Parallel Two-Grid Semismooth Newton-Krylov-Schwarz Method for Nonlinear Complementarity Problems

Abstract

We develop scalable parallel domain decomposition algorithms for nonlinear complementarity problems including, for example, obstacle problems and free boundary value problems. Semismooth Newton is a popular approach for such problems, however, the method is not suitable for large scale calculations because the number of Newton iterations is not scalable with respect to the grid size; i.e., when the grid is refined, the number of Newton iterations often increases drastically. In this paper, we introduce a family of NewtonKrylov-Schwarz methods based on a smoothed grid sequencing method, a semismooth inexact Newton method, and a two-grid restricted overlapping Schwarz preconditioner. We show numerically that such an approach is totally scalable in the sense that the number of Newton iterations and the number of linear iterations are both nearly independent of the grid size and the number of processors. In addition, the method is not sensitive to the sharp discontinuity often associated with obstacle problems. We present numerical results for several large scale calculations obtained on machines with hundreds of processors.

Nonlinearly preconditioned semismooth Newton methods for variational inequality solution of two-phase flow in porous media

“…We briefly review the framework of the semismooth Newton algorithm for solving variational inequality problems [17,27,46,48,56], which serves as the basis of the proposed method. The semismooth algorithm is based on a reformulation of (13) as a semismooth system of equations satisfying a semismooth property using the NCP-function [28].…”

“…Given an initial guess X 0 ∈ R N and let X k be the current approximation at the k th Newton iteration. Then the class of semismooth Newton with backtracking (SNB) algorithms [3,46,56] consists of the following steps to find the next approximation X k+1 .…”

Nonlinear Preconditioning Techniques for Full-Space Lagrange--Newton Solution of PDE-Constrained Optimization Problems

“…Then, an inexact Newton method is employed to solve a large sparse nonlinear system of equations derived from the first-order optimality condition. The family of continuation methods, such as the grid-sequencing approach [2,34] or the parameter continuation approach [3,32], is quite robust for these difficult nonlinear problems but not efficient; see their applications for PDE-constrained optimization problems [7,48,49,51]. First, at each Newton iteration, a large sparse saddle point system needs to be solved, which is often highly ill-conditioned [5].…”

“…Newton-type methods enjoy fast convergence when the nonlinearity in the system is well-balanced; however, for some problems, such as the control of incompressible flows, even linear convergence is difficult to achieve and a long stagnation period often appears in the iteration history. The family of continuation methods, such as the grid-sequencing approach [2,34] or the parameter continuation approach [3,32], is quite robust for these difficult nonlinear problems but not efficient; see their applications for PDE-constrained optimization problems [7,48,49,51]. The system has nine field variables, and each field variable plays a different role in the nonlinearity of the system.…”