November 19 – Joëlle Ferzly: Semismooth and smoothing Newton methods for nonlinear systems with complementarity constraints: adaptivity and inexact resolution

Joëlle Ferzly: Thursday 19 November at 11:00 via zoom (meeting ID: 966 1837 4193 and code: zFG6Lb)

We are interested in nonlinear algebraic systems with complementarity constraints stemming from numerical discretizations of nonlinear complementarity problems. The particularity is that they are nondifferentiable, so that classical linearization schemes like the Newton method cannot be applied directly. To approximate the solution of such nonlinear systems, an iterative linearization algorithm like the semismooth Newton-min can be used. We consider smoothing methods, where the nondifferentiable nonlinearity is smoothed. In particular, a smoothing Newton algorithm based on the smoothed min or Fischer-Burmeister function, and a smoothing interior-point algorithm. The corresponding linear system is approximately solved using any iterative linear algebraic solver. We derive an a posteriori error estimate that allows to distinguish the smoothing, linearization, and algebraic error components. These ingredients are then used to formulate adaptive criteria for stopping the linear and nonlinear solver. This leads us to propose an adaptive algorithm ensuring important savings in terms of the number of cumulated algebraic iterations. We apply our analysis to the system of variational inequalities describing the contact between two membranes. We will show that the proposed algorithm, in combination with the GMRES algebraic solver, is promising in comparison with other methods.

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