Public Member Functions
| ||Ifpack_CrsRick (const Epetra_CrsMatrix &A, const Ifpack_IlukGraph &Graph)|
| ||Ifpack_CrsRick constuctor with variable number of indices per row. |
| ||Ifpack_CrsRick (const Ifpack_CrsRick &Matrix)|
| ||Copy constructor. |
|virtual ||~Ifpack_CrsRick ()|
| ||Ifpack_CrsRick Destructor. |
|int ||InitValues ()|
| ||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. |
|void ||SetRelaxValue (double RelaxValue)|
| ||Set RILU(k) relaxation parameter. |
|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 ||Factor ()|
| ||Compute ILU factors L and 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_Vector &x, Epetra_Vector &y) const |
| ||Returns the result of a Ifpack_CrsRick forward/back solve on a Epetra_Vector x in y. |
|int ||Solve (bool Trans, const Epetra_MultiVector &X, Epetra_MultiVector &Y) const |
| ||Returns the result of a Ifpack_CrsRick 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 ||GetRelaxValue ()|
| ||Get RILU(k) relaxation parameter. |
|double ||GetAbsoluteThreshold ()|
| ||Get absolute threshold value. |
|double ||GetRelativeThreshold ()|
| ||Get relative threshold value. |
|Epetra_CombineMode ||GetOverlapMode ()|
| ||Get overlap mode type. |
|int ||NumGlobalRows () const |
| ||Returns the number of global matrix rows. |
|int ||NumGlobalCols () const |
| ||Returns the number of global matrix columns. |
|int ||NumGlobalNonzeros () const |
| ||Returns the number of nonzero entries in the global graph. |
|virtual int ||NumGlobalDiagonals () const |
| ||Returns the number of diagonal entries found in the global input graph. |
|int ||NumMyRows () const |
| ||Returns the number of local matrix rows. |
|int ||NumMyCols () const |
| ||Returns the number of local matrix columns. |
|int ||NumMyNonzeros () const |
| ||Returns the number of nonzero entries in the local graph. |
|virtual int ||NumMyDiagonals () const |
| ||Returns the number of diagonal entries found in the local input graph. |
|int ||IndexBase () const |
| ||Returns the index base for row and column indices for this graph. |
|const Ifpack_IlukGraph & ||Graph () const |
| ||Returns the address of the Ifpack_IlukGraph associated with this factored matrix. |
|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 Ifpack_IlukGraph & ||Graph_|
|Epetra_CrsMatrix * ||U_|
|Epetra_Vector * ||D_|
|Epetra_MultiVector * ||OverlapX_|
|Epetra_MultiVector * ||OverlapY_|
|ostream & ||operator<< (ostream &os, const Ifpack_CrsRick &A)|
| ||<< operator will work for Ifpack_CrsRick. |
|char * ||Label () const |
| ||Returns a character string describing the operator. |
|int ||SetUseTranspose (bool UseTranspose)|
| ||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. |
Ifpack_CrsRick: A class for constructing and using an incomplete lower/upper (ILU) factorization of a given Epetra_CrsMatrix.
The Ifpack_CrsRick class computes a "Relaxed" ILU factorization with level k fill of a given Epetra_CrsMatrix. The factorization that is produced is a function of several parameters:
The pattern of the matrix - All fill is derived from the original matrix nonzero structure. Level zero fill is defined as the original matrix pattern (nonzero structure), even if the matrix value at an entry is stored as a zero. (Thus it is possible to add entries to the ILU factors by adding zero entries the original matrix.)
Level of fill - Starting with the original matrix pattern as level fill of zero, the next level of fill is determined by analyzing the graph of the previous level and determining nonzero fill that is a result of combining entries that were from previous level only (not the current level). This rule limits fill to entries that are direct decendents from the previous level graph. Fill for level k is determined by applying this rule recursively. For sufficiently large values of k, the fill would eventually be complete and an exact LU factorization would be computed. Level of fill is defined during the construction of the Ifpack_IlukGraph object.
Level of overlap - All Ifpack preconditioners work on parallel distributed memory computers by using the row partitioning the user input matrix to determine the partitioning for local ILU factors. If the level of overlap is set to zero, the rows of the user matrix that are stored on a given processor are treated as a self-contained local matrix and all column entries that reach to off-processor entries are ignored. Setting the level of overlap to one tells Ifpack to increase the size of the local matrix by adding rows that are reached to by rows owned by this processor. Increasing levels of overlap are defined recursively in the same way. For sufficiently large levels of overlap, the entire matrix would be part of each processor's local ILU factorization process. Level of overlap is defined during the construction of the Ifpack_IlukGraph object.
Once the factorization is computed, applying the factorization \(LUy = x\) results in redundant approximations for any elements of y that correspond to rows that are part of more than one local ILU factor. The OverlapMode (changed by calling SetOverlapMode()) defines how these redundancies are handled using the Epetra_CombineMode enum. The default is to zero out all values of y for rows that were not part of the original matrix row distribution.
Fraction of relaxation - Ifpack_CrsRick computes the ILU factorization row-by-row. As entries at a given row are computed, some number of them will be dropped because they do match the prescribed sparsity pattern. The relaxation factor determines how these dropped values will be handled. If the RelaxValue (changed by calling SetRelaxValue()) is zero, then these extra entries will by dropped. This is a classical ILU approach. If the RelaxValue is 1, then the sum of the extra entries will be added to the diagonal. This is a classical Modified ILU (MILU) approach. If RelaxValue is between 0 and 1, then RelaxValue times the sum of extra entries will be added to the diagonal.
For most situations, RelaxValue should be set to zero. For certain kinds of problems, e.g., reservoir modeling, there is a conservation principle involved such that any operator should obey a zero row-sum property. MILU was designed for these cases and you should set the RelaxValue to 1. For other situations, setting RelaxValue to some nonzero value may improve the stability of factorization, and can be used if the computed ILU factors are poorly conditioned.
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_CrsRick objects
Constructing Ifpack_CrsRick objects is a multi-step process. The basic steps are as follows:
Create Ifpack_CrsRick 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_CrsRick 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_CrsRick constructor.
Definition at line 192 of file Ifpack_CrsRick.h.