NOX::EpetraNew::FiniteDifferenceColoring Class Reference

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

#include <NOX_EpetraNew_FiniteDifferenceColoring.H>

Inheritance diagram for NOX::EpetraNew::FiniteDifferenceColoring:

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Collaboration diagram for NOX::EpetraNew::FiniteDifferenceColoring:
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List of all members.

Public Member Functions

 FiniteDifferenceColoring (NOX::Parameter::List &printingParams, Interface::Required &i, const Epetra_Vector &initialGuess, Epetra_MapColoring &colorMap, 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 (Interface::Required &i, const Epetra_Vector &initialGuess, Epetra_MapColoring &colorMap, vector< Epetra_IntVector > &columns, bool parallelColoring=false, bool distance1=false, double beta=1.0e-6, double alpha=1.0e-4)
 Constructor.
 FiniteDifferenceColoring (NOX::Parameter::List &printingParams, Interface::Required &i, const Epetra_Vector &initialGuess, Epetra_CrsGraph &rawGraph, Epetra_MapColoring &colorMap, 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 (Interface::Required &i, const Epetra_Vector &initialGuess, Epetra_CrsGraph &rawGraph, Epetra_MapColoring &colorMap, vector< Epetra_IntVector > &columns, bool parallelColoring=false, bool distance1=false, double beta=1.0e-6, double alpha=1.0e-4)
 Constructor.
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 ()
 Create containers for using color and index maps in parallel coloring.

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
const Epetra_MapColoringcolorMap
 Color Map created by external algorithm.
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_MapcMap
 Coloring Map created by external algorithm.
Epetra_ImportImporter
 Importer needed for mapping Color Map to unColored Map.
Epetra_VectorcolorVect
 Color vector based on Color Map containing perturbations.
Epetra_VectorbetaColorVect
 Color vector based on Color Map containing beta value(s).
Epetra_VectormappedColorVect
 Color vector based on unColorred Map containing perturbations.
Epetra_VectorxCol_perturb
 Perturbed solution vector based on column map.
const Epetra_BlockMapcolumnMap
 Overlap Map (Column Map of Matrix Graph) needed for parallel.
Epetra_ImportrowColImporter
 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, 2nd 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.


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