LOCA's Homotopy Algorithm. More...

#include <LOCA_Homotopy_DeflatedGroup.H>

Inheritance diagram for LOCA::Homotopy::DeflatedGroup:

Collaboration diagram for LOCA::Homotopy::DeflatedGroup:

Public Member Functions
	DeflatedGroup (const Teuchos::RCP< LOCA::GlobalData > &global_data, const Teuchos::RCP< Teuchos::ParameterList > &topParams, const Teuchos::RCP< Teuchos::ParameterList > &hParams, const Teuchos::RCP< LOCA::Homotopy::AbstractGroup > &grp, const Teuchos::RCP< const NOX::Abstract::Vector > &start_vec, const std::vector< Teuchos::RCP< const NOX::Abstract::Vector > > &prev_solns, const double identity_sign=1.0)
	Constructor. More...

	DeflatedGroup (const DeflatedGroup &source, NOX::CopyType type=NOX::DeepCopy)
	Copy constructor.

virtual	~DeflatedGroup ()
	Destructor.

double	getHomotopyParam () const
	Get homotopy parameter.

Implementation of NOX::Abstract::Group virtual methods
virtual NOX::Abstract::Group &	operator= (const NOX::Abstract::Group &source)
	Assignment operator.

virtual Teuchos::RCP < NOX::Abstract::Group >	clone (NOX::CopyType type=NOX::DeepCopy) const
	Clone function.

virtual void	setX (const NOX::Abstract::Vector &y)
	Set the solution vector to y.

virtual void	computeX (const NOX::Abstract::Group &g, const NOX::Abstract::Vector &d, double step)
	Compute and return solution vector, x, where this.x = grp.x + step * d.

virtual NOX::Abstract::Group::ReturnType	computeF ()
	Compute extended continuation equations.

virtual NOX::Abstract::Group::ReturnType	computeJacobian ()
	Compute extended continuation jacobian.

virtual NOX::Abstract::Group::ReturnType	computeGradient ()
	Gradient is not defined for this system.

virtual NOX::Abstract::Group::ReturnType	computeNewton (Teuchos::ParameterList &params)
	Compute Newton direction for extended continuation system.

virtual NOX::Abstract::Group::ReturnType	applyJacobian (const NOX::Abstract::Vector &input, NOX::Abstract::Vector &result) const
	Applies Jacobian for extended system.

virtual NOX::Abstract::Group::ReturnType	applyJacobianTranspose (const NOX::Abstract::Vector &input, NOX::Abstract::Vector &result) const
	Jacobian transpose not defined for this system.

virtual NOX::Abstract::Group::ReturnType	applyJacobianInverse (Teuchos::ParameterList &params, const NOX::Abstract::Vector &input, NOX::Abstract::Vector &result) const
	Applies Jacobian inverse for extended system.

virtual NOX::Abstract::Group::ReturnType	applyJacobianMultiVector (const NOX::Abstract::MultiVector &input, NOX::Abstract::MultiVector &result) const
	Applies Jacobian for extended system.

virtual NOX::Abstract::Group::ReturnType	applyJacobianTransposeMultiVector (const NOX::Abstract::MultiVector &input, NOX::Abstract::MultiVector &result) const
	Jacobian transpose not defined for this system.

virtual NOX::Abstract::Group::ReturnType	applyJacobianInverseMultiVector (Teuchos::ParameterList &params, const NOX::Abstract::MultiVector &input, NOX::Abstract::MultiVector &result) const
	Applies Jacobian inverse for extended system.

virtual bool	isF () const
	Return `true` if extended residual is valid.

virtual bool	isJacobian () const
	Return `true` if the extended Jacobian is valid.

virtual bool	isGradient () const
	Always returns false.

virtual bool	isNewton () const
	Return `true` if the extended Newton direction is valid.

virtual const NOX::Abstract::Vector &	getX () const
	Return extended solution vector.

virtual const NOX::Abstract::Vector &	getF () const
	Return extended residual.

virtual double	getNormF () const
	Return 2-norm of extended residual.

virtual const NOX::Abstract::Vector &	getGradient () const
	Gradient is never valid.

virtual const NOX::Abstract::Vector &	getNewton () const
	Return extended Newton direction.

virtual Teuchos::RCP< const NOX::Abstract::Vector >	getXPtr () const
	Return RCP to extended solution vector.

virtual Teuchos::RCP< const NOX::Abstract::Vector >	getFPtr () const
	Return RCP to extended residual.

virtual Teuchos::RCP< const NOX::Abstract::Vector >	getGradientPtr () const
	Gradient is never valid.

virtual Teuchos::RCP< const NOX::Abstract::Vector >	getNewtonPtr () const
	Return RCP to extended Newton direction.

virtual double	getNormNewtonSolveResidual () const
	Returns 2-norm of extended Newton solve residual.

Implementation of LOCA::Extended::MultiAbstractGroup
virtual methods
virtual Teuchos::RCP< const LOCA::MultiContinuation::AbstractGroup >	getUnderlyingGroup () const
	Return underlying group.

virtual Teuchos::RCP < LOCA::MultiContinuation::AbstractGroup >	getUnderlyingGroup ()
	Return underlying group.

Implementation of LOCA::MultiContinuation::AbstractGroup
virtual methods
virtual void	copy (const NOX::Abstract::Group &source)
	Assignment operator.

virtual void	setParamsMulti (const std::vector< int > &paramIDs, const NOX::Abstract::MultiVector::DenseMatrix &vals)
	Set parameters indexed by (integer) paramIDs.

virtual void	setParams (const ParameterVector &p)
	Set the parameter vector in the group to p (pVector = p).

virtual void	setParam (int paramID, double val)
	Set parameter indexed by (integer) paramID.

virtual void	setParam (std::string paramID, double val)
	Set parameter indexed by (std::string) paramID.

virtual const ParameterVector &	getParams () const
	Return a const reference to the ParameterVector owned by the group.

virtual double	getParam (int paramID) const
	Return copy of parameter indexed by (integer) paramID.

virtual double	getParam (std::string paramID) const
	Return copy of parameter indexed by (std::string) paramID.

virtual NOX::Abstract::Group::ReturnType	computeDfDpMulti (const std::vector< int > &paramIDs, NOX::Abstract::MultiVector &dfdp, bool isValidF)

virtual void	preProcessContinuationStep (LOCA::Abstract::Iterator::StepStatus stepStatus)
	Perform any preprocessing before a continuation step starts. More...

virtual void	postProcessContinuationStep (LOCA::Abstract::Iterator::StepStatus stepStatus)
	Perform any postprocessing after a continuation step finishes. More...

virtual void	projectToDraw (const NOX::Abstract::Vector &x, double *px) const
	Projects solution to a few scalars for multiparameter continuation.

virtual int	projectToDrawDimension () const
	Returns the dimension of the project to draw array.

virtual double	computeScaledDotProduct (const NOX::Abstract::Vector &a, const NOX::Abstract::Vector &b) const
	Compute a scaled dot product.

virtual void	printSolution (const double conParam) const
	Function to print out solution and parameter after successful step.

virtual void	printSolution (const NOX::Abstract::Vector &x, const double conParam) const
	Function to print out a vector and parameter after successful step.

virtual void	scaleVector (NOX::Abstract::Vector &x) const
	Scales a vector using scaling vector.

Implementation of
LOCA::BorderedSystem::AbstractGroup virtual methods
virtual int	getBorderedWidth () const
	Return the total width of the bordered rows/columns.

virtual Teuchos::RCP< const NOX::Abstract::Group >	getUnborderedGroup () const
	Get bottom-level unbordered group.

virtual bool	isCombinedAZero () const
	Indicates whether combined A block is zero.

virtual bool	isCombinedBZero () const
	Indicates whether combined B block is zero.

virtual bool	isCombinedCZero () const
	Indicates whether combined C block is zero.

virtual void	extractSolutionComponent (const NOX::Abstract::MultiVector &v, NOX::Abstract::MultiVector &v_x) const

virtual void	extractParameterComponent (bool use_transpose, const NOX::Abstract::MultiVector &v, NOX::Abstract::MultiVector::DenseMatrix &v_p) const

virtual void	loadNestedComponents (const NOX::Abstract::MultiVector &v_x, const NOX::Abstract::MultiVector::DenseMatrix &v_p, NOX::Abstract::MultiVector &v) const

virtual void	fillA (NOX::Abstract::MultiVector &A) const
	Fill the combined A block as described above.

virtual void	fillB (NOX::Abstract::MultiVector &B) const
	Fill the combined B block as described above.

virtual void	fillC (NOX::Abstract::MultiVector::DenseMatrix &C) const
	Fill the combined C block as described above.

Public Member Functions inherited from LOCA::Extended::MultiAbstractGroup
	MultiAbstractGroup ()
	Default constructor.

virtual	~MultiAbstractGroup ()
	Destructor.

virtual Teuchos::RCP < NOX::Abstract::Group >	getNestedGroup ()
	Override from NOX::Abstract::Group base class. Calls getUnderlyingGroup() from this class.

virtual Teuchos::RCP< const NOX::Abstract::Group >	getNestedGroup () const
	Override from NOX::Abstract::Group base class. Calls getUnderlyingGroup() from this class.

virtual Teuchos::RCP< const LOCA::MultiContinuation::AbstractGroup >	getBaseLevelUnderlyingGroup () const
	Return base-level underlying group. More...

virtual Teuchos::RCP < LOCA::MultiContinuation::AbstractGroup >	getBaseLevelUnderlyingGroup ()
	Return base-level underlying group. More...

Public Member Functions inherited from NOX::Abstract::Group
	Group ()
	Constructor. More...

virtual	~Group ()
	Destructor.

virtual NOX::Abstract::Group::ReturnType	applyRightPreconditioning (bool useTranspose, Teuchos::ParameterList &params, const NOX::Abstract::Vector &input, NOX::Abstract::Vector &result) const
	Apply right preconditiong to the given input vector. More...

virtual NOX::Abstract::Group::ReturnType	applyRightPreconditioningMultiVector (bool useTranspose, Teuchos::ParameterList &params, const NOX::Abstract::MultiVector &input, NOX::Abstract::MultiVector &result) const
	applyRightPreconditioning for multiple right-hand sides More...

virtual const NOX::Abstract::Vector &	getScaledX () const

virtual void	logLastLinearSolveStats (NOX::SolverStats &stats) const
	Adds statistics from last linear solve to the SovlerStats object.

virtual NOX::Abstract::Group::ReturnType	getNormLastLinearSolveResidual (double &residual) const
	Return the norm of the last linear solve residual as the result of either a call to computeNewton() or applyJacobianInverse(). More...

Public Member Functions inherited from LOCA::MultiContinuation::AbstractGroup
	AbstractGroup ()
	Default constructor.

virtual	~AbstractGroup ()
	Destructor.

Public Member Functions inherited from LOCA::BorderedSystem::AbstractGroup
	AbstractGroup ()
	Constructor.

virtual	~AbstractGroup ()
	Destructor.

Protected Member Functions
virtual void	resetIsValid ()
	Resets all isValid flags to false.

virtual void	setupViews ()
	Sets up multivector views.

void	setHomotopyParam (double param)
	Set homotopy parameter.

void	setStepperParameters (Teuchos::ParameterList &topParams)
	Creates and sets the "Stepper" parameter sublist.

Protected Attributes
Teuchos::RCP< LOCA::GlobalData >	globalData
	Pointer LOCA global data object.

Teuchos::RCP < LOCA::Parameter::SublistParser >	parsedParams
	Parsed top-level parameters.

Teuchos::RCP < Teuchos::ParameterList >	homotopyParams
	Pitchfork parameter list.

Teuchos::RCP < LOCA::Homotopy::AbstractGroup >	grpPtr
	Pointer to base group that defines .

Teuchos::RCP < LOCA::BorderedSystem::AbstractGroup >	bordered_grp
	Pointer to base group as a bordered group.

LOCA::MultiContinuation::ExtendedMultiVector	xMultiVec
	Stores the extended solution vector.

LOCA::MultiContinuation::ExtendedMultiVector	fMultiVec
	Stores the extended residual vector.

LOCA::MultiContinuation::ExtendedMultiVector	newtonMultiVec
	Stores the extended Newton vector.

LOCA::MultiContinuation::ExtendedMultiVector	gradientMultiVec
	Stores the extended gradient vector.

Teuchos::RCP < LOCA::MultiContinuation::ExtendedVector >	xVec
	Stores view of first column of xMultiVec.

Teuchos::RCP < LOCA::MultiContinuation::ExtendedVector >	fVec
	Stores view of first column of fMultiVec.

Teuchos::RCP < LOCA::MultiContinuation::ExtendedVector >	newtonVec
	Stores view of first column of newtonMultiVec.

Teuchos::RCP < LOCA::MultiContinuation::ExtendedVector >	gradientVec
	Stores view of first column of gradientMultiVec.

Teuchos::RCP< const NOX::Abstract::Vector >	startVec
	Stores the starting vector, $x^\ast$ .

double	identitySign

std::vector< Teuchos::RCP < const NOX::Abstract::Vector > >	solns
	Stores array of deflated solution vectors.

Teuchos::RCP < NOX::Abstract::Vector >	distVec
	Temporary vector for storing distance.

Teuchos::RCP < NOX::Abstract::MultiVector >	totalDistMultiVec
	Temporary vector for storing sum of distances.

Teuchos::RCP < NOX::Abstract::Vector >	totalDistVec
	Temporary vector for storing sum of distances.

Teuchos::RCP < NOX::Abstract::MultiVector >	underlyingF
	underlying group's F vector as a multivector

Teuchos::RCP < LOCA::BorderedSolver::JacobianOperator >	jacOp

Teuchos::RCP < LOCA::BorderedSolver::AbstractStrategy >	borderedSolver
	Stores bordered solver strategy.

Teuchos::RCP < NOX::Abstract::MultiVector::DenseMatrix >	minusOne
	Stores -1.

int	numSolns
	Number of previous solutions.

std::vector< double >	distances
	Array of distances from previous solutions.

double	distProd
	Product of distances.

std::vector< int >	index_f
	Stores indices for getting f part of fMultiVec.

LOCA::ParameterVector	paramVec
	Copy of the ParameterVector for the underlying grpPtr. More...

double	conParam
	Value of the homotopy continuation parameter. More...

int	conParamID
	Continuatioin parameter ID number from the ParameterVector.

const std::string	conParamLabel
	Contains the std::string used to identify the homotopy parameter in the ParameterVector object.

bool	augmentJacForHomotopyNotImplemented
	Tracks whether the LOCA::Homotopy::Group method augmentJacobianForHomotopy is implemented. If not, the augmentation is applied during the applyJacobian assuming a matrix-free implementation.

bool	isValidF
	Is residual vector valid.

bool	isValidJacobian
	Is Jacobian matrix valid.

bool	isValidNewton
	Is Newton vector valid.

bool	isValidGradient
	Is Gradient vector valid.

bool	isBordered
	Flag that indicates whether underlying group is a bordered group.

Additional Inherited Members
Public Types inherited from NOX::Abstract::Group
enum	ReturnType { Ok, NotDefined, BadDependency, NotConverged, Failed }
	The computation of, say, the Newton direction in computeNewton() may fail in many different ways, so we have included a variety of return codes to describe the failures. Of course, we also have a code for success. More...

Detailed Description

LOCA's Homotopy Algorithm.

The HomotopyGroup is a concrete implementation of the LOCA::Continuation::AbstractGroup that modifies the set of nonlinear equations to be solved to allow for Homotopy to be applied to the system. This object should be used in conjunction with the LOCA::Stepper object to drive the continuation. This algorithm solves a system of nonlinear equations supplied by the user ( $ F(x) $ ) through continuation. An artificial parameter $\lambda$ is used to control the continuation. The idea is to solve a simple equation starting at $\lambda$ = 0 and, using the solution from the previous step, solve systems of equations that gets progressively closer to the true system of interest ( at $\lambda$ = 1.0 we recover the original equations $ F(x) $ ). By constraining the definition of $g(x, \lambda)$ and using artificial parameter contiuation, the continuation branch should be free of multiplicity and bifurcation phenomena.

The modified system of equations, $g(x, \lambda)$ , supplied by the HomotopyGroup is defined as:

$g(x, \lambda) = \lambda F(x) + (1.0 - \lambda)(x - a)(S)$

where $ x$ is the solution vector, $\lambda$ is an artificial parameter, $ F(x) $ is the set of nonlinear equations the user supplies, $ g(x) $ is the corresponding set of homotopy equations that LOCA will solve, $ a $ is a random vector, and $ S $ is a scaling factor used to switch sign of the last term (typically valued 1.0 or -1.0).

This group requires the loca Stepper for continuation from $\lambda$ = 0.0 (a simple set of equations to solve) to $\lambda$ = 1.0 (the set of equations requested by the user, $ F(x) $ ). The Homotopy::Group will generate the Stepper parameter sublist in the parameter list that is passed in to the constructor. The user is free to modify this list (it sets default values) before passing it into the stepper object but should NOT change the starting and stopping values for the continuation parameter.

References:

ALGORITHM 652 HOMPACK: A Suite of Codes for Globally Convergent Homotopy Algorithms, Watson, L.T., Billups, S.C, and Morgan, A.P., ACM Transactions on Mathematical Software, Vol. 13, No. 3, September 1987, pp281-310.

Constructor & Destructor Documentation

LOCA::Homotopy::DeflatedGroup::DeflatedGroup	(	const Teuchos::RCP< LOCA::GlobalData > &	global_data,
		const Teuchos::RCP< Teuchos::ParameterList > &	topParams,
		const Teuchos::RCP< Teuchos::ParameterList > &	hParams,
		const Teuchos::RCP< LOCA::Homotopy::AbstractGroup > &	grp,
		const Teuchos::RCP< const NOX::Abstract::Vector > &	start_vec,
		const std::vector< Teuchos::RCP< const NOX::Abstract::Vector > > &	prev_solns,
		const double	identity_sign = `1.0`
	)

Constructor.

Parameters

global_data	[in] Global data object
topParams	[in] Parsed top-level parameter list.
hParams	[in] Homotopy parameters
grp	[in] Group representing .

References LOCA::ParameterVector::addParameter(), bordered_grp, borderedSolver, conParam, conParamID, conParamLabel, LOCA::ParameterVector::getIndex(), LOCA::Extended::Vector::getScalar(), LOCA::MultiContinuation::ExtendedVector::getXVec(), globalData, grpPtr, homotopyParams, isBordered, LOCA::GlobalData::locaFactory, minusOne, paramVec, parsedParams, LOCA::Parameter::SublistParser::parseSublists(), Teuchos::rcp(), setStepperParameters(), setupViews(), startVec, and xVec.

Member Function Documentation

NOX::Abstract::Group::ReturnType LOCA::Homotopy::DeflatedGroup::computeDfDpMulti	(	const std::vector< int > &	paramIDs,
		NOX::Abstract::MultiVector &	dfdp,
		bool	isValidF
	)

virtual

Compute $\partial F/\partial p$ for each parameter $ p$ indexed by paramIDs. The first column of dfdp holds F, which is valid if isValidF is true. Otherwise F must be computed.

Implements LOCA::MultiContinuation::AbstractGroup.

References LOCA::Extended::MultiVector::getScalars(), LOCA::MultiContinuation::ExtendedMultiVector::getXMultiVec(), NOX::Abstract::Group::Ok, Teuchos::SerialDenseMatrix< OrdinalType, ScalarType >::putScalar(), NOX::Abstract::MultiVector::scale(), NOX::Abstract::MultiVector::subView(), and NOX::Abstract::MultiVector::update().

void LOCA::Homotopy::DeflatedGroup::extractParameterComponent	(	bool	use_transpose,
		const NOX::Abstract::MultiVector &	v,
		NOX::Abstract::MultiVector::DenseMatrix &	v_p
	)		const

virtual

Given the vector v, extract the parameter components of all of the nested subvectors in v down to the solution component for the unbordered group.

Implements LOCA::BorderedSystem::AbstractGroup.

References Teuchos::SerialDenseMatrix< OrdinalType, ScalarType >::assign(), LOCA::Extended::MultiVector::getScalars(), LOCA::MultiContinuation::ExtendedMultiVector::getXMultiVec(), Teuchos::SerialDenseMatrix< OrdinalType, ScalarType >::numCols(), Teuchos::SerialDenseMatrix< OrdinalType, ScalarType >::numRows(), and Teuchos::View.

void LOCA::Homotopy::DeflatedGroup::extractSolutionComponent	(	const NOX::Abstract::MultiVector &	v,
		NOX::Abstract::MultiVector &	v_x
	)		const

virtual

Given the vector v, extract the underlying solution component corresponding to the unbordered group.

Implements LOCA::BorderedSystem::AbstractGroup.

References LOCA::MultiContinuation::ExtendedMultiVector::getXMultiVec().

void LOCA::Homotopy::DeflatedGroup::loadNestedComponents	(	const NOX::Abstract::MultiVector &	v_x,
		const NOX::Abstract::MultiVector::DenseMatrix &	v_p,
		NOX::Abstract::MultiVector &	v
	)		const