Concrete implementation for creating an Epetra_RowMatrix Jacobian via finite differencing of the residual using coloring. More...

#include <NOX_Epetra_FiniteDifferenceColoring.H>

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List of all members.

Public Member Functions
	FiniteDifferenceColoring (Teuchos::ParameterList &printingParams, const Teuchos::RCP< Interface::Required > &i, const NOX::Epetra::Vector &initialGuess, const Teuchos::RCP< Epetra_MapColoring > &colorMap, const Teuchos::RCP< std::vector< Epetra_IntVector > > &columns, bool parallelColoring=false, bool distance1=false, double beta=1.0e-6, double alpha=1.0e-4)
	Constructor with output control.
	FiniteDifferenceColoring (Teuchos::ParameterList &printingParams, const Teuchos::RCP< Interface::Required > &i, const NOX::Epetra::Vector &initialGuess, const Teuchos::RCP< Epetra_CrsGraph > &rawGraph, const Teuchos::RCP< Epetra_MapColoring > &colorMap, const Teuchos::RCP< std::vector< Epetra_IntVector > > &columns, bool parallelColoring=false, bool distance1=false, double beta=1.0e-6, double alpha=1.0e-4)
	Constructor with output control.
virtual	~FiniteDifferenceColoring ()
	Pure virtual destructor.
virtual bool	computeJacobian (const Epetra_Vector &x, Epetra_Operator &Jac)
	Compute Jacobian given the specified input vector, x. Returns true if computation was successful.
virtual bool	computeJacobian (const Epetra_Vector &x)
	Compute Jacobian given the specified input vector, x. Returns true if computation was successful.
virtual void	createColorContainers ()
	Output the coloring map, index map and underlying matrix.
Protected Types
enum	ColoringType { NOX_SERIAL, NOX_PARALLEL }
Protected Attributes
ColoringType	coloringType
	Enum flag for type of coloring being used.
bool	distance1
	bool flag for specifying special case of distance-1 coloring
Teuchos::RCP< const Epetra_MapColoring >	colorMap
	Color Map created by external algorithm.
Teuchos::RCP< std::vector < Epetra_IntVector > >	columns
	vector of Epetra_IntVectors containing columns corresponding to a given row and color
int	numColors
	Number of colors in Color Map.
int	maxNumColors
	Max Number of colors on all procs in Color Map.
int *	colorList
	List of colors in Color Map.
std::list< int >	listOfAllColors
	List of colors in Color Map.
std::map< int, int >	colorToNumMap
	Inverse mapping from color id to its position in colorList.
Epetra_Map *	cMap
	Coloring Map created by external algorithm.
Epetra_Import *	Importer
	Importer needed for mapping Color Map to unColored Map.
Epetra_Vector *	colorVect
	Color vector based on Color Map containing perturbations.
Epetra_Vector *	betaColorVect
	Color vector based on Color Map containing beta value(s)
Epetra_Vector *	mappedColorVect
	Color vector based on unColorred Map containing perturbations.
Epetra_Vector *	xCol_perturb
	Perturbed solution vector based on column map.
const Epetra_BlockMap *	columnMap
	Overlap Map (Column Map of Matrix Graph) needed for parallel.
Epetra_Import *	rowColImporter
	An Import object needed in parallel to map from row-space to column-space.

Detailed Description

Concrete implementation for creating an Epetra_RowMatrix Jacobian via finite differencing of the residual using coloring.

The Jacobian entries are calculated via 1st or 2nd order finite differencing. This requires $ N + 1 $ or $ 2N + 1 $ calls to computeF(), respectively, where $ N $ is the number of colors.

$J_{ij} = \frac{\partial F_i}{\partial x_j} = \frac{F_i(x+\delta\mathbf{e}_j) - F_i(x)}{\delta}$

where $J$ is the Jacobian, $F$ is the function evaluation, $x$ is the solution vector, and $\delta$ is a small perturbation to the $x_j$ entry.

Instead of perturbing each $N_{dof}$ problem degrees of freedom sequentially and then evaluating all $N_{dof}$ functions for each perturbation, coloring allows several degrees of freedom (all belonging to the same color) to be perturbed at the same time. This reduces the total number of function evaluations needed to compute $\mathbf{J}$ from $N_{dof}^2$ as is required using FiniteDifference to $N\cdot N_{dof}$ , often representing substantial computational savings.

Coloring is based on a user-supplied color map generated using an appropriate algorithm, eg greedy-algorithm - Y. Saad, "Iterative Methods for Sparse %Linear Systems, 2<sup>nd</sup> ed.," chp. 3, SIAM, 2003.. Use can be made of the coloring algorithm provided by the EpetraExt package in Trilinos. The 1Dfem_nonlinearColoring and Brusselator example problems located in the nox/epetra-examples subdirectory demonstrate use of the EpetraExt package, and the 1Dfem_nonlinearColoring directory also contains a stand-alone coloring algorithm very similar to that in EpetraExt.

The perturbation, $\delta$ , is calculated using the following equation:

$\delta = \alpha * | x_j | + \beta$

where $\alpha$ is a scalar value (defaults to 1.0e-4) and $\beta$ is another scalar (defaults to 1.0e-6).

Since both FiniteDifferenceColoring and FiniteDifference inherit from the Epetra_RowMatrix class, they can be used as preconditioning matrices for AztecOO preconditioners.

As for FiniteDifference, 1st order accurate Forward and Backward differences as well as 2nd order accurate Centered difference can be specified using setDifferenceMethod with the appropriate enumerated type passed as the argument.

Using FiniteDifferenceColoring in Parallel

Two ways of using this class in a distributed parallel environment are currently supported. From an application standpoint, the two approaches differ only in the status of the solution iterate used in the residual fill. If an object of this class is contructed with parallelColoring = true the solution iterate will be passe back in a non-ghosted form. On the contrary, setting this parameter to false in the constructor will cause the solution iterate to be in a ghosted form when calling back for a residual fill. When using the second approach, the user should be aware that the perturbed vector used to compute residuals has already been scattered to a form consistent with the column space of the Epetra_CrsGraph. In practice, this means that the perturbed vector used by computeF() has already been scattered to a ghosted or overlapped state. The application should then not perform this step but rather simply use the vector provided with the possible exception of requiring a local index reordering to bring the column-space based vector in sync with a potentially different ghosted index ordering. See the Brusselator and %1Dfem_nonlinearColoring example problems for details.

Special Case for Approximate Jacobian Construction

Provision is made for a simplified and cheaper use of coloring that currently provides only for the diagonal of the Jacobian to be computed. This is based on using a first-neighbors coloring of the original Jacobian graph using the Epetra_Ext MapColoring class with the distance1 argument set to true. This same argument should also be set to true in the constructor to this class. The result will be a diagonal Jacobian filled in a much more efficient manner.

Member Function Documentation

void FiniteDifferenceColoring::createColorContainers ( ) [virtual]

Output the coloring map, index map and underlying matrix.

Create containers for using color and index maps in parallel coloring

References colorList, colorMap, colorToNumMap, listOfAllColors, maxNumColors, and numColors.

Referenced by FiniteDifferenceColoring().

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

NOX_Epetra_FiniteDifferenceColoring.H
NOX_Epetra_FiniteDifferenceColoring.C

Public Member Functions

Protected Types

Protected Attributes

Detailed Description

Member Function Documentation