#include <NOX_LineSearch_NonlinearCG.H>

Inheritance diagram for NOX::LineSearch::NonlinearCG:

Collaboration diagram for NOX::LineSearch::NonlinearCG:

Public Member Functions
	NonlinearCG (const Teuchos::RCP< NOX::GlobalData > &gd, Teuchos::ParameterList &params)
	Constructor.

	~NonlinearCG ()
	Destructor.

bool	reset (const Teuchos::RCP< NOX::GlobalData > &gd, Teuchos::ParameterList &params)

bool	compute (NOX::Abstract::Group &newgrp, double &step, const NOX::Abstract::Vector &dir, const NOX::Solver::Generic &s)
	Perform a line search.

Public Member Functions inherited from NOX::LineSearch::Generic
	Generic ()
	Default constructor.

virtual	~Generic ()
	Destructor.

Detailed Description

Use NonlinearCG linesearch.

This is a simple linesearch intended to be used with NOX::Direction::NonlinearCG, which provides search direction , in computing an update to the current solution vector $x_{new} = x_{old} + \lambda d$ . It is designed to compute a step length $\lambda$ consistent with the exact linesearch of Linear CG for linear problems, and it avoids use of matrices by employing a directional derivative (details below). The step length, $\lambda$ is computed from a single evaluation of,

$\lambda = - \frac{F(x_{old})^T d}{d^T J(x_{old})d}$

where is the n x n Jacobian matrix. Explicit construction of is avoided by performing the product using a directional derivative (cf NOX::Epetra::MatrixFree):

$J(x_{old})d \approx \frac{F(x_{old} + \delta d) - F(x_{old})}{\delta}$

where $\delta = 10^{-6} (10^{-6} + ||x_{old}|| / ||d||)$ .

Derivation / Theory:

This linesearch is derived by attempting to achieve in a single step, the following minimization:

$\min_\lambda \phi(\lambda)\equiv\phi (x_{old}+ \lambda d)$

where $\phi$ is a merit function chosen (but never explicitly given) so that an equivalence to Linear CG holds, ie $\nabla\phi(x) \leftrightarrow F(x)$ . The minimization above can now be cast as an equation:

$\phi ' (\lambda) = \nabla\phi (x_{old}+ \lambda d)^T d = F(x_{old}+ \lambda d)^T d = 0~~.$

An approximate solution to this equation can be obtained from a second-order expansion of

$\phi(\lambda)$

,

$\phi(\lambda)\approx\phi (0) + \phi ' (0)\lambda + \phi '' (0) \frac{\lambda^2}{2}$

from which it immediately follows

$\%lambda_{min} \approx - \frac{\phi ' (0)}{\phi '' (0)} = - \frac{F(x_{old})^T d}{d^T J(x_{old})d}$

Input Parameters

The NonlinearCG linesearch is selected using:

"Line Search"

"Method" = "%NonlinearCG" [required]

Currently, no adjustable parameters exist for this linesarch.

References

linesearch is adapted from ideas presented in Section 14.2 of:

Jonathan Richard Shewchuk, "An Introduction to the Conjugate Gradient Method Without the Agonizing Pain</A>," 1994. Though presented within the context of nonlinear optimization, the connection to solving nonlinear equation systems is made via the equivalence $f'(x) \leftrightarrow F(x)$ . \author Russ Hooper, Org. 9233, Sandia National Labs

Member Function Documentation

◆ compute()

bool NOX::LineSearch::NonlinearCG::compute	(	NOX::Abstract::Group &	grp,
		double &	step,
		const NOX::Abstract::Vector &	dir,
		const NOX::Solver::Generic &	s
	)

virtual

Perform a line search.

On input:

Parameters

grp	The initial solution vector, $x_{\rm old}$ .
dir	A vector of directions to be used in the line search, .
s	The nonlinear solver.

On output:

Parameters

step	The distance the direction was scaled, $\lambda$ .
grp	The `grp` is updated with a new solution, $x_{\rm new}$ , resulting from the linesearch. Normally, for a single direction line search, this is computed as:

$x_{\rm new} = x_{\rm old} + \lambda d.$

Ideally, $\|F(x_{\rm new})\| < \|F(x_{\rm old})\|$ (e.g the final direction is a descent direction).

Note that the dir object is a std::vector. For typical line searches as described in the above equation, this vector is of size one. We have used a std::vector to allow for special cases of multi-directional line searches such as the Bader/Schnabel curvillinear line search.

Return value is true for a successful line search computation.

Implements NOX::LineSearch::Generic.

References NOX::Abstract::Group::computeF(), NOX::Abstract::Group::computeX(), NOX::Utils::fill(), NOX::Abstract::Group::getF(), NOX::Solver::Generic::getPreviousSolutionGroup(), NOX::Utils::InnerIteration, and NOX::Abstract::Vector::innerProduct().

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

NOX_LineSearch_NonlinearCG.H
NOX_LineSearch_NonlinearCG.C

Public Member Functions

Detailed Description

Member Function Documentation

◆ compute()