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Concrete implementation for creating an Epetra_RowMatrix Jacobian via finite differencing of the residual using coloring. More...
#include <NOX_Epetra_FiniteDifferenceColoring.H>
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< vector< Epetra_IntVector > > &columns, bool parallelColoring=false, bool distance1=false, double beta=1.0e6, double alpha=1.0e4)  
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< vector< Epetra_IntVector > > &columns, bool parallelColoring=false, bool distance1=false, double beta=1.0e6, double alpha=1.0e4)  
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 distance1 coloring  
Teuchos::RCP< const Epetra_MapColoring >  colorMap 
Color Map created by external algorithm.  
Teuchos::RCP< 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.  
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 rowspace to columnspace. 
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 or calls to computeF(), respectively, where is the number of colors.
where is the Jacobian, is the function evaluation, is the solution vector, and is a small perturbation to the entry.
Instead of perturbing each problem degrees of freedom sequentially and then evaluating all 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 from as is required using FiniteDifference to , often representing substantial computational savings.
Coloring is based on a usersupplied color map generated using an appropriate algorithm, eg greedyalgorithm  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/epetraexamples subdirectory demonstrate use of the EpetraExt package, and the 1Dfem_nonlinearColoring directory also contains a standalone coloring algorithm very similar to that in EpetraExt.
The perturbation, , is calculated using the following equation:
where is a scalar value (defaults to 1.0e4) and is another scalar (defaults to 1.0e6).
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 nonghosted 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 columnspace 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 firstneighbors 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.
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().