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

TL;DR

This paper introduces a novel, globally convergent, parallel solution strategy for large-scale non-linear programming problems, combining local asynchronous solutions with a global recombination step, and analyzes its convergence and numerical performance.

Contribution

It presents a new globally convergent parallel method based on ASPIN with controlled non-linear preconditioning and trust-region strategies.

Findings

Ensures global convergence for large-scale non-linear problems.

Demonstrates effectiveness on 3D non-linear elasticity problems.

Achieves parallel efficiency on massively parallel computers.

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.