Ifpack2 Templated Preconditioning Package Version 1.0
Ifpack2::RILUK< MatrixType > Class Template Reference

A class for constructing and using an incomplete lower/upper (ILU) factorization of a given Tpetra::RowMatrix. More...

#include <Ifpack2_RILUK_decl.hpp>

Inheritance diagram for Ifpack2::RILUK< MatrixType >:
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List of all members.

## Public Member Functions

RILUK (const Teuchos::RCP< const MatrixType > &A_in)
RILUK constuctor with variable number of indices per row.
virtual ~RILUK ()
Ifpack2_RILUK Destructor.
void SetRelaxValue (magnitudeType RelaxValue)
Set RILU(k) relaxation parameter.
void SetAbsoluteThreshold (magnitudeType Athresh)
Set absolute threshold value.
void SetRelativeThreshold (magnitudeType Rthresh)
Set relative threshold value.
void SetOverlapMode (Tpetra::CombineMode OverlapMode)
Set overlap mode type.
void setParameters (const Teuchos::ParameterList &parameterlist)
Set parameters using a Teuchos::ParameterList object.
void initialize ()
Computes all (graph-related) data necessary to initialize the preconditioner.
bool isInitialized () const
Returns true if the preconditioner has been successfully initialized, false otherwise.
int getNumInitialize () const
Returns the number of calls to initialize().
void compute ()
Compute ILU factors L and U using the specified diagonal perturbation thresholds and relaxation parameters.
bool isComputed () const
If compute() is completed, this query returns true, otherwise it returns false.
int getNumCompute () const
Returns the number of calls to compute().
int getNumApply () const
Returns the number of calls to Apply().
double getInitializeTime () const
Returns the time spent in Initialize().
double getComputeTime () const
Returns the time spent in Compute().
double getApplyTime () const
Returns the time spent in Apply().
void apply (const Tpetra::MultiVector< Scalar, LocalOrdinal, GlobalOrdinal, Node > &X, Tpetra::MultiVector< Scalar, LocalOrdinal, GlobalOrdinal, Node > &Y, Teuchos::ETransp mode=Teuchos::NO_TRANS, Scalar alpha=Teuchos::ScalarTraits< Scalar >::one(), Scalar beta=Teuchos::ScalarTraits< Scalar >::zero()) const
Returns the result of a RILUK forward/back solve on a Tpetra::MultiVector<Scalar,LocalOrdinal,GlobalOrdinal,Node> X in Y.
int Multiply (const Tpetra::MultiVector< Scalar, LocalOrdinal, GlobalOrdinal, Node > &X, Tpetra::MultiVector< Scalar, LocalOrdinal, GlobalOrdinal, Node > &Y, Teuchos::ETransp mode=Teuchos::NO_TRANS) const
Returns the result of multiplying U, D and L in that order on an Tpetra::MultiVector<Scalar,LocalOrdinal,GlobalOrdinal,Node> X in Y.
magnitudeType computeCondEst (Teuchos::ETransp mode) const
Returns the maximum over all the condition number estimate for each local ILU set of factors.
magnitudeType computeCondEst (CondestType CT=Ifpack2::Cheap, LocalOrdinal MaxIters=1550, magnitudeType Tol=1e-9, const Teuchos::Ptr< const Tpetra::RowMatrix< Scalar, LocalOrdinal, GlobalOrdinal, Node > > &Matrix=Teuchos::null)
Computes the condition number estimate and returns its value.
magnitudeType getCondEst () const
Returns the computed condition number estimate, or -1.0 if not computed.
Teuchos::RCP< const
Tpetra::RowMatrix< Scalar,
LocalOrdinal, GlobalOrdinal,
Node > >
getMatrix () const
Returns a pointer to the input matrix.
magnitudeType GetRelaxValue () const
Get RILU(k) relaxation parameter.
magnitudeType getAbsoluteThreshold () const
Get absolute threshold value.
magnitudeType getRelativeThreshold () const
Get relative threshold value.
Tpetra::CombineMode getOverlapMode ()
Get overlap mode type.
int getGlobalNumEntries () const
Returns the number of nonzero entries in the global graph.
const Teuchos::RCP
< Ifpack2::IlukGraph
< LocalOrdinal, GlobalOrdinal,
Node > > &
getGraph () const
Returns the Ifpack2::IlukGraph associated with this factored matrix.
const MatrixType & getL () const
Returns the L factor associated with this factored matrix.
const Tpetra::Vector< Scalar,
LocalOrdinal, GlobalOrdinal,
Node > &
getD () const
Returns the D factor associated with this factored matrix.
const MatrixType & getU () const
Returns the U factor associated with this factored matrix.
const Teuchos::RCP< const
Tpetra::Map< LocalOrdinal,
GlobalOrdinal, Node > > &
getDomainMap () const
Returns the Tpetra::Map object associated with the domain of this operator.
const Teuchos::RCP< const
Tpetra::Map< LocalOrdinal,
GlobalOrdinal, Node > > &
getRangeMap () const
Returns the Tpetra::Map object associated with the range of this operator.

## Detailed Description

### template<class MatrixType> class Ifpack2::RILUK< MatrixType >

A class for constructing and using an incomplete lower/upper (ILU) factorization of a given Tpetra::RowMatrix.

Ifpack2::RILUK computes a "Relaxed" ILU factorization with level k fill of a given Tpetra::RowMatrix.

For a complete list of valid parameters, see Ifpack2::RILUK::setParameters.

The factorization that is produced is a function of several parameters:

1. 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.)

2. 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.

3. Level of overlap - All Ifpack2 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 Ifpack2_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 Tpetra::CombineMode enum. The default is to zero out all values of y for rows that were not part of the original matrix row distribution.

4. Fraction of relaxation - Ifpack2_RILUK 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.

5. 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 $$(LU)^{-1}$$.

The condition of a matrix $$B$$, called $$cond_p(B)$$, is defined as $$cond_p(B) = \|B\|_p\|B^{-1}\|_p$$ in some appropriate norm $$p$$. $$cond_p(B)$$ 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 $$B$$ or $$B^{-1}$$ may be meaningless.

The $$\infty$$-norm of a vector $$y$$ is defined as the maximum of the absolute values of the vector entries, and the $$\infty$$-norm of a matrix C is defined as $$\|C\|_\infty = \max_{\|y\|_\infty = 1} \|Cy\|_\infty$$. A crude lower bound for the $$cond_\infty(C)$$ is $$\|C^{-1}e\|_\infty$$ where $$e = (1, 1, \ldots, 1)^T$$. It is a lower bound because $$cond_\infty(C) = \|C\|_\infty\|C^{-1}\|_\infty \ge \|C^{-1}\|_\infty \ge |C^{-1}e\|_\infty$$.

For our purposes, we want to estimate $$cond_\infty(LU)$$, where $$L$$ and $$U$$ are our incomplete factors. Edmond in his Ph.D. thesis demonstrates that $$\|(LU)^{-1}e\|_\infty$$ provides an effective estimate for $$cond_\infty(LU)$$. Furthermore, since finding $$z$$ such that $$LUz = y$$ is a basic kernel for applying the preconditioner, computing this estimate of $$cond_\infty(LU)$$ is performed by setting $$y = e$$, calling the solve kernel to compute $$z$$ and then computing $$\|z\|_\infty$$.

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 $$A$$ 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 $$A$$ 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 $$(d_1, d_2, \ldots, d_n)$$ with $$d_i = sgn(d_i)\alpha + d_i\rho$$, $$i=1, 2, \ldots, n$$, where $$n$$ is the matrix dimension and $$sgn(d_i)$$ returns the sign of the diagonal entry. This has the effect of forcing the diagonal values to have minimal magnitude of $$\alpha$$ and to increase each by an amount proportional to $$\rho$$, and still keep the sign of the original diagonal entry.

Counting Floating Point Operations

Each Ifpack2::RILUK object keeps 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 Teuchos::Time class, one can get accurate parallel performance numbers. The ResetFlops() function resets the floating point counter.

## Constructor & Destructor Documentation

template<class MatrixType >
 Ifpack2::RILUK< MatrixType >::RILUK ( const Teuchos::RCP< const MatrixType > & A_in )

RILUK constuctor with variable number of indices per row.

Creates a RILUK object and allocates storage.

Parameters:
 In Graph_in - Graph generated by IlukGraph.
template<class MatrixType >
 Ifpack2::RILUK< MatrixType >::~RILUK ( )  [virtual]

Ifpack2_RILUK Destructor.

## Member Function Documentation

template<class MatrixType>
 void Ifpack2::RILUK< MatrixType >::SetRelaxValue ( magnitudeType RelaxValue )  [inline]

Set RILU(k) relaxation parameter.

template<class MatrixType>
 void Ifpack2::RILUK< MatrixType >::SetAbsoluteThreshold ( magnitudeType Athresh )  [inline]

Set absolute threshold value.

template<class MatrixType>
 void Ifpack2::RILUK< MatrixType >::SetRelativeThreshold ( magnitudeType Rthresh )  [inline]

Set relative threshold value.

template<class MatrixType>
 void Ifpack2::RILUK< MatrixType >::SetOverlapMode ( Tpetra::CombineMode OverlapMode )  [inline]

Set overlap mode type.

template<class MatrixType >
 void Ifpack2::RILUK< MatrixType >::setParameters ( const Teuchos::ParameterList & parameterlist )  [virtual]

Set parameters using a Teuchos::ParameterList object.

• "fact: iluk level-of-fill" (int)
• "fact: iluk level-of-overlap" (int)
Not currently supported.
• "fact: absolute threshold" (magnitude-type)
• "fact: relative threshold" (magnitude-type)
• "fact: relax value" (magnitude-type)
template<class MatrixType >
 void Ifpack2::RILUK< MatrixType >::initialize ( )  [virtual]

Computes all (graph-related) data necessary to initialize the preconditioner.

template<class MatrixType>
 bool Ifpack2::RILUK< MatrixType >::isInitialized ( ) const [inline, virtual]

Returns true if the preconditioner has been successfully initialized, false otherwise.

template<class MatrixType>
 int Ifpack2::RILUK< MatrixType >::getNumInitialize ( ) const [inline, virtual]

Returns the number of calls to initialize().

template<class MatrixType >
 void Ifpack2::RILUK< MatrixType >::compute ( )  [virtual]

Compute ILU factors L and U using the specified diagonal perturbation thresholds and relaxation parameters.

This function computes the RILU(k) factors L and U using the current:

1. Ifpack2_IlukGraph specifying the structure of L and U.
2. Value for the RILU(k) relaxation parameter.
3. Value for the a priori diagonal threshold values.

initialize() must be called before the factorization can proceed.

template<class MatrixType>
 bool Ifpack2::RILUK< MatrixType >::isComputed ( ) const [inline, virtual]

If compute() is completed, this query returns true, otherwise it returns false.

template<class MatrixType>
 int Ifpack2::RILUK< MatrixType >::getNumCompute ( ) const [inline, virtual]

Returns the number of calls to compute().

template<class MatrixType>
 int Ifpack2::RILUK< MatrixType >::getNumApply ( ) const [inline, virtual]

Returns the number of calls to Apply().

template<class MatrixType>
 double Ifpack2::RILUK< MatrixType >::getInitializeTime ( ) const [inline, virtual]

Returns the time spent in Initialize().

template<class MatrixType>
 double Ifpack2::RILUK< MatrixType >::getComputeTime ( ) const [inline, virtual]

Returns the time spent in Compute().

template<class MatrixType>
 double Ifpack2::RILUK< MatrixType >::getApplyTime ( ) const [inline, virtual]

Returns the time spent in Apply().

template<class MatrixType >
 void Ifpack2::RILUK< MatrixType >::apply ( const Tpetra::MultiVector< Scalar, LocalOrdinal, GlobalOrdinal, Node > & X, Tpetra::MultiVector< Scalar, LocalOrdinal, GlobalOrdinal, Node > & Y, Teuchos::ETransp mode = Teuchos::NO_TRANS, Scalar alpha = Teuchos::ScalarTraits::one(), Scalar beta = Teuchos::ScalarTraits::zero() ) const [virtual]

Returns the result of a RILUK forward/back solve on a Tpetra::MultiVector<Scalar,LocalOrdinal,GlobalOrdinal,Node> X in Y.

Parameters:
 In Trans -If true, solve transpose problem. In X - A Tpetra::MultiVector of dimension NumVectors to solve for. Out Y -A Tpetra::MultiVector of dimension NumVectorscontaining result.
Returns:
Integer error code, set to 0 if successful.
template<class MatrixType >
 int Ifpack2::RILUK< MatrixType >::Multiply ( const Tpetra::MultiVector< Scalar, LocalOrdinal, GlobalOrdinal, Node > & X, Tpetra::MultiVector< Scalar, LocalOrdinal, GlobalOrdinal, Node > & Y, Teuchos::ETransp mode = Teuchos::NO_TRANS ) const

Returns the result of multiplying U, D and L in that order on an Tpetra::MultiVector<Scalar,LocalOrdinal,GlobalOrdinal,Node> X in Y.

Parameters:
 In Trans -If true, multiply by L^T, D and U^T in that order. In X - A Tpetra::MultiVector of dimension NumVectors to solve for. Out Y -A Tpetra::MultiVector of dimension NumVectorscontaining result.
Returns:
Integer error code, set to 0 if successful.
template<class MatrixType >
 Teuchos::ScalarTraits< typename MatrixType::scalar_type >::magnitudeType Ifpack2::RILUK< MatrixType >::computeCondEst ( Teuchos::ETransp mode ) const

Returns the maximum over all the condition number estimate for each local ILU set of factors.

This functions computes a local condition number estimate on each processor and return the maximum over all processors of the estimate.

Parameters:
 In Trans -If true, solve transpose problem. Out ConditionNumberEstimate - The maximum across all processors of the infinity-norm estimate of the condition number of the inverse of LDU.
template<class MatrixType>
 magnitudeType Ifpack2::RILUK< MatrixType >::computeCondEst ( CondestType CT = Ifpack2::Cheap, LocalOrdinal MaxIters = 1550, magnitudeType Tol = 1e-9, const Teuchos::Ptr< const Tpetra::RowMatrix< Scalar, LocalOrdinal, GlobalOrdinal, Node > > & Matrix = Teuchos::null )  [inline, virtual]

Computes the condition number estimate and returns its value.

template<class MatrixType>
 magnitudeType Ifpack2::RILUK< MatrixType >::getCondEst ( ) const [inline, virtual]

Returns the computed condition number estimate, or -1.0 if not computed.

template<class MatrixType>
 Teuchos::RCP > Ifpack2::RILUK< MatrixType >::getMatrix ( ) const [inline, virtual]

Returns a pointer to the input matrix.

template<class MatrixType>
 magnitudeType Ifpack2::RILUK< MatrixType >::GetRelaxValue ( ) const [inline]

Get RILU(k) relaxation parameter.

template<class MatrixType>
 magnitudeType Ifpack2::RILUK< MatrixType >::getAbsoluteThreshold ( ) const [inline]

Get absolute threshold value.

template<class MatrixType>
 magnitudeType Ifpack2::RILUK< MatrixType >::getRelativeThreshold ( ) const [inline]

Get relative threshold value.

template<class MatrixType>
 Tpetra::CombineMode Ifpack2::RILUK< MatrixType >::getOverlapMode ( )  [inline]

Get overlap mode type.

template<class MatrixType>
 int Ifpack2::RILUK< MatrixType >::getGlobalNumEntries ( ) const [inline]

Returns the number of nonzero entries in the global graph.

template<class MatrixType>
 const Teuchos::RCP >& Ifpack2::RILUK< MatrixType >::getGraph ( ) const [inline]

Returns the Ifpack2::IlukGraph associated with this factored matrix.

template<class MatrixType>
 const MatrixType& Ifpack2::RILUK< MatrixType >::getL ( ) const [inline]

Returns the L factor associated with this factored matrix.

template<class MatrixType>
 const Tpetra::Vector& Ifpack2::RILUK< MatrixType >::getD ( ) const [inline]

Returns the D factor associated with this factored matrix.

template<class MatrixType>
 const MatrixType& Ifpack2::RILUK< MatrixType >::getU ( ) const [inline]

Returns the U factor associated with this factored matrix.

template<class MatrixType>
 const Teuchos::RCP >& Ifpack2::RILUK< MatrixType >::getDomainMap ( ) const [inline, virtual]

Returns the Tpetra::Map object associated with the domain of this operator.

template<class MatrixType>
 const Teuchos::RCP >& Ifpack2::RILUK< MatrixType >::getRangeMap ( ) const [inline, virtual]

Returns the Tpetra::Map object associated with the range of this operator.

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