Bramble-Pasciak preconditioned solver for generalized saddle point problems. More...

#include <bramblePasciak.hh>

Inheritance diagram for vectorial::BramblePasciak:

Public Types
typedef Cmplxtype< Real >::type	c_type
	Real type of data type. More...

typedef Realtype< Real >::type	r_type
	Real type of data type. More...

typedef Real	type
	Type of data, e.g. matrix entries. More...

Public Member Functions
	BramblePasciak (concepts::Operator< Real > &A, concepts::Operator< Real > &B, concepts::Operator< Real > &Bt, concepts::Operator< Real > &C, concepts::Operator< Real > &W, Real maxeps, int maxit=0, uint relres=false)
	Constructor. More...

virtual const uint	dimX () const
	Returns the size of the image space of the operator (number of rows of the corresponding matrix) More...

virtual const uint	dimY () const
	Returns the size of the source space of the operator (number of columns of the corresponding matrix) More...

Real	epsilon () const
	Returns the residual. More...

uint	iterations () const
	Returns the number of iterations. More...

virtual void	operator() ()
	Application operator without argument. More...

virtual void	operator() (const concepts::Function< Real > &fncY, concepts::Function< Real > &fncX)

void	operator() (const concepts::Vector< Real > &fncY, concepts::Vector< Real > &fncX)

virtual void	operator() (const Function< c_type > &fncY, Function< c_type > &fncX)
	Application operator for complex function `fncY`. More...

virtual void	operator() (const Function< r_type > &fncY, Function< Real > &fncX)
	Application operator for real function `fncY`. More...

virtual void	show_messages ()

Protected Member Functions
std::ostream &	info (std::ostream &os) const

Protected Attributes
uint	dimX_
	Dimension of image space and the source space. More...

uint	dimY_

Private Attributes
concepts::Operator< Real > &	A_
	Upper left submatrix. More...

concepts::Operator< Real > &	B_
	Upper right submatrix. More...

uint	bdim_ [2]
	Dimensions of partly matrices (in image and source space) More...

concepts::Operator< Real > &	Bt_
	Lower left submatrix. More...

concepts::Operator< Real > &	C_
	Lower right submatrix. More...

Real	eps_
	Current residual. More...

uint	it_
	Number of iterations. More...

Real	maxeps_
	Convergence criterion. More...

uint	maxit_
	Maximal number of iterations until abortion. More...

bool	relres_
	false: absolute residual, true: relative residual More...

concepts::Operator< Real > &	W_
	Preconditioner for A. More...

Detailed Description

Bramble-Pasciak preconditioned solver for generalized saddle point problems.

$\left(\begin{array}{lr}A&B\\B^{\top}&-C\end{array}\right) \left(\begin{array}{c}U\\P\end{array}\right) = \left(\begin{array}{c}L_u\\L_p\end{array}\right)$

A has to be symmetric and positive definite, C has to be symmetric and positive semidefinite, and B has to satisfy the discrete inf-sup-condition.

The main idea is to transform the equations and to provide an inner product in which the transformed matrix is symmetric and positive definite. The transformed system is then solved by conjugate gradients in this inner product. See [1] for more details.

The class is an operator on a vectorial::Space consisting of two spaces, say spc1 and spc2. A is spc1->spc1, B is spc2->spc1 and C is spc2->spc2. The entire vectorial space has to be provided at construction time.

Constructing an object of this class does not solve the given system. Use the application operator to solve the system. If you want to specify a starting vector for the cg iterations, set fncX before calling the application operator to this starting value. fncX also holds the result after the solve.

The application operator throws NoConvergence if the desired residual maxeps is not reached within the given number of iterations maxit.

See also: J. H. Bramble, J. E. Pasciak: A Preconditioning Technique for Indefinite Systems Resulting from Mixed Approximations of Elliptic Problems. Mathematics of Computation, Vol. 50, Nr. 181, pp. 1-17, 1988.; J. H. Bramble, J. E. Pasciak: Corrigenda for A Preconditioning Technique for Indefinite Systems Resulting from Mixed Approximations of Elliptic Problems. Mathematics of Computation, Vol. 51, Nr. 183, pp. 387-388, 1988.

Definition at line 54 of file bramblePasciak.hh.

Member Typedef Documentation

◆ c_type

typedef Cmplxtype<Real >::type concepts::Operator< Real >::c_type

inherited

Real type of data type.

Definition at line 49 of file compositions.hh.

◆ r_type

typedef Realtype<Real >::type concepts::Operator< Real >::r_type

inherited

Real type of data type.

Definition at line 47 of file compositions.hh.

◆ type

typedef Real concepts::Operator< Real >::type

inherited

Type of data, e.g. matrix entries.

Definition at line 45 of file compositions.hh.

Constructor & Destructor Documentation

◆ BramblePasciak()

vectorial::BramblePasciak::BramblePasciak	(	concepts::Operator< Real > &	A,
		concepts::Operator< Real > &	B,
		concepts::Operator< Real > &	Bt,
		concepts::Operator< Real > &	C,
		concepts::Operator< Real > &	W,
		Real	maxeps,
		int	maxit = `0`,
		uint	relres = `false`
	)

Constructor.

Parameters

A	Upper left submatrix
B	Upper right submatrix
Bt	Lower left submatrix
C	Lower right submatrix
W	Preconditioner for A. Must approximate A^-1 and satisfy c0 (A U,U) < (W^-1 U,U) <= c1 (A U,U) for all U with constants 0 < c0 < c1 < 1.
spc	Entire vectorial space
maxeps	Maximal residual
maxit	Maximal number of iterations
relres	Relative residual

Member Function Documentation

◆ dimX()

virtual const uint concepts::Operator< Real >::dimX ( ) const

inlinevirtualinherited

Returns the size of the image space of the operator (number of rows of the corresponding matrix)

Definition at line 93 of file compositions.hh.

◆ dimY()

virtual const uint concepts::Operator< Real >::dimY ( ) const

inlinevirtualinherited

Returns the size of the source space of the operator (number of columns of the corresponding matrix)

Definition at line 98 of file compositions.hh.

◆ epsilon()

Real vectorial::BramblePasciak::epsilon ( ) const

inline

Returns the residual.

Calling this method makes only sense after a linear system has been solved.

Definition at line 85 of file bramblePasciak.hh.

◆ info()

std::ostream& vectorial::BramblePasciak::info ( std::ostream & os ) const

protectedvirtual

Reimplemented from concepts::Operator< Real >.

◆ iterations()

uint vectorial::BramblePasciak::iterations ( ) const

inline

Returns the number of iterations.

Calling this method makes only sense after a linear system has been solved.

Definition at line 80 of file bramblePasciak.hh.

◆ operator()() [1/5]

virtual void concepts::Operator< Real >::operator() ( )

virtualinherited

Application operator without argument.

Reimplemented in concepts::VecOperator< Real >.

◆ operator()() [2/5]

virtual void vectorial::BramblePasciak::operator()	(	const concepts::Function< Real > &	fncY,
		concepts::Function< Real > &	fncX
	)

virtual

◆ operator()() [3/5]

void vectorial::BramblePasciak::operator()	(	const concepts::Vector< Real > &	fncY,
		concepts::Vector< Real > &	fncX
	)

◆ operator()() [4/5]

virtual void concepts::Operator< Real >::operator()	(	const Function< c_type > &	fncY,
		Function< c_type > &	fncX
	)

virtualinherited

Application operator for complex function fncY.

Computes fncX = A(fncY) where A is this operator. fncX becomes complex.

In derived classes its enough to implement the operator() for complex Operator's. If a real counterpart is not implemented, the function fncY is splitted into real and imaginary part and the application operator for real functions is called for each. Then the result is combined.

If in a derived class the operator() for complex Operator's is not implemented, a exception is thrown from here.

Reimplemented in concepts::VecOperator< Real >.

◆ operator()() [5/5]

virtual void concepts::Operator< Real >::operator()	(	const Function< r_type > &	fncY,
		Function< F > &	fncX
	)

virtualinherited

Application operator for real function fncY.

Computes fncX = A(fncY) where A is this operator.

fncX becomes the type of the operator, for real data it becomes real, for complex data it becomes complex.

In derived classes its enough to implement the operator() for real Operator's. If a complex counterpart is not implemented, the function fncY is transformed to a complex function and then the application operator for complex functions is called.

If in a derived class the operator() for real Operator's is not implemented, a exception is thrown from here.

Reimplemented in concepts::VecOperator< Real >.

◆ show_messages()

virtual void concepts::Operator< Real >::show_messages ( )

inlinevirtualinherited

Definition at line 100 of file compositions.hh.

Member Data Documentation

◆ A_

concepts::Operator<Real>& vectorial::BramblePasciak::A_

private

Upper left submatrix.

Definition at line 92 of file bramblePasciak.hh.

◆ B_

concepts::Operator<Real>& vectorial::BramblePasciak::B_

private

Upper right submatrix.

Definition at line 94 of file bramblePasciak.hh.

◆ bdim_

uint vectorial::BramblePasciak::bdim_[2]

private

Dimensions of partly matrices (in image and source space)

Definition at line 90 of file bramblePasciak.hh.

◆ Bt_

concepts::Operator<Real>& vectorial::BramblePasciak::Bt_

private

Lower left submatrix.

Definition at line 96 of file bramblePasciak.hh.

◆ C_

concepts::Operator<Real>& vectorial::BramblePasciak::C_

private

Lower right submatrix.

Definition at line 98 of file bramblePasciak.hh.

◆ dimX_

uint concepts::Operator< Real >::dimX_

protectedinherited

Dimension of image space and the source space.

Definition at line 104 of file compositions.hh.

◆ dimY_

uint concepts::Operator< Real >::dimY_

protectedinherited

Definition at line 104 of file compositions.hh.

◆ eps_

Real vectorial::BramblePasciak::eps_

private

Current residual.

Definition at line 106 of file bramblePasciak.hh.

◆ it_

uint vectorial::BramblePasciak::it_

private

Number of iterations.

Definition at line 108 of file bramblePasciak.hh.

◆ maxeps_

Real vectorial::BramblePasciak::maxeps_

private

Convergence criterion.

Definition at line 102 of file bramblePasciak.hh.

◆ maxit_

uint vectorial::BramblePasciak::maxit_

private

Maximal number of iterations until abortion.

Definition at line 104 of file bramblePasciak.hh.

◆ relres_

bool vectorial::BramblePasciak::relres_

private

false: absolute residual, true: relative residual

Definition at line 110 of file bramblePasciak.hh.

◆ W_

concepts::Operator<Real>& vectorial::BramblePasciak::W_

private

Preconditioner for A.

Definition at line 100 of file bramblePasciak.hh.

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

vectorial/bramblePasciak.hh

vectorial::BramblePasciak Class Reference

Public Types

Public Member Functions

Protected Member Functions

Protected Attributes

Private Attributes

Detailed Description

Member Typedef Documentation

◆ c_type

◆ r_type

◆ type

Constructor & Destructor Documentation

◆ BramblePasciak()

Member Function Documentation

◆ dimX()

◆ dimY()

◆ epsilon()

◆ info()

◆ iterations()

◆ operator()() [1/5]

◆ operator()() [2/5]

◆ operator()() [3/5]

◆ operator()() [4/5]

◆ operator()() [5/5]

◆ show_messages()

Member Data Documentation

◆ A_

◆ B_

◆ bdim_

◆ Bt_

◆ C_

◆ dimX_

◆ dimY_

◆ eps_

◆ it_

◆ maxeps_

◆ maxit_

◆ relres_

◆ W_