was partially supported by Ministerio de Economía, Industria y Competitividad (MINECO) of Spain and European Regional Development Fund (ERDF), grant MTM2014-59179-C2-1-P. F.J. Aragón Artacho was supported by the Ramón y Cajal program by MINECO and ERDF (RYC-2013-13327) and R. Campoy was supported by MINECO and European Social Fund (BES-2015-073360) under the program “Ayudas para contratos predoctorales para la formación de doctores 2015.”
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