We propose a stochastic Forward Douglas-Rachford Splitting framework for finding a zero point of the sum of three maximally monotone operators in real separable Hilbert space, where one of them is cocoercive. We first prove the weak almost sure convergence of the proposed method. We then characterize the rate of convergence in expectation in the case of strongly monotone operators. Finally, we ...

In the hierarchy of freeness construction, free independence was reduced to tensor independence in the weak sense of convergence of moments. In this paper we show how to reduce free independence to tensor independence in the strong sense. We construct a suitable unital *-algebra of closed operators ‘affiliated’ with a given unital *-algebra and call the associated closure ‘monotone’. Then we pr...

An iterative scheme for solving ill-posed nonlinear operator equations with monotone operators is introduced and studied in this paper. A discrete version of the Dynamical Systems Method (DSM) algorithm for stable solution of ill-posed operator equations with monotone operators is proposed and its convergence is proved. A discrepancy principle is proposed and justified. A priori and a posterior...

In this paper we prove a generalization of the iterated homogenization theorem for monotone operators, proved by Lions et al. in [20] and [21]. Our results enable us to homogenize more realistic models of multiscale structures.

In this paper, a common solution problem is investigated based on a Bregman projection. Strong convergence of the monotone projection algorithm for monotone operators and bifunctions is obtained in a reflexive Banach space.

The subject is the overview of the use of quasi-entropy in finite dimensional spaces. Matrix monotone functions and relative modular operators are used. The origin is the relative entropy and the f -divergence, monotone metrics, covariance and the χ2 divergence are the most important particular cases. The extension of the monotone metric to two variables is a new concept.

lorentz characterized the almost convergence through the concept of uniform convergence of de lavallée-poussin mean. in this paper, we generalize the notion of almost convergence by using the concept ofinvariant mean and the generalized de la vallée-poussin mean. we determine the bounded linear operators forthe generalized σ-conservative, σ-regular and σ-coercive matrices.

We propose a splitting algorithm for solving a coupled system of primal-dual monotone inclusions in real Hilbert spaces. The proposed algorithm has a structure identical to that of the forward-backward algorithm with variable metric. The operators involved in the problem formulation are used separately in the sense that single-valued operators are used individually and approximately in the forw...

Recent characterizations of various types of differentiable generalized monotone maps by Karamardian– Schaible–Crouzeix and their strengthened versions by Crouzeix–Ferland are extended to the nonsmooth case. For nondifferentiable locally Lipschitz maps necessary and/or sufficient conditions for quasimonotonicity, pseudomonotonicity and strict/ strong pseudomonotonicity are derived. To accomplis...

We present a stability result for a wide class doubly nonlinear equations, featuring general maximal monotone operators, and (possibly) nonconvex and nonsmooth energy functionals. The limit analysis resides on the reformulation of the differential evolution as a scalar energy-conservation equation with the aid of the so-called Fitzpatrick theory for the representation of monotone operators. In ...