Inexact Newton Utilities More...

#include <NOX_Direction_Utils_InexactNewton.H>

Public Member Functions
	InexactNewton ()
	Constructor.

	InexactNewton (const Teuchos::RCP< NOX::GlobalData > &gd, Teuchos::ParameterList &directionSublist)
	Constructor.

virtual	~InexactNewton ()
	Destructor.

virtual bool	reset (const Teuchos::RCP< NOX::GlobalData > &gd, Teuchos::ParameterList &directionSublist)
	Reset the utilities.

virtual double	computeForcingTerm (const NOX::Abstract::Group &soln, const NOX::Abstract::Group &oldSoln, int niter, const NOX::Solver::Generic &solver, double eta_last=-1.0)

Detailed Description

Inexact Newton Utilities

If we use an iterative linear solver for a Newton-based solve, then this is called an inexact Newton method. The tolerance used to terminate the linear solve is called the forcing term. The forcing term may be constant, or it may be adjustable. In either case, at iteration $k$ we require,

$\frac{\|J_k d_k - (-F_k)\|}{\|F_k\|} \leq \eta_k.$

Here $\eta_k$ is the forcing term for iteration $k$ .

Note: This solution tolerance is to be enforced by the user's implementation of NOX::Abstract::Group::computeNewton; it is passed in as the "Tolerance" in the parameter list for that function.

Adjustable forcing terms were introduced by Eisenstat and Walker (1982); here they are implemented as described in Pernice and Walker (1998). We have two choices for adjustable forcing terms:

Type 1

$\eta_k = \left\vert \frac{\| F_k \| - \|J_{k-1} d_{k-1} - (-F_{k-1}) \| } {\|F_{k-1}\|} \right\vert$

With the following safeguards imposed:

$\max\{\eta_{k-1}^{\frac{1 + \sqrt{5}}{2}}, \eta_{\min} \} \leq \eta_k \leq \eta_{\max}$
Type 2

$\eta_k = \gamma \left( \frac{\|F_k\|}{\|F_{k-1}\|} \right)^\alpha$

With the following safeguards imposed:

$\max\{\gamma \eta_{k-1}^{\alpha}, \eta_{\min} \} \leq \eta_k \leq \eta_{\max}$

Parameters

"Forcing Term Method" - Method to compute the forcing term, i.e., the tolerance for the linear solver. Choices are:
- "Constant" [default]
- "Type 1"
- "Type 2"
"Forcing Term Initial Tolerance" - $\eta_0$ (initial linear solver tolerance). Defaults to 0.1.
"Forcing Term Minimum Tolerance" - $\eta_{\min}$ . Defaults to 1.0e-6.
"Forcing Term Maximum Tolerance" - $\eta_{\max}$ . Defaults to 0.01.
"Forcing Term Alpha" - $\alpha$ (used only by "Type 2"). Defaults to 1.5.
"Forcing Term Gamma" - $\gamma$ (used only by "Type 2"). Defaults to 0.9.
"Forcing Term User Defined Norm" (NOX::Parameter::UserNorm derived object) - If using a Type 1 or Type 2 adjustable forcing term, the norms used to calculate $\eta_k$ should be based on the same norm used in the convergence test of the linear solver. Essentially this means that the norm must account for LEFT scaling of any kind that is applied to the linear system during a solve. If set, the computation of $\eta_k$ will be done using a used defined function that is passed in through a NOX::Parameter::Arbitrary derived object. It will take the arbitrary object and cast it to a NOX::Parameter::UserNorm object and use it for norm computations. If this parameter is not set, this method uses the L-2 Norm for any norm computations of $\eta_k$ .
"Set Tolerance in Parameter List" When calling computeForcingTerm() the value of the forcing term will be set in the parmaeter list pointed to by the InexactNewton object. It will be set under paramsPtr->sublist(<directionMethod>).sublist("Linear Solver").set("Tolerance", eta_k). Defaults to true.

Note: When using a forcing term, it's critically important the the residual of the original system is used in the comparison. This can be an issue if scaling or left preconditioning is applied to the linear system.

References

Michael Pernice and Homer F. Walker, NITSOL: A Newton Iterative Solver for Nonlinear Systems, SISC 19(Jan 1998):302-318.
S. C. Eisenstat and H. F. Walker, Globally convergent inexact Newton methods, SINUM 19(1982):400-408

Member Function Documentation

double NOX::Direction::Utils::InexactNewton::computeForcingTerm	(	const NOX::Abstract::Group &	soln,
		const NOX::Abstract::Group &	oldSoln,
		int	niter,
		const NOX::Solver::Generic &	solver,
		double	eta_last = `-1.0`
	)

virtual

Called each iteration to reset the forcing term (ie, the convergence tolerance for the linear solver).

if the user supplied eta_last then it will use this value for eta_km1 instead of looking for it in the "Linear Solver" sublist.

References NOX::Abstract::Group::applyJacobian(), NOX::Abstract::Vector::clone(), NOX::Utils::Details, NOX::Abstract::Group::getF(), NOX::Abstract::Group::getNormF(), NOX::Abstract::Group::getX(), Teuchos::is_null(), NOX::Abstract::Group::isJacobian(), NOX::ShapeCopy, and NOX::Abstract::Vector::update().

The documentation for this class was generated from the following files:

NOX_Direction_Utils_InexactNewton.H
NOX_Direction_Utils_InexactNewton.C

Public Member Functions

Detailed Description

Member Function Documentation