Public Member Functions
| ||Ifpack_CrsIct (const Epetra_CrsMatrix &A, double Droptol=1.0E-4, int Lfil=20)|
| ||Ifpack_CrsIct constuctor with variable number of indices per row. |
| ||Ifpack_CrsIct (const Ifpack_CrsIct &IctOperator)|
| ||Copy constructor. |
|virtual ||~Ifpack_CrsIct ()|
| ||Ifpack_CrsIct Destructor. |
|void ||SetAbsoluteThreshold (double Athresh)|
| ||Set absolute threshold value. |
|void ||SetRelativeThreshold (double Rthresh)|
| ||Set relative threshold value. |
|void ||SetOverlapMode (Epetra_CombineMode OverlapMode)|
| ||Set overlap mode type. |
|int ||SetParameters (const Teuchos::ParameterList ¶meterlist, bool cerr_warning_if_unused=false)|
| ||Set parameters using a Teuchos::ParameterList object. |
|int ||InitValues (const Epetra_CrsMatrix &A)|
| ||Initialize L and U with values from user matrix A. |
|bool ||ValuesInitialized () const |
| ||If values have been initialized, this query returns true, otherwise it returns false. |
|int ||Factor ()|
| ||Compute IC factor U using the specified graph, diagonal perturbation thresholds and relaxation parameters. |
|bool ||Factored () const |
| ||If factor is completed, this query returns true, otherwise it returns false. |
|int ||Solve (bool Trans, const Epetra_MultiVector &X, Epetra_MultiVector &Y) const |
| ||Returns the result of a Ifpack_CrsIct forward/back solve on a Epetra_MultiVector X in Y. |
|int ||Multiply (bool Trans, const Epetra_MultiVector &X, Epetra_MultiVector &Y) const |
| ||Returns the result of multiplying U, D and U^T in that order on an Epetra_MultiVector X in Y. |
|int ||Condest (bool Trans, double &ConditionNumberEstimate) const |
| ||Returns the maximum over all the condition number estimate for each local ILU set of factors. |
|double ||GetAbsoluteThreshold ()|
| ||Get absolute threshold value. |
|double ||GetRelativeThreshold ()|
| ||Get relative threshold value. |
|Epetra_CombineMode ||GetOverlapMode ()|
| ||Get overlap mode type. |
|int ||NumGlobalNonzeros () const |
| ||Returns the number of nonzero entries in the global graph. |
|int ||NumMyNonzeros () const |
| ||Returns the number of nonzero entries in the local graph. |
|const Epetra_Vector & ||D () const |
| ||Returns the address of the D factor associated with this factored matrix. |
|const Epetra_CrsMatrix & ||U () const |
| ||Returns the address of the U factor associated with this factored matrix. |
Protected Member Functions
|void ||SetFactored (bool Flag)|
|void ||SetValuesInitialized (bool Flag)|
|bool ||Allocated () const |
|int ||SetAllocated (bool Flag)|
Private Member Functions
|int ||Allocate ()|
|const Epetra_CrsMatrix & ||A_|
|const Epetra_Comm & ||Comm_|
< Epetra_CrsMatrix >
< Epetra_Vector >
< Epetra_MultiVector >
< Epetra_MultiVector >
|void * ||Aict_|
|void * ||Lict_|
|double * ||Ldiag_|
|ostream & ||operator<< (ostream &os, const Ifpack_CrsIct &A)|
| ||<< operator will work for Ifpack_CrsIct. |
|const char * ||Label () const |
| ||Returns a character string describing the operator. |
|int ||SetUseTranspose (bool UseTranspose_in)|
| ||If set true, transpose of this operator will be applied. |
|int ||Apply (const Epetra_MultiVector &X, Epetra_MultiVector &Y) const |
| ||Returns the result of a Epetra_Operator applied to a Epetra_MultiVector X in Y. |
|int ||ApplyInverse (const Epetra_MultiVector &X, Epetra_MultiVector &Y) const |
| ||Returns the result of a Epetra_Operator inverse applied to an Epetra_MultiVector X in Y. |
|double ||NormInf () const |
| ||Returns 0.0 because this class cannot compute Inf-norm. |
|bool ||HasNormInf () const |
| ||Returns false because this class cannot compute an Inf-norm. |
|bool ||UseTranspose () const |
| ||Returns the current UseTranspose setting. |
|const Epetra_Map & ||OperatorDomainMap () const |
| ||Returns the Epetra_Map object associated with the domain of this operator. |
|const Epetra_Map & ||OperatorRangeMap () const |
| ||Returns the Epetra_Map object associated with the range of this operator. |
|const Epetra_Comm & ||Comm () const |
| ||Returns the Epetra_BlockMap object associated with the range of this matrix operator. |
Ifpack_CrsIct: A class for constructing and using an incomplete Cholesky factorization of a given Epetra_CrsMatrix.
The Ifpack_CrsIct class computes a threshold based incomplete LDL^T factorization of a given Epetra_CrsMatrix. The factorization that is produced is a function of several parameters:
Maximum number of entries per row/column in factor - The factorization will contain at most this number of nonzero terms in each row/column of the factorization.
Diagonal perturbation - Prior to computing the factorization, it is possible to modify the diagonal entries of the matrix for which the factorization will be computing. If the absolute and relative perturbation values are zero and one, respectively, the factorization will be compute for the original user matrix A. Otherwise, the factorization will computed for a matrix that differs from the original user matrix in the diagonal values only. Below we discuss the details of diagonal perturbations. The absolute and relative threshold values are set by calling SetAbsoluteThreshold() and SetRelativeThreshold(), respectively.
Estimating Preconditioner Condition Numbers
For ill-conditioned matrices, we often have difficulty computing usable incomplete factorizations. The most common source of problems is that the factorization may encounter a small or zero pivot, in which case the factorization can fail, or even if the factorization succeeds, the factors may be so poorly conditioned that use of them in the iterative phase produces meaningless results. Before we can fix this problem, we must be able to detect it. To this end, we use a simple but effective condition number estimate for .
The condition of a matrix , called , is defined as in some appropriate norm . gives some indication of how many accurate floating point digits can be expected from operations involving the matrix and its inverse. A condition number approaching the accuracy of a given floating point number system, about 15 decimal digits in IEEE double precision, means that any results involving or may be meaningless.
The -norm of a vector is defined as the maximum of the absolute values of the vector entries, and the -norm of a matrix C is defined as . A crude lower bound for the is where . It is a lower bound because .
For our purposes, we want to estimate , where and are our incomplete factors. Edmond in his Ph.D. thesis demonstrates that provides an effective estimate for . Furthermore, since finding such that is a basic kernel for applying the preconditioner, computing this estimate of is performed by setting , calling the solve kernel to compute and then computing .
A priori Diagonal Perturbations
Given the above method to estimate the conditioning of the incomplete factors, if we detect that our factorization is too ill-conditioned we can improve the conditioning by perturbing the matrix diagonal and restarting the factorization using this more diagonally dominant matrix. In order to apply perturbation, prior to starting the factorization, we compute a diagonal perturbation of our matrix and perform the factorization on this perturbed matrix. The overhead cost of perturbing the diagonal is minimal since the first step in computing the incomplete factors is to copy the matrix into the memory space for the incomplete factors. We simply compute the perturbed diagonal at this point.
The actual perturbation values we use are the diagonal values with , , where is the matrix dimension and returns the sign of the diagonal entry. This has the effect of forcing the diagonal values to have minimal magnitude of and to increase each by an amount proportional to , and still keep the sign of the original diagonal entry.
Constructing Ifpack_CrsIct objects
Constructing Ifpack_CrsIct objects is a multi-step process. The basic steps are as follows:
Create Ifpack_CrsIct instance, including storage, via constructor.
Enter values via one or more Put or SumInto functions.
Complete construction via FillComplete call.
Note that, even after a matrix is constructed, it is possible to update existing matrix entries. It is not possible to create new entries.
Counting Floating Point Operations
Each Ifpack_CrsIct object keep track of the number of serial floating point operations performed using the specified object as the this argument to the function. The Flops() function returns this number as a double precision number. Using this information, in conjunction with the Epetra_Time class, one can get accurate parallel performance numbers. The ResetFlops() function resets the floating point counter.
- A Epetra_Map is required for the Ifpack_CrsIct constructor.
Definition at line 159 of file Ifpack_CrsIct.h.