Dune::SeqDILU< M, X, Y, l > Class Template Reference

Sequential DILU preconditioner. More...

#include <dune/istl/preconditioners.hh>

Public Types
using	matrix_type = M
	The matrix type the preconditioner is for.
using	block_type = typename matrix_type::block_type
	block type of matrix
using	domain_type = X
	The domain type of the preconditioner.
using	range_type = Y
	The range type of the preconditioner.
using	field_type = typename X::field_type
	The field type of the preconditioner.
using	scalar_field_type = Simd::Scalar< field_type >
	scalar type underlying the field_type
using	real_field_type = typename FieldTraits< scalar_field_type >::real_type
	real scalar type underlying the field_type

Public Member Functions
	SeqDILU (const M &A, real_field_type w)
	Constructor. More...
	SeqDILU (const std::shared_ptr< const AssembledLinearOperator< M, X, Y > > &A, const ParameterTree &configuration)
	Constructor. More...
	SeqDILU (const M &A, const ParameterTree &config)
	Constructor. More...
void	pre (X &x, Y &b) override
	Prepare the preconditioner. More...
void	apply (X &v, const Y &d) override
	Apply the preconditioner. More...
void	post (X &x) override
	Clean up. More...
SolverCategory::Category	category () const override
	Category of the preconditioner (see SolverCategory::Category)

Protected Attributes
const M &	_A_
	The matrix we operate on.
const real_field_type	_w
	The relaxation factor to use.
const bool	wNotIdentity_
	true if w != 1.0

Detailed Description

template<class M, class X, class Y, int l = 1>
class Dune::SeqDILU< M, X, Y, l >

Sequential DILU preconditioner.

Wraps the naked ISTL generic DILU preconditioner into the solver framework.

The Diagonal Incomplete LU factorization (DILU) is a simplified variant of the ILU(0) factorization, where only the diagonal elements are factorised. This preconditioner can be written as

M = (D + L_A) D^{-1} (D + U_A),

where L_A and U_A are the strictly lower and upper parts of A and D is the diagonal matrix containing the generated pivot values. The matrix M has the property

diag(A) = diag(M) = diag((D + L_A) D^{-1} (D + U_A)) = diag(D + L_A D^{-1} U_A)

such that the diagonal matrix D can be constructed:

D_11 = A_11 D_22 = A22 - L_A_{21} D_{11}^{-1} U_A_{12} and etc.

Note that when applying the preconditioner, M is never explicitly created but instead a lower and upper triangualr solve is performed using the values of A and D^-1. When applying the preconditioner, we are working with the residual d = b - Ax and update v = x_{n+1} - x_n, such that:

v = M^{-1} d = (D + U_A)^{-1} D (D + L_A)^{-1} d

define y = (D + L_A)^{-1} d

lower triangular solve: (D + L_A) y = d upper triangular solve: (D + U_A) v = D y

This means that the preconditioner only requires an additional storage of the diagonal matrix D^{-1}

For more details, see: R. Barrett et al., "Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods", 1994. Available from: https://www.netlib.org/templates/templates.pdf

Template Parameters

M	The matrix type to operate on
X	Type of the update
Y	Type of the defect
l	Ignored. Just there to have the same number of template arguments as other preconditioners.

Constructor & Destructor Documentation

SeqDILU() [1/3]

template<class M , class X , class Y , int l = 1>

Dune::SeqDILU< M, X, Y, l >::SeqDILU ( const M &  A, real_field_type  w  )

inline

Constructor.

Constructor invoking DILU gets all parameters to operate the prec.

Parameters

A	The matrix to operate on.
w	The relaxation factor.

SeqDILU() [2/3]

template<class M , class X , class Y , int l = 1>

Dune::SeqDILU< M, X, Y, l >::SeqDILU ( const std::shared_ptr< const AssembledLinearOperator< M, X, Y > > &  A, const ParameterTree &  configuration  )

inline

Constructor.

Parameters

A	The assembled linear operator to use.
configuration	ParameterTree containing preconditioner parameters.

ParameterTree Key	Meaning
relaxation	The relaxation factor. default=1.0

See ISTL_Factory for the ParameterTree layout and examples.

SeqDILU() [3/3]

template<class M , class X , class Y , int l = 1>

Dune::SeqDILU< M, X, Y, l >::SeqDILU ( const M &  A, const ParameterTree &  config  )

inline

Constructor.

Parameters

A	The matrix to operate on.
config	ParameterTree containing preconditioner parameters.

ParameterTree Key	Meaning
relaxation	The relaxation factor. default=1.0

See ISTL_Factory for the ParameterTree layout and examples.

Member Function Documentation

apply()

template<class M , class X , class Y , int l = 1>

void Dune::SeqDILU< M, X, Y, l >::apply ( X &  v, const Y &  d  )

inlineoverridevirtual

Apply the preconditioner.

Apply one step of the preconditioner to the system A(v)=d.

On entry v=0 and d=b-A(x) (although this might not be computed in that way. On exit v contains the update, i.e one step computes \( v = M^{-1} d \) where \( M \) is the approximate inverse of the operator \( A \) characterizing the preconditioner.

Parameters

[out]	v	The update to be computed
	d	The current defect.

Implements Dune::Preconditioner< X, Y >.

post()

template<class M , class X , class Y , int l = 1>

void Dune::SeqDILU< M, X, Y, l >::post ( X &  x )

inlineoverridevirtual

Clean up.

Clean up.

This method is called after the last apply call for the linear system to be solved. Memory may be deallocated safely here. x is the solution of the linear equation.

Parameters

x	The right hand side of the equation.

Implements Dune::Preconditioner< X, Y >.

pre()

template<class M , class X , class Y , int l = 1>

void Dune::SeqDILU< M, X, Y, l >::pre ( X &  x, Y &  b  )

inlineoverridevirtual

Prepare the preconditioner.

Prepare the preconditioner.

A solver solves a linear operator equation A(x)=b by applying one or several steps of the preconditioner. The method pre() is called before the first apply operation. b and x are right hand side and solution vector of the linear system respectively. It may. e.g., scale the system, allocate memory or compute a (I)LU decomposition. Note: The ILU decomposition could also be computed in the constructor or with a separate method of the derived method if several linear systems with the same matrix are to be solved.

Note

if a preconditioner is copied (e.g. for a second thread) again the pre() method has to be called to ensure proper memory management.

X x(0.0);
Y b = ...; // rhs
Preconditioner<X,Y> prec(...);
prec.pre(x,b);   // prepare the preconditioner
prec.apply(x,b); // can be called multiple times now...
prec.post(x);    // cleanup internal state

Parameters

x	The left hand side of the equation.
b	The right hand side of the equation.

Implements Dune::Preconditioner< X, Y >.

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

• dune/istl/preconditioners.hh