Uzawa algorithm with conjugate directions for generalized saddle point problems. More...

#include <cgUzawa.hh>

Inheritance diagram for vectorial::CGUzawa:

Collaboration diagram for vectorial::CGUzawa:

Public Types
typedef Cmplxtype< Real >::type	c_type
	Real type of data type.
typedef Realtype< Real >::type	r_type
	Real type of data type.
typedef Real	type
	Type of data, e.g. matrix entries.
Public Member Functions
	CGUzawa (concepts::Operator< Real > &A, concepts::Operator< Real > &B, concepts::Operator< Real > &Bt, concepts::Operator< Real > &C, concepts::Operator< Real > &Ai, Real maxeps, int maxit=0, uint relres=false)
	Constructor.
virtual const uint	dimX () const
	Returns the size of the image space of the operator (number of rows of the corresponding matrix)
virtual const uint	dimY () const
	Returns the size of the source space of the operator (number of columns of the corresponding matrix)
Real	epsilon () const
	Returns the residual.
uint	iterations () const
	Returns the number of (outer) iterations.
virtual void	operator() (const Function< c_type > &fncY, Function< c_type > &fncX)
	Application operator for complex function `fncY`.
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< r_type > &fncY, Function< Real > &fncX)
	Application operator for real function `fncY`.
Protected Member Functions
std::ostream &	info (std::ostream &os) const
	Returns information in an output stream.
Protected Attributes
uint	dimX_
	Dimension of image space and the source space.
uint	dimY_
Private Attributes
concepts::Operator< Real > &	A_
	Upper left submatrix.
concepts::Operator< Real > &	Ai_
	Solver for A.
concepts::Operator< Real > &	B_
	Upper right submatrix.
uint	bdim_ [2]
	Dimensions of partly matrices (in image and source space)
concepts::Operator< Real > &	Bt_
	Lower left submatrix.
concepts::Operator< Real > &	C_
	Lower right submatrix.
Real	eps_
	Current residual.
uint	it_
	Number of iterations.
Real	maxeps_
	Convergence criterion.
uint	maxit_
	Maximal number of iterations until abortion.
bool	relres_
	false: absolute residual, true: relative residual

Detailed Description

Uzawa algorithm with conjugate directions 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 eliminate $U$ from the above equation, resulting in the equation $(B^{\top} A^{-1} B + C)P = B^{\top} A^{-1}L_u - L_p$ . This reduced system is solved with conjugate gradients. The action of A^-1 is a suitable solver that has to be provided at construction time. Then, $U$ is obtained from $A \ U = L_u - B \ P$ . 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:: D. Braess: Finite Elemente: Theorie, schnelle Löser und Anwendungen in der Elastitzitätstheorie. Springer, Berlin, 1997.

Definition at line 49 of file cgUzawa.hh.

Member Typedef Documentation

typedef Cmplxtype<Real >::type concepts::Operator< Real >::c_type [inherited]

Real type of data type.

Reimplemented in concepts::VecOperator< Real >, concepts::DenseMatrix< Real >, concepts::DiagonalMatrix< Real >, concepts::Matrix< Real >, concepts::Permutation< Real >, and concepts::SparseMatrix< Real >.

Definition at line 47 of file compositions.hh.

typedef Realtype<Real >::type concepts::Operator< Real >::r_type [inherited]

Real type of data type.

Reimplemented in concepts::VecOperator< Real >, concepts::DenseMatrix< Real >, concepts::DiagonalMatrix< Real >, concepts::Matrix< Real >, concepts::Permutation< Real >, and concepts::SparseMatrix< Real >.

Definition at line 45 of file compositions.hh.

typedef Real concepts::Operator< Real >::type [inherited]

Type of data, e.g. matrix entries.

Definition at line 43 of file compositions.hh.

Constructor & Destructor Documentation

vectorial::CGUzawa::CGUzawa	(	concepts::Operator< Real > &	A,
		concepts::Operator< Real > &	B,
		concepts::Operator< Real > &	Bt,
		concepts::Operator< Real > &	C,
		concepts::Operator< Real > &	Ai,
		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
Ai	Solver for A
spc	Entire vectorial space
maxeps	Maximal residual
maxit	Maximal number of iterations
relres	Relative residual

Member Function Documentation

virtual const uint concepts::Operator< Real >::dimX ( ) const [inline, virtual, inherited]

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

Examples:: hpFEM2d-simple.cc, and hpFEM2d.cc.

Definition at line 87 of file compositions.hh.

virtual const uint concepts::Operator< Real >::dimY ( ) const [inline, virtual, inherited]

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

Definition at line 92 of file compositions.hh.

Real vectorial::CGUzawa::epsilon ( ) const [inline]

Returns the residual.

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

Definition at line 80 of file cgUzawa.hh.

std::ostream& vectorial::CGUzawa::info ( std::ostream & os ) const [protected, virtual]

Returns information in an output stream.

Reimplemented from concepts::Operator< Real >.

uint vectorial::CGUzawa::iterations ( ) const [inline]

Returns the number of (outer) iterations.

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

Definition at line 75 of file cgUzawa.hh.

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

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 >, concepts::DenseMatrix< Real >, concepts::DiagonalMatrix< Real >, concepts::Matrix< Real >, concepts::Permutation< Real >, and concepts::SparseMatrix< Real >.

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

virtual void vectorial::CGUzawa::operator()	(	const concepts::Function< Real > &	fncY,
		concepts::Function< Real > &	fncX
	)		`[virtual]`

virtual void concepts::Operator< Real >::operator()	(	const Function< r_type > &	fncY,
		Function< Real > &	fncX
	)		`[virtual, inherited]`

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 >, concepts::DenseMatrix< Real >, concepts::DiagonalMatrix< Real >, concepts::Matrix< Real >, concepts::Permutation< Real >, and concepts::SparseMatrix< Real >.