Monotone+Skew Decomposition of Inclusion Problems: Algorithms and Applications

Abstract

The principle underlying this talk is the basic observation that the problem of simultaneously solving a large class of composite monotone inclusions and their duals can be reduced to that of finding a zero of the sum of a maximally monotone operator and a linear skew-adjoint operator. An algorithmic framework is developed for solving this generic problem in a Hilbert space setting. New primal-dual splitting algorithms are derived from this framework for inclusions involving composite monotone operators, and convergence results are established. These algorithms draw their simplicity and efficacy from the fact that they operate in a fully decomposed fashion in the sense that the monotone operators and the linear transformations involved are activated separately at each iteration. Applications to composite variational problems are demonstrated. In particular, new domain decomposition methods for partial differential equations will be presented.

Parts of this talk result from joint work with H. Attouch and L. Briceno-Arias.

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