On the Globalization of ASPIN Employing Trust-Region Control Strategies -- Convergence Analysis and Numerical Examples

Abstract: The parallel solution of large scale non-linear programming problems, which arise for example from the discretization of non-linear partial differential equations, is a highly demanding task. Here, a novel solution strategy is presented, which is inherently parallel and globally convergent. Each global non-linear iteration step consists of asynchronous solutions of local non-linear programming problems followed by a global recombination step. The recombination step, which is the solution of a quadratic programming problem, is designed in a way such that it ensures global convergence. As it turns out, the new strategy can be considered as a globalized additively preconditioned inexact Newton (ASPIN) method. However, in our approach the influence of ASPIN's non-linear preconditioner on the gradient is controlled in order to ensure a sufficient decrease condition. Two different control strategies are described and analyzed. Convergence to first-order critical points of our non-linear solution strategy is shown under standard trust-region assumptions. The strategy is investigated along difficult minimization problems arising from non-linear elasticity in 3D solved on a massively parallel computer with several thousand cores.