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In this paper we aim to minimize the sum of two nonsmooth (possibly alsononconvex) functions in separate variables connected by a smooth couplingfunction. To tackle this problem we chose a continuous forward-backwardapproach and introduce a dynamical system which is formulated by means of thepartial gradients of the smooth coupling function and the proximal pointoperator of the two nonsmooth functions. Moreover, we consider variable ratesof implicitness of the resulting system. We discuss the existence anduniqueness of a solution and carry out the asymptotic analysis of itsconvergence behaviour to a critical point of the optimization problem, when aregularization of the objective function fulfills the Kurdyka-Lojasiewiczproperty. We further provide convergence rates for the solution trajectory interms of the Lojasiewicz exponent. We conclude this work with numericalsimulations which confirm and validate the analytical results.

A forward-backward dynamical approach for nonsmooth problems with block structure coupled by a smooth function