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#ifndef NOX_MERITFUNCTION_SUMOFSQUARES_H
#define NOX_MERITFUNCTION_SUMOFSQUARES_H
#include "NOX_Common.H" // for std::ostream
#include "Teuchos_RCP.hpp"
#include "NOX_Utils.H"
#include "NOX_MeritFunction_Generic.H"
#include "NOX_LineSearch_Utils_Slope.H"
namespace NOX {
namespace MeritFunction {
//! Sum of squares merit function.
/*!
A basic merit function used in many nonlinear equation solvers:
\f[
f = \frac{1}{2} \| F(x) \| ^2
\f]
Where the norm is the 2-Norm using the NOX::Abstract::Vector's
inner product.
This is the default merit function used in nox.
This merit function is taken from: J. E. Dennis Jr. and Robert B.
Schnabel, "Numerical Methods for Unconstrained Optimization and
Nonlinear Equations," Prentice Hall, 1983
*/
class SumOfSquares : public virtual NOX::MeritFunction::Generic {
public:
//! Constructor.
SumOfSquares(const Teuchos::RCP<NOX::Utils>& u);
//! Destructor.
virtual ~SumOfSquares();
//! Computes the merit function, \f$ f(x) = \frac{1}{2}\| F(x) \|^2 \f$.
virtual double computef(const NOX::Abstract::Group& grp) const;
//! Computes the gradient, \f$ g = \nabla f(x) = J(x)^T F(x) \f$.
virtual void computeGradient(const NOX::Abstract::Group& group,
NOX::Abstract::Vector& result) const;
//! Computes the slope, \f$ s(x,d) = d^T \nabla f(x) = d^T J(x)^T F(x) \f$.
/*! If the Jacobian is not computed in the \c grp object, then the
slope can be approximated using directional derivatives. More
information can be found in the method computeSlopeWithoutJac.
*/
virtual double computeSlope(const NOX::Abstract::Vector& dir,
const NOX::Abstract::Group& grp) const;
//! Computes the quadratic model, \f$ m(x,d) = f(x) + \nabla f(x)^T d + d^T \nabla^2 f(x) d \f$.
/*!
We approximate \f$ \nabla^2f(x) \approx J^TJ \f$:
\f[
m(d) = f(x) + (J(x)^T F)^T d + \frac{1}{2} d^T B d
\f]
*/
virtual double computeQuadraticModel(const NOX::Abstract::Vector& dir,
const NOX::Abstract::Group& grp) const;
//! Computes the vector in the steepest descent direction that minimizes, the quadratic model.
/*!
Computes the vector \c result:
\f[
result = \frac{\nabla f^T \nabla f}{\nabla f^T B \nabla f} \nabla f = -\frac{(J^T F)^T (J^T F)}{(J J^T F)^T (J J^T F)} J^T F
\f]
*/
virtual void computeQuadraticMinimizer(const NOX::Abstract::Group& grp,
NOX::Abstract::Vector& result) const;
virtual const std::string& name() const;
private:
//! Disallow default ctor.
SumOfSquares() {};
//! This is a variant of the computeSlope() method above optimized to work with out having to compute an explicit Jacobian.
/*!
Calculates and returns
\f[
\zeta = d^T \nabla f(x) = d^TJ^TF
\f]
Here \f$d\f$ represents the input parameter \c dir \f$\nabla
f(x)\f$ is the gradient associated with the given group (for
nonlinear solves this equates to \f$ J^TF \f$ where \f$ J \f$ is
the Jacobian and \f$ F \f$ is the original nonlinear function).
We can rewrite this equation as:
\f[ d^TJ^TF = F^TJd \f]
which allows us to use directional derivatives to estimate \f$ J^TF \f$:
\f[ F^TJd = F^T \frac{F(x + \eta d) - F(x)}{\eta} \f]
This may allow for faster computations of the slope if the
Jacobian is expensive to evaluate.
where \f$\eta\f$ is a scalar perturbation calculated by:
\f[ \eta = \lambda * (\lambda + \frac{\| x\|}{\| d\|} ) \f]
\f$ \lambda \f$ is a constant fixed at 1.0e-6.
*/
virtual double
computeSlopeWithoutJacobian(const NOX::Abstract::Vector& dir,
const NOX::Abstract::Group& grp) const;
//! This is a variant of the computeSlope() method above that works when the Jacobian operator is available but doesn't support transpose
/*!
Calculates and returns
\f[
\zeta = d^T \nabla f(x) = (Jd)^TF
\f]
Variables are as defined above. This alternative form for the slope
is provided to support Jacobian operators which do not
support the transpose operation.
*/
virtual double
computeSlopeWithoutJacobianTranspose(const NOX::Abstract::Vector& dir,
const NOX::Abstract::Group& grp) const;
private:
//!Printing utilities.
Teuchos::RCP<Utils> utils;
//! Temporary vector for computations.
mutable Teuchos::RCP<NOX::Abstract::Vector> tmpVecPtr;
//! Temporary vector for computations.
/*! Only allocated if the method computeJacobianWithOutJac is called. */
mutable Teuchos::RCP<NOX::Abstract::Group> tmpGrpPtr;
//! Name of this function
std::string meritFunctionName;
};
} // namespace MeritFunction
} // namespace NOX
#endif
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