Abstract
This paper investigates the randomized version of the Kaczmarz method to solve linear systems in the case where the adjoint of the system matrix is not exact—a situation we refer to as “mismatched adjoint”. We show that the method may still converge both in the over- and underdetermined consistent case under appropriate conditions, and we calculate the expected asymptotic rate of linear convergence. Moreover, we analyze the inconsistent case and obtain results for the method with mismatched adjoint as for the standard method. Finally, we derive a method to compute optimized probabilities for the choice of the rows and illustrate our findings with numerical examples.
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Notes
If \(\langle a_{i} \,,\, v_{i}\rangle = 0\), Algorithm 1 is not defined and in the case of \(\langle a_{i} \,,\, v_{i}\rangle <0\), the probabilities \(p_{i}\) below in Remark 2.4 would not be non-negative. However, in the case \(\langle a_{i} \,,\, v_{i}\rangle <0\) we could switch the sign of the \(v_{i}\)s, and the expressions in (2.1) and (2.2) would not change).
The code to produce the figures in this article is available at https://github.com/dirloren/rkma.
We could also add a perturbation to A - the numerical results would be rather similar. Setting entries to zero would be numerically beneficial if A would have many small entries, but this was not be the motivation here.
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Acknowledgements
The authors thank Emil Sidky and Per Christian Hansen for valuable discussions and the reviewers for helpful remarks.
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Communicated by Lothar Reichel.
This material was based upon work partially supported by the National Science Foundation under Grant DMS-1127914 to the Statistical and Applied Mathematical Sciences Institute. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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Lorenz, D.A., Rose, S. & Schöpfer, F. The randomized Kaczmarz method with mismatched adjoint. Bit Numer Math 58, 1079–1098 (2018). https://doi.org/10.1007/s10543-018-0717-x
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DOI: https://doi.org/10.1007/s10543-018-0717-x