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In this paper we consider from two different aspects the proximal alternating
direction method of multipliers (ADMM) in Hilbert spaces. We first consider the
application of the proximal ADMM to solve well-posed linearly constrained
two-block separable convex minimization problems in Hilbert spaces and obtain
new and improved non-ergodic convergence rate results, including linear and
sublinear rates under certain regularity conditions. We next consider proximal
ADMM as a regularization method for solving linear ill-posed inverse problems
in Hilbert spaces. When the data is corrupted by additive noise, we establish,
under a benchmark source condition, a convergence rate result in terms of the
noise level when the number of iteration is properly chosen.
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