The goal of this thesis is to develop new splitting techniques for set-valued operators to solve structured monotone inclusion problems in Hilbert spaces. Duality plays a central role in this work. It allows us to obtain decompositions which would not be available through a purely primal approach. We develop several fixed and variable metric algorithms in a unified framework, and show in particular that many existing methods are special cases of the forward-backward method formulated in a suitable product space. The proposed methods are applied to variational inequalities, minimization problems, inverse problems, signal processing problems, feasibility problems, and best approximation problems. Next, we introduce the notion of a variable metric quasi-Fejér sequence and analyze its asymptotic properties. These results allow us to obtain extensions of splitting schemes to problems in which the metric varies at each iteration.
