Detailed Description

template<class T>
class gismo::gsMultiGridOp< T >

Multigrid preconditioner.

This class implements an generic geometric multigrid framework. This class implements a preconditioner which can be used, e.g., in an iterative solver, like gsConjugateGradient. If a simple multigrid iteration is required, use this class as a preconditioner in gsGradientMethod with step size 1.

The class expects transfer matrices, which can be constructed with the class gsGridHierarchy.

The intergrid transfers (restriction and prolongation) and the stiffness matrix are stored as gsLinearOperator, which means that also matrix-free variants are possible. If linear operators are provided, operators representing the siffness matrix have to be provided for all grid levles. If the integrid transfers and the stiffness matrix are provided as a sparse matrix, the matrices for the coarser levels are computed automatically based on the Galerkin principle.

For all levels, a smoother has to be provided by defining setSmoother().

The solver for the coarsest grid level is created automatically if not provided by the caller.

The solver can be configured to use V- or W-cycles using setNumCycles(). The number of pre- and post-smoothing steps can be configured with setNumPreSmooth() and setNumPostSmooth(), respectively.

Also full multigrid and cascadic multigrid algorithms for computing an initial guess are provided.

Inheritance diagram for gsMultiGridOp< T >:

Collaboration diagram for gsMultiGridOp< T >:

Public Types
typedef gsPreconditionerOp< T >	Base
	Direct base class.

typedef gsLinearOperator< T >::Ptr	BasePtr
	Base class.

typedef gsMatrix< T >	Matrix
	Matrix type.

typedef gsLinearOperator< T >::Ptr	OpPtr
	Shared pointer to gsLinearOperator.

typedef gsPreconditionerOp< T > ::Ptr	PrecondPtr
	Shared pointer to gsPreconditionerOp.

typedef memory::shared_ptr < gsMultiGridOp >	Ptr
	Shared pointer for gsMultiGridOp.

typedef gsSparseMatrix< T >	SpMatrix
	Sparse matrix type.

typedef memory::shared_ptr < SpMatrix >	SpMatrixPtr
	Smart pointer to sparse matrix type.

typedef gsSparseMatrix< T, RowMajor >	SpMatrixRowMajor
	Matrix type for transfers.

typedef memory::shared_ptr < SpMatrixRowMajor >	SpMatrixRowMajorPtr
	Smart pointer to matrix type for transfers.

typedef memory::unique_ptr < gsMultiGridOp >	uPtr
	Unique pointer for gsMultiGridOp.

Public Member Functions
void	apply (const gsMatrix< T > &input, gsMatrix< T > &x) const
	apply the operator on the input vector and store the result in x More...

void	cascadicMultiGrid (const std::vector< Matrix > &rhs, const std::vector< Matrix > &fixedValues, Matrix &result) const
	Perform one cascadic multigrid cycle and store the resulting solution vector in result. More...

T	coarseGridCorrectionDamping () const
	Get the damping of for the coarse-grid correction.

OpPtr	coarseSolver () const
	Get the solver for the coarsest problem (level 0)

index_t	cols () const override
	Returns the number of columns of the operator.

T	estimateLargestEigenvalueOfPreconditionedSystem (index_t steps=10) const
	Estimates the largest eigenvalue of \( PA \). More...

index_t	finestLevel () const
	The index of the finest level (0-based)

void	fullMultiGrid (const std::vector< Matrix > &rhs, const std::vector< Matrix > &dirichletIntp, Matrix &result) const
	Perform one full multigrid cycle and store the resulting solution vector in result. More...

	gsMultiGridOp (SpMatrix fineMatrix, std::vector< SpMatrixRowMajor > transferMatrices, OpPtr coarseSolver=OpPtr())
	Constructor. More...

	gsMultiGridOp (SpMatrixPtr fineMatrix, std::vector< SpMatrixRowMajorPtr > transferMatrices, OpPtr coarseSolver=OpPtr())
	Constructor. More...

	gsMultiGridOp (const std::vector< OpPtr > &ops, const std::vector< OpPtr > &prolongation, const std::vector< OpPtr > &restriction, OpPtr coarseSolver=OpPtr())
	Constructor for a matix-free variant. More...

const SpMatrix &	matrix (index_t lvl) const
	Stiffness matrix for given level.

const SpMatrix &	matrix () const
	Stiffness matrix for finest level.

void	multiGridStep (index_t level, const Matrix &rhs, Matrix &x) const
	Perform one multigrid cycle, starting from the given level. More...

index_t	nDofs (index_t lvl) const
	Number of dofs for the given level.

index_t	nDofs () const
	Number of dofs for the finest level.

index_t	numCycles (index_t lvl) const
	Get number of coarse grid steps to be applied.

index_t	numLevels () const
	Number of levels in the multigrid construction.

index_t	numOfSweeps ()
	Get the number of sweeps to be applied in the member function apply.

void	prolongVector (index_t lc, const Matrix &coarse, Matrix &fine) const
	Prolong a vector of coefficients from the level lc, given by coarse, to the next finer level.

void	restrictVector (index_t lf, const Matrix &fine, Matrix &coarse) const
	Restrict a vector of coefficients from the level lf, given by fine, to the next coarser level.

index_t	rows () const override
	Returns the number of rows of the operator.

void	setCoarseGridCorrectionDamping (T damping)
	Set the damping of for the coarse-grid correction.

void	setCoarseSolver (const OpPtr &sol)
	Set the solver for the coarsest problem (level 0)

void	setNumCycles (index_t n)

void	setNumCycles (index_t lvl, index_t n)

void	setNumOfSweeps (index_t n)
	Set the number of sweeps to be applied in the member function apply.

void	setNumPostSmooth (index_t n)
	Set number of post-smoothing steps to perform.

index_t	setNumPostSmooth () const
	Get number of post-smoothing steps to perform.

void	setNumPreSmooth (index_t n)
	Set number of pre-smoothing steps to perform.

index_t	setNumPreSmooth () const
	Get number of pre-smoothing steps to perform.

void	setOptions (const gsOptionList &opt) override
	Set the options based on a gsOptionList.

void	setSmoother (index_t lvl, const PrecondPtr &sm)

void	setSymmSmooth (bool s)

void	setUnderlyingOp (index_t lvl, OpPtr op)
	Set underlying operator (=stiffness matrix) for certain level.

PrecondPtr	smoother (index_t lvl) const

void	smoothingStep (index_t level, const Matrix &rhs, Matrix &x) const
	Apply smoothing steps on the corresponding level. More...

void	smoothingStep (const Matrix &rhs, Matrix &x) const
	Apply smoothing steps on finest grid. More...

void	solveCoarse (const Matrix &rhs, Matrix &result) const
	Solve the problem with direct solver on coarsest level.

void	step (const Matrix &rhs, Matrix &x) const override
	Perform one multigrid cycle. More...

void	stepT (const Matrix &rhs, Matrix &x) const override
	Perform one transposed multigrid cycle. More...

bool	symmSmooth () const
	Get symmetric post-smoothing option.

OpPtr	underlyingOp (index_t lvl) const
	Underlying operator (=stiffness matrix) for given level.

OpPtr	underlyingOp () const override
	Underlying operator (=stiffness matrix) for finest level.

Static Public Member Functions
static gsOptionList	defaultOptions ()
	Returns a list of default options.

static gsIdentityOp< T >	Identity (const index_t dim)
	Identity operator.

static uPtr	make (SpMatrix fineMatrix, std::vector< SpMatrixRowMajor > transferMatrices, OpPtr coarseSolver=OpPtr())
	Make function returning smart pointer. More...

static uPtr	make (SpMatrixPtr fineMatrix, std::vector< SpMatrixRowMajorPtr > transferMatrices, OpPtr coarseSolver=OpPtr())
	Make function returning smart pointer. More...

static uPtr	make (const std::vector< OpPtr > &ops, const std::vector< OpPtr > &prolongation, const std::vector< OpPtr > &restriction, OpPtr coarseSolver=OpPtr())
	Make function returning a shared pointer for a matix-free variant. More...

Private Member Functions
void	init (SpMatrixPtr fineMatrix, std::vector< SpMatrixRowMajorPtr > transferMatrices)
	Init function that is used by matrix based constructors.

void	initCoarseSolver () const
	Init solver on coarsest grid level.

Private Attributes
OpPtr	m_coarseSolver
	Solver for the coarsest-grid problem.

T	m_damping
	Damping for the coarse-grid correction.

index_t	m_nLevels
	Number of levels.

gsVector< index_t >	m_numCycles
	Number of coarse-grid steps (1=V-cycle, 2=W-cycle)

index_t	m_numPostSmooth
	Number of post-smoothing steps.

index_t	m_numPreSmooth
	Number of pre-smoothing steps.

std::vector< OpPtr >	m_ops
	Underlying operators (=stiffness matrix) on each level.

std::vector< OpPtr >	m_prolong
	Prolongation matrix for each grid transition.

std::vector< OpPtr >	m_restrict
	Restriction matrix for each grid transition.

std::vector< PrecondPtr >	m_smoother
	Smoothers on each level.

bool	m_symmSmooth
	Symmmetric smoothing.

Constructor & Destructor Documentation

gsMultiGridOp	(	SpMatrix	fineMatrix,
		std::vector< SpMatrixRowMajor >	transferMatrices,
		OpPtr	coarseSolver = `OpPtr()`
	)

Constructor.

Parameters

fineMatrix	Stiffness matrix on the finest grid
transferMatrices	Intergrid transfer matrices representing restriction and prolongation operators
coarseSolver	Linear operator representing the exact solver on the coarsest grid level, defaulted to a direct solver (PartialPivLUSolver)

gsMultiGridOp	(	SpMatrixPtr	fineMatrix,
		std::vector< SpMatrixRowMajorPtr >	transferMatrices,
		OpPtr	coarseSolver = `OpPtr()`
	)

Constructor.

Parameters

fineMatrix	Stiffness matrix (as smart pointers) on the finest grid
transferMatrices	Intergrid transfer matrices representing restriction and prolongation operators
coarseSolver	Linear operator representing the exact solver on the coarsest grid level, defaulted to a direct solver (PartialPivLUSolver)

gsMultiGridOp	(	const std::vector< OpPtr > &	ops,
		const std::vector< OpPtr > &	prolongation,
		const std::vector< OpPtr > &	restriction,
		OpPtr	coarseSolver = `OpPtr()`
	)

Constructor for a matix-free variant.

Parameters

ops	Linear operators representing the stiffness matrix on all levels
prolongation	Linear operators representing the prolongation operators
restriction	Linear operators representing the restriction operators
coarseSolver	Linear operator representing the exact solver on the coarsest grid level, defaulted to a direct solver (PartialPivLUSolver)

Member Function Documentation

void apply	(	const gsMatrix< T > &	input,
		gsMatrix< T > &	x
	)		const

inlinevirtualinherited

apply the operator on the input vector and store the result in x

Parameters

input	Input vector
x	result vector

Implements gsLinearOperator< T >.

Reimplemented in gsPreconditionerFromOp< T >.

void cascadicMultiGrid	(	const std::vector< Matrix > &	rhs,
		const std::vector< Matrix > &	fixedValues,
		Matrix &	result
	)		const

Perform one cascadic multigrid cycle and store the resulting solution vector in result.

Parameters

[in]	rhs	The right-hand-sides for all grid levels
[in]	fixedValues	The values for the fixed dofs (like eliminated Dirichlet values)
[out]	result	The destination for the result

T estimateLargestEigenvalueOfPreconditionedSystem ( index_t steps = 10 ) const

inlineinherited

Estimates the largest eigenvalue of \( PA \).

Parameters

steps Number of steps to be performed.

void fullMultiGrid	(	const std::vector< Matrix > &	rhs,
		const std::vector< Matrix > &	dirichletIntp,
		Matrix &	result
	)		const

Perform one full multigrid cycle and store the resulting solution vector in result.

Parameters

[in]	rhs	The right-hand-sides for all grid levels
[in]	fixedValues	The values for the fixed dofs (like eliminated Dirichlet values)
[out]	result	The destination for the result

static uPtr make	(	SpMatrix	fineMatrix,
		std::vector< SpMatrixRowMajor >	transferMatrices,
		OpPtr	coarseSolver = `OpPtr()`
	)

inlinestatic

Make function returning smart pointer.

Parameters

fineMatrix	Stiffness matrix on the finest grid
transferMatrices	Intergrid transfer matrices representing restriction and prolongation operators
coarseSolver	Linear operator representing the exact solver on the coarsest grid level, defaulted to a direct solver (PartialPivLUSolver)

static uPtr make	(	SpMatrixPtr	fineMatrix,
		std::vector< SpMatrixRowMajorPtr >	transferMatrices,
		OpPtr	coarseSolver = `OpPtr()`
	)

inlinestatic

Make function returning smart pointer.

Parameters

fineMatrix	Stiffness matrix (as smart pointers) on the finest grid
transferMatrices	Intergrid transfer matrices representing restriction and prolongation operators
coarseSolver	Linear operator representing the exact solver on the coarsest grid level, defaulted to a direct solver (PartialPivLUSolver)

static uPtr make	(	const std::vector< OpPtr > &	ops,
		const std::vector< OpPtr > &	prolongation,
		const std::vector< OpPtr > &	restriction,
		OpPtr	coarseSolver = `OpPtr()`
	)

inlinestatic

Make function returning a shared pointer for a matix-free variant.

Parameters

ops	Linear operators representing the stiffness matrix on all levels
prolongation	Linear operators representing the prolongation operators
restriction	Linear operators representing the restriction operators
coarseSolver	Linear operator representing the exact solver on the coarsest grid level, defaulted to a direct solver (PartialPivLUSolver)

void multiGridStep	(	index_t	level,
		const Matrix &	rhs,
		Matrix &	x
	)		const

Perform one multigrid cycle, starting from the given level.

Parameters

[in]	level	Level
[in]	rhs	Right-hand side
[in,out]	x	Current iterate vector

void setNumCycles ( index_t n )

inline

Set number of coarse grid steps to be applied

Parameters

n	Number of levels (1 for V-cycle, 2 for W-cycle)

void setNumCycles	(	index_t	lvl,
		index_t	n
	)

inline

Set number of coarse grid steps to be applied

Parameters

lvl	Level of coarser grid (multiGridStep on level lvl+1 calls n times multiGridStep for level lvl)
n	Number of calls (1 for V-cycle, 2 for W-cycle)

void setSmoother	(	index_t	lvl,
		const PrecondPtr &	sm
	)

Set the smoother

Parameters

lvl	The corresponding level
sm	The smoother

void setSymmSmooth ( bool s )

inline

Set symmetric post-smoothing option

Parameters

s	Iff true (default), stepT is called for post-smoothing

PrecondPtr smoother ( index_t lvl ) const

inline

Get the smoother

Parameters

lvl	The corresponding level

void smoothingStep	(	index_t	level,
		const Matrix &	rhs,
		Matrix &	x
	)		const

Apply smoothing steps on the corresponding level.

Parameters

[in]	level	Level
[in]	rhs	Right hand side
[in,out]	x	Current iterate vector

void smoothingStep	(	const Matrix &	rhs,
		Matrix &	x
	)		const

inline

Apply smoothing steps on finest grid.

Parameters

[in]	rhs	Right hand side
[in,out]	x	Current iterate vector

void step	(	const Matrix &	rhs,
		Matrix &	x
	)		const

inlineoverridevirtual

Perform one multigrid cycle.

Parameters

[in]	rhs	Right-hand side
[in,out]	x	Current iterate vector

Implements gsPreconditionerOp< T >.

void stepT	(	const Matrix &	rhs,
		Matrix &	x
	)		const

inlineoverridevirtual

Perform one transposed multigrid cycle.

Parameters

[in]	rhs	Right-hand side
[in,out]	x	Current iterate vector

This assumes that the stiffness matrix is symmetric, that SymmSmooth is true and that the smoothers implement stepT correctly.

Reimplemented from gsPreconditionerOp< T >.

Detailed Description

template<class T> class gismo::gsMultiGridOp< T >

Public Types

Public Member Functions

Static Public Member Functions

Private Member Functions

Private Attributes

Constructor & Destructor Documentation

Member Function Documentation

template<class T>
class gismo::gsMultiGridOp< T >