Inexact Newton Methods with Restricted Additive Schwarz Based Nonlinear Elimination for Problems with High Local Nonlinearity

@article{Cai2011InexactNM,
  title={Inexact Newton Methods with Restricted Additive Schwarz Based Nonlinear Elimination for Problems with High Local Nonlinearity},
  author={Xiao-Chuan Cai and Xuefeng Li},
  journal={SIAM J. Sci. Comput.},
  year={2011},
  volume={33},
  pages={746-762}
}
  • X. CaiXuefeng Li
  • Published 1 April 2011
  • Computer Science, Mathematics
  • SIAM J. Sci. Comput.
The classical inexact Newton algorithm is an efficient and popular technique for solving large sparse nonlinear systems of equations. When the nonlinearities in the system are well balanced, a near quadratic convergence is often observed; however, if some of the equations are much more nonlinear than the others in the system, the convergence is much slower. The slow convergence (or sometimes divergence) is often determined by the small number of equations in the system with the highest… 
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31 References

TWO-LEVEL NONLINEAR ELIMINATION BASED PRECONDITIONERS FOR INEXACT NEWTON METHODS WITH APPLICATION IN SHOCKED DUCT FLOW CALCULATION

This paper introduces a two-level nonlinear elimination algorithm, in which a subset of equations is first identified that prevents Newton from having the fast convergence and then iteratively eliminated them from the global nonlinear system of equations.

Nonlinearly Preconditioned Inexact Newton Algorithms

A nonlinear additive Schwarz-based parallel nonlinear preconditioner is proposed and studied and it is shown numerically that the new method converges well even for some difficult problems, such as high Reynolds number flows, where a traditional inexact Newton method fails.

A parallel nonlinear additive Schwarz preconditioned inexact Newton algorithm for incompressible Navier-Stokes equations

A Nonlinear Additive Schwarz Preconditioned Inexact Newton Method for Shocked Duct Flow

The nonlinear additive Schwarz algorithm is used as the preconditioner and the performance of PIN for a compressible shock tube problem, which is known to be a diÆcult test case for inexact Newton type algorithms.

Globally Convergent Inexact Newton Methods

The primary goal is to introduce and analyze new inexact Newton methods, but consideration is also given to “globally convergence” features designed to improve convergence from arbitrary starting points.

A class of parallel two-level nonlinear Schwarz preconditioned inexact Newton algorithms ☆

Nonlinearly Preconditioned Krylov Subspace Methods for Discrete Newton Algorithms

A nonlinear SSOR type preconditioning is derived which numerical experiments show to be as effective as the linear SSOR preconditionsing that uses the Jacobian explicitly.

Convergence Theory of Nonlinear Newton-Krylov Algorithms

Some convergence theory for nonlinear Krylov subspace methods is presented, to analyze these methods when they are combined with global strategies such as linesearch techniques and model trust region algorithms.

Domain Decomposition Methods for Monotone Nonlinear Elliptic Problems

This paper considers additive Schwarz algorithms used together with the classical inexact Newton methods for the numerical solution of some non linear strongly elliptic equations discretized by the nite element methods.

GLOBAL CONVERGENCE OF INEXACT NEWTON METHODS FOR TRANSONIC FLOW

In computational fluid dynamics, non-linear differential equations are essential to represent important effects such as shock waves in transonic flow. Discretized versions of these non-linear…