  In this work, we develop a systematic framework for computing the resolvent
of the sum of two or more monotone operators which only activates each operator
in the sum individually. The key tool in the development of this framework is
the notion of the "strengthening" of a set-valued operator, which can be viewed
as a type of regularisation that preserves computational tractability. After
deriving a number of iterative schemes through this framework, we demonstrate
their application to best approximation problems, image denoising and elliptic
PDEs.
