  The circumcentered Douglas--Rachford method (C--DRM), introduced by Behling,
Bello Cruz and Santos, is an acceleration of the well-known Douglas-Rachford
method (DRM) for finding the best approximation onto the intersection of
finitely many affine subspaces. Inspired by the C--DRM, we introduced the more
flexible circumcentered reflection method (CRM) and circumcentered isometry
method (CIM). The CIM essentially chooses the closest point to the solution
among all of the points in an associated affine hull as its iterate and is a
generalization of the CRM. The circumcentered--reflection method introduced by
Behling et al. to generalize the C--DRM is a special class of our CRM.
  We consider the CIM induced by a set of finitely many isometries for finding
the best approximation onto the intersection of fixed point sets of the
isometries which turns out to be an intersection of finitely many affine
subspaces. We extend our previous linear convergence results on CRMs in
finite-dimensional spaces from reflections to isometries. In order to better
accelerate the symmetric method of alternating projections (MAP), the
accelerated symmetric MAP first applies another operator to the initial point.
(Similarly, to accelerate the DRM, the C--DRM first applies another operator to
the initial point as well.) Motivated by these facts, we show results on the
linear convergence of CIMs in Hilbert spaces with first applying another
operator to the initial point. In particular, under some restrictions, our
results imply that some CRMs attain the known linear convergence rate of the
accelerated symmetric MAP in Hilbert spaces. We also exhibit a class of CRMs
converging to the best approximation in Hilbert spaces with a convergence rate
no worse than the sharp convergence rate of MAP. The fact that some CRMs attain
the linear convergence rate of MAP or accelerated symmetric MAP is entirely
new.
