Luiz-Rafael Santos — Page 2

arXiv2017

On the linear convergence of the circumcentered-reflection method

In order to accelerate the Douglas--Rachford method we recently developed the circumcentered--reflection method, which provides the closest iterate to the solution among all points relying on successive reflections, for the best approximation problem related to two affine subspaces. We now prove that this is still the case when considering a family of finitely many affine subspaces. This property yields linear convergence and incites embedding of circumcenters within classical reflection and projection based methods for more general feasibility problems.

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arXiv2017

Circumcentering the Douglas--Rachford method

We introduce and study a geometric modification of the Douglas-Rach\-ford method called the Circumcentered-Douglas-Rachford method. This method iterates by taking the intersection of bisectors of reflection steps for solving certain classes of feasibility problems. The convergence analysis is established for best approximation problems involving two (affine) subspaces and both our theoretical and numerical results compare favorably to the original Douglas-Rachford method. Under suitable conditions, it is shown that the linear rate of convergence of the Circumcentered-Douglas-Rachford method is at least the cosine of the Friedrichs angle between the (affine) subspaces, which is known to be the sharp rate for the Douglas-Rachford method. We also present a preliminary discussion on the Circumcentered-Douglas-Rachford method applied to the many set case and to examples featuring non-affine convex sets.

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