Iterative refinement (IR) is an iterative method that uses another coarse method to approximate the error of the current solution via the current residual. More...

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

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

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

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Classes
class	Factory

struct	parameters_type

Public Types
using	value_type = ValueType

using	transposed_type = Ir< 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...

std::shared_ptr< const LinOp >	get_solver () const
	Returns the solver operator used as the inner solver. More...

void	set_solver (std::shared_ptr< const LinOp > new_solver)
	Sets the solver operator used as the inner solver. More...

Ir &	operator= (const Ir &)
	Copy-assigns an IR solver. More...

Ir &	operator= (Ir &&)
	Move-assigns an IR solver. More...

	Ir (const Ir &)
	Copy-constructs an IR solver. More...

	Ir (Ir &&)
	Move-constructs an IR solver. 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::EnableSolverBase< Ir< 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< Ir< 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...

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...

Friends
class	EnableApplyWithInitialGuess< Ir >

Detailed Description

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

Iterative refinement (IR) is an iterative method that uses another coarse method to approximate the error of the current solution via the current residual.

Moreover, it can be also considered as preconditioned Richardson iteration with relaxation factor = 1.

Let \( x_k \) be the approximation of the solution of \( A x = b \) after \( k \) iterations and let \( x \) denote the exact solution. The residual and the error of \( x_k \) are

\[ r_k = b - A x_k, \qquad e_k = x - x_k, \]

and they are linked by the residual equation \( A e_k = A x - A x_k = b - A x_k = r_k \). Knowing \( e_k \) exactly would give the exact solution in a single update \( x = x_k + e_k \). Instead of solving \( A e_k = r_k \) exactly, IR approximates \( e_k \) by \( \tilde e_k \) using a cheap inner solver, applies that correction with a relaxation factor \( \alpha \),

\[ x_{k+1} = x_k + \alpha \tilde e_k, \]

and repeats the process with the updated iterate. Written with the names used below, \( x_k \) is solution, \( r_k \) is residual and \( \tilde e_k \) is error:

solution = initial_guess
while not converged:
    residual = b - A solution
    error = solver(A, residual)
    solution = solution + relaxation_factor * error

With relaxation_factor equal to 1 (default), the solver is Iterative Refinement, with relaxation_factor equal to a value other than 1, the solver is a Richardson iteration, with possibility for additional preconditioning.

Assume \( \alpha = 1 \) and that solver has accuracy \( c \), i.e. \( \| e_k - \tilde e_k \| \le c \| e_k \| \). Then iterative refinement converges with a convergence rate of \( c \): from \( e_k - \tilde e_k = (x - x_k) - \tilde e_k = x - x_{k+1} \) it follows that \( \| x - x_{k+1} \| \le c \| x - x_k \| \).

Unless otherwise specified via the solver factory parameter, this implementation uses the identity operator (i.e. the solver that approximates the solution of a system \( A x = b \) by setting \( x := b \)) as the default inner solver. It leaves the residual unchanged, \( \tilde e_k = r_k \), so the iteration reduces to the Richardson iteration \( x_{k+1} = x_k + \alpha r_k \), which converges for every initial guess if and only if \( | 1 - \alpha \lambda | < 1 \) holds for every eigenvalue \( \lambda \) of \( A \).

References

Saad, Y. Iterative Methods for Sparse Linear Systems. 2nd ed. SIAM, 2003. https://doi.org/10.1137/1.9780898718003

Template Parameters

ValueType precision of matrix elements

Constructor & Destructor Documentation

◆ Ir() [1/2]

template<typename ValueType = default_precision>

gko::solver::Ir< ValueType >::Ir ( const Ir< ValueType > & )

Copy-constructs an IR solver.

Inherits the executor, shallow-copies inner solver, stopping criterion and system matrix.

◆ Ir() [2/2]

template<typename ValueType = default_precision>

gko::solver::Ir< ValueType >::Ir ( Ir< ValueType > && )

Move-constructs an IR solver.

Preserves the executor, moves inner solver, stopping criterion and system matrix. The moved-from object is empty (0x0 and nullptr inner solver, stopping criterion and system matrix)

Member Function Documentation

◆ apply_uses_initial_guess()

template<typename ValueType = default_precision>

bool gko::solver::Ir< 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.

References gko::solver::provided.

◆ conj_transpose()

template<typename ValueType = default_precision>

std::unique_ptr<LinOp> gko::solver::Ir< 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.

◆ get_solver()

template<typename ValueType = default_precision>

std::shared_ptr<const LinOp> gko::solver::Ir< ValueType >::get_solver ( ) const

inline

Returns the solver operator used as the inner solver.

Returns: the solver operator used as the inner solver

◆ operator=() [1/2]

template<typename ValueType = default_precision>

Ir& gko::solver::Ir< ValueType >::operator= ( const Ir< ValueType > & )

Copy-assigns an IR solver.

Preserves the executor, shallow-copies inner solver, stopping criterion and system matrix. If the executors mismatch, clones inner solver, stopping criterion and system matrix onto this executor.

◆ operator=() [2/2]

template<typename ValueType = default_precision>

Ir& gko::solver::Ir< ValueType >::operator= ( Ir< ValueType > && )

Move-assigns an IR solver.

Preserves the executor, moves inner solver, stopping criterion and system matrix. If the executors mismatch, clones inner solver, stopping criterion and system matrix onto this executor. The moved-from object is empty (0x0 and nullptr inner solver, stopping criterion and system matrix)

◆ parse()

template<typename ValueType = default_precision>

static parameters_type gko::solver::Ir< 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

◆ set_solver()

template<typename ValueType = default_precision>

void gko::solver::Ir< ValueType >::set_solver ( std::shared_ptr< const LinOp > new_solver )

Sets the solver operator used as the inner solver.

Parameters

new_solver the new inner solver

◆ transpose()

template<typename ValueType = default_precision>

std::unique_ptr<LinOp> gko::solver::Ir< 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/ir.hpp

Classes

Public Types

Public Member Functions

Static Public Member Functions

Friends

Detailed Description

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

Constructor & Destructor Documentation

◆ Ir() [1/2]

◆ Ir() [2/2]

Member Function Documentation

◆ apply_uses_initial_guess()

◆ conj_transpose()

◆ get_solver()

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ parse()

◆ set_solver()

◆ transpose()

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