Hybrid Krylov methods for nonlinear systems of equations
SIAM Journal on Scientific and Statistical Computing
Mathematical Programming: Series A and B
Convergence analysis of some algorithms for solving nonsmooth equations
Mathematics of Operations Research
Monotone multigrid methods for elliptic variational inequalities I
Numerische Mathematik
Choosing the forcing terms in an inexact Newton method
SIAM Journal on Scientific Computing - Special issue on iterative methods in numerical linear algebra; selected papers from the Colorado conference
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
A semismooth equation approach to the solution of nonlinear complementarity problems
Mathematical Programming: Series A and B
A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems
Mathematical Programming: Series A and B - Special issue on computational nonsmooth optimization
A New Nonsmooth Equations Approach to Nonlinear Complementarity Problems
SIAM Journal on Control and Optimization
Parallel Newton--Krylov--Schwarz Algorithms for the Transonic Full Potential Equation
SIAM Journal on Scientific Computing
Penalty methods for American options with stochastic volatility
Journal of Computational and Applied Mathematics
A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems
SIAM Journal on Scientific Computing
Quadratic Convergence for Valuing American Options Using a Penalty Method
SIAM Journal on Scientific Computing
An Algebraic Convergence Theory for Restricted Additive Schwarz Methods Using Weighted Max Norms
SIAM Journal on Numerical Analysis
Convergence Theory of Restricted Multiplicative Schwarz Methods
SIAM Journal on Numerical Analysis
The Primal-Dual Active Set Strategy as a Semismooth Newton Method
SIAM Journal on Optimization
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
SIAM Journal on Numerical Analysis
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
SIAM Journal on Scientific Computing
An algorithm for the fast solution of symmetric linear complementarity problems
Numerische Mathematik
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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 Newton-Krylov-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.