It solves, over any real Hilbert space ℋ, monotone inclusion problems of the form

find *x* ∈ zer ( *B* + ∑_{i=1}^{n} *A _{i}* ) := {

- all considered operators are maximal monotone,
*B*is cocoercive,- all the
*A*'s are simple, in the sense that we can compute easily their resolvant:_{i}*J*= (Id +_{Ai}*A*)_{i}^{-1}.

In particular, it enables minimization over ℋ of convex problems of the form

min_{x∈ℋ} *F*(*x*) := *f*(*x*) + ∑_{i=1}^{n} *g _{i}*(

- all considered functions belong to Γ
_{0}(ℋ), the class of lower semi-continuous, proper, convex functions from ℋ to ]−∞,+∞], *f*is differentiable with Lipschitz-continuous gradient,- all the
*g*'s are simple in the sense that we can compute easily their Moreau's proximity operator: prox_{i}_{gi}(*x*) = argmin_{y∈ℋ}

||1 2 *x*−*y*||^{2}+*g*(_{i}*y*) .

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Some supplementary results: deblurring task - inpainting task - composite task - composite task with TV regularization