PIPE_CG or the pipelined conjugate gradient method is an iterative type Krylov subspace method which is suitable for symmetric positive definite methods. More...

#include <ginkgo/core/solver/pipe_cg.hpp>

Inheritance diagram for gko::solver::PipeCg< ValueType >:

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Collaboration diagram for gko::solver::PipeCg< ValueType >:

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Classes
class	Factory

struct	parameters_type

Public Types
using	value_type = ValueType

using	transposed_type = PipeCg< ValueType >

Public Member Functions
std::unique_ptr< LinOp >	transpose () const override
	Returns a LinOp representing the transpose of the Transposable object. More...

std::unique_ptr< LinOp >	conj_transpose () const override
	Returns a LinOp representing the conjugate transpose of the Transposable object. More...

bool	apply_uses_initial_guess () const override
	Return true as iterative solvers use the data in x as an initial guess. More...

const parameters_type &	get_parameters () const

Public Member Functions inherited from gko::LinOp
void	apply (ptr_param< const LinOp > b, ptr_param< LinOp > x) const
	Applies a linear operator to a vector (or a sequence of vectors). More...

void	apply (ptr_param< const LinOp > alpha, ptr_param< const LinOp > b, ptr_param< const LinOp > beta, ptr_param< LinOp > x) const
	Performs the operation x = alpha * op(b) + beta * x. More...

const dim< 2 > &	get_size () const noexcept
	Returns the size of the operator. More...

LinOp &	operator= (const LinOp &)=default
	Copy-assigns a LinOp. More...

LinOp &	operator= (LinOp &&other)
	Move-assigns a LinOp. More...

	LinOp (const LinOp &)=default
	Copy-constructs a LinOp. More...

	LinOp (LinOp &&other)
	Move-constructs a LinOp. More...

Public Member Functions inherited from gko::PolymorphicObject
virtual void	validate_data () const
	Throws gko::InvalidData exception if we found the data inside the object does not fulfill certain property up to our knowledge.

PolymorphicObject &	operator= (const PolymorphicObject &)

std::shared_ptr< const Executor >	get_executor () const noexcept
	Returns the Executor of the object. More...

Public Member Functions inherited from gko::log::EnableLogging< PolymorphicObject >
void	add_logger (std::shared_ptr< const Logger > logger) override

void	remove_logger (const Logger *logger) override

void	remove_logger (ptr_param< const Logger > logger)

const std::vector< std::shared_ptr< const Logger > > &	get_loggers () const override

void	clear_loggers () override

Public Member Functions inherited from gko::log::Loggable
void	remove_logger (ptr_param< const Logger > logger)

Public Member Functions inherited from gko::solver::EnablePreconditionedIterativeSolver< ValueType, PipeCg< ValueType > >
	EnablePreconditionedIterativeSolver (std::shared_ptr< const LinOp > system_matrix, std::shared_ptr< const stop::CriterionFactory > stop_factory, std::shared_ptr< const LinOp > preconditioner)

	EnablePreconditionedIterativeSolver (std::shared_ptr< const LinOp > system_matrix, const FactoryParameters &params)

Public Member Functions inherited from gko::solver::EnableSolverBase< PipeCg< ValueType > >
EnableSolverBase &	operator= (const EnableSolverBase &other)
	Creates a shallow copy of the provided system matrix, clones it onto this executor if executors don't match.

EnableSolverBase &	operator= (EnableSolverBase &&other)
	Moves the provided system matrix, clones it onto this executor if executors don't match. More...

	EnableSolverBase (std::shared_ptr< const LinOp > system_matrix)

	EnableSolverBase (const EnableSolverBase &other)
	Creates a shallow copy of the provided system matrix.

	EnableSolverBase (EnableSolverBase &&other)
	Moves the provided system matrix. More...

int	get_num_workspace_ops () const override

std::vector< std::string >	get_workspace_op_names () const override

std::vector< int >	get_workspace_scalars () const override
	Returns the IDs of all scalars (workspace vectors with system dimension-independent size, usually 1 x num_rhs).

std::vector< int >	get_workspace_vectors () const override
	Returns the IDs of all vectors (workspace vectors with system dimension-dependent size, usually system_matrix_size x num_rhs).

Public Member Functions inherited from gko::solver::SolverBase< LinOp >
std::shared_ptr< const LinOp >	get_system_matrix () const
	Returns the system matrix, with its concrete type, used by the solver. More...

Public Member Functions inherited from gko::solver::EnableIterativeBase< PipeCg< ValueType > >
EnableIterativeBase &	operator= (const EnableIterativeBase &other)
	Creates a shallow copy of the provided stopping criterion, clones it onto this executor if executors don't match.

EnableIterativeBase &	operator= (EnableIterativeBase &&other)
	Moves the provided stopping criterion, clones it onto this executor if executors don't match. More...

	EnableIterativeBase (std::shared_ptr< const stop::CriterionFactory > stop_factory)

	EnableIterativeBase (const EnableIterativeBase &other)
	Creates a shallow copy of the provided stopping criterion.

	EnableIterativeBase (EnableIterativeBase &&other)
	Moves the provided stopping criterion. More...

void	set_stop_criterion_factory (std::shared_ptr< const stop::CriterionFactory > new_stop_factory) override
	Sets the stopping criterion of the solver. More...

Public Member Functions inherited from gko::solver::IterativeBase
std::shared_ptr< const stop::CriterionFactory >	get_stop_criterion_factory () const
	Gets the stopping criterion factory of the solver. More...

Public Member Functions inherited from gko::solver::EnablePreconditionable< PipeCg< ValueType > >
void	set_preconditioner (std::shared_ptr< const LinOp > new_precond) override
	Sets the preconditioner operator used by the Preconditionable. More...

EnablePreconditionable &	operator= (const EnablePreconditionable &other)
	Creates a shallow copy of the provided preconditioner, clones it onto this executor if executors don't match.

EnablePreconditionable &	operator= (EnablePreconditionable &&other)
	Moves the provided preconditioner, clones it onto this executor if executors don't match. More...

	EnablePreconditionable (std::shared_ptr< const LinOp > preconditioner)

	EnablePreconditionable (const EnablePreconditionable &other)
	Creates a shallow copy of the provided preconditioner.

	EnablePreconditionable (EnablePreconditionable &&other)
	Moves the provided preconditioner. More...

Public Member Functions inherited from gko::Preconditionable
virtual std::shared_ptr< const LinOp >	get_preconditioner () const
	Returns the preconditioner operator used by the Preconditionable. More...

Static Public Member Functions
static auto	build () -> decltype(Factory ::create())

static parameters_type	parse (const config::pnode &config, const config::registry &context, const config::type_descriptor &td_for_child=config::make_type_descriptor< ValueType >())
	Create the parameters from the property_tree. More...

Detailed Description

template<typename ValueType = default_precision>
class gko::solver::PipeCg< ValueType >

PIPE_CG or the pipelined conjugate gradient method is an iterative type Krylov subspace method which is suitable for symmetric positive definite methods.

It improves upon the CG method by allowing computation of inner products and norms to be overlapped with operator and preconditioner application. The pipelined method scales up to the 10^6 nodes of the assumed exascale machine, while its standard counterpart level off about one order of magnitude earlier, as suggested in the referenced paper (see below).

Mathematically the iterates \( x_k \) are the same as those produced by CG: the algorithm minimizes the energy-norm error over the Krylov subspace \( x_0 + \mathcal{K}_k(A, r_0) \) and uses the same Fletcher-Reeves \( \beta_k \). CG, however, needs two global reductions per iteration that cannot be merged: \( \rho_k = \langle r_k, z_k \rangle \) is needed to build \( p_k \), and only afterwards can \( q_k = A p_k \) and the second reduction \( \langle p_k, q_k \rangle \) be computed.

The pipelined variant removes that second synchronization point by carrying an extra vector \( w_k = A z_k \) — the operator applied to the preconditioned residual \( z_k = M r_k \), which plain CG never forms. It gives access to \( \delta_k = \langle w_k, z_k \rangle = \langle A z_k, z_k \rangle \), from which the denominator of \( \alpha_k \) follows by a recurrence instead of a second reduction:

\[ \langle p_k, A p_k \rangle = \delta_k - \left| \frac{\rho_k}{\rho_{k-1}} \right|^2 \langle p_{k-1}, A p_{k-1} \rangle . \]

Because \( \rho_k \) and \( \delta_k \) are available at the same point of the iteration, they are computed in a single global reduction that can be overlapped with the operator and preconditioner applies updating the auxiliary vectors \( w_k \), \( q_k = A p_k \), \( m_k = M w_k \) and \( n_k = A m_k \), all of which are advanced by short recurrences. In exact arithmetic the trajectory is identical to CG; in finite arithmetic the decoupling of dependencies amplifies round-off and the residuals can deviate from the classical iterates.

Possible issues:

Numerical instability: Due to the rearrangement of the operations, the method is known to be less stable than standard PCG.
The method is suitable for cases where a large number of iterations need to be performed and when the solver is distributed over a large number of distributed nodes. The advantage of lesser number of reductions that need to be performed comes at the cost of increased vector operations, and the cost of increased storage of vectors.
As the CG itself, this method performs very well for symmetric positive definite matrices but it is in general not suitable for general matrices.

References

Sanan, P., Schnepp, S. M., May, D. A. Pipelined, Flexible Krylov Subspace Methods. SIAM Journal on Scientific Computing, 38 (5), 2016. https://doi.org/10.1137/15M1049130

Template Parameters

ValueType precision of matrix elements

Member Function Documentation

◆ apply_uses_initial_guess()

template<typename ValueType = default_precision>

bool gko::solver::PipeCg< ValueType >::apply_uses_initial_guess ( ) const

inlineoverridevirtual

Return true as iterative solvers use the data in x as an initial guess.

Returns: true as iterative solvers use the data in x as an initial guess.

Reimplemented from gko::LinOp.

◆ conj_transpose()

template<typename ValueType = default_precision>

std::unique_ptr<LinOp> gko::solver::PipeCg< ValueType >::conj_transpose ( ) const

overridevirtual

Returns a LinOp representing the conjugate transpose of the Transposable object.

Returns: a pointer to the new conjugate transposed object

Implements gko::Transposable.

◆ parse()

template<typename ValueType = default_precision>

static parameters_type gko::solver::PipeCg< ValueType >::parse	(	const config::pnode &	config,
		const config::registry &	context,
		const config::type_descriptor &	td_for_child = `config::make_type_descriptor< ValueType >()`
	)

static

Create the parameters from the property_tree.

Because this is directly tied to the specific type, the value/index type settings within config are ignored and type_descriptor is only used for children configs.

Parameters

config	the property tree for setting
context	the registry
td_for_child	the type descriptor for children configs. The default uses the value type of this class.

Returns: parameters

◆ transpose()

template<typename ValueType = default_precision>

std::unique_ptr<LinOp> gko::solver::PipeCg< ValueType >::transpose ( ) const

overridevirtual

Returns a LinOp representing the transpose of the Transposable object.

Returns: a pointer to the new transposed object

Implements gko::Transposable.

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

ginkgo/core/solver/pipe_cg.hpp

Classes

Public Types

Public Member Functions

Static Public Member Functions

Detailed Description

template<typename ValueType = default_precision> class gko::solver::PipeCg< ValueType >

Member Function Documentation

◆ apply_uses_initial_guess()

◆ conj_transpose()

◆ parse()

◆ transpose()

template<typename ValueType = default_precision>
class gko::solver::PipeCg< ValueType >