Reference documentation for deal.II version Git fcc0cac376 2021-05-09 08:07:52 +0200
SolverGMRES< VectorType > Class Template Reference

#include <deal.II/lac/solver_gmres.h>

Inheritance diagram for SolverGMRES< VectorType >:
[legend]

## Public Types

using vector_type = VectorType

## Public Member Functions

SolverGMRES (const SolverGMRES< VectorType > &)=delete

template<typename MatrixType , typename PreconditionerType >
void solve (const MatrixType &A, VectorType &x, const VectorType &b, const PreconditionerType &preconditioner)

boost::signals2::connection connect_condition_number_slot (const std::function< void(double)> &slot, const bool every_iteration=false)

boost::signals2::connection connect_eigenvalues_slot (const std::function< void(const std::vector< std::complex< double >> &)> &slot, const bool every_iteration=false)

boost::signals2::connection connect_hessenberg_slot (const std::function< void(const FullMatrix< double > &)> &slot, const bool every_iteration=true)

boost::signals2::connection connect_krylov_space_slot (const std::function< void(const internal::SolverGMRESImplementation::TmpVectors< VectorType > &)> &slot)

boost::signals2::connection connect_re_orthogonalization_slot (const std::function< void(int)> &slot)

boost::signals2::connection connect (const std::function< SolverControl::State(const unsigned int iteration, const double check_value, const VectorType &current_iterate)> &slot)

template<class Archive >
void serialize (Archive &ar, const unsigned int version)

Subscriptor functionality

Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class.

void subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

void unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

unsigned int n_subscriptions () const

template<typename StreamType >
void list_subscribers (StreamType &stream) const

void list_subscribers () const

## Static Public Member Functions

static ::ExceptionBaseExcTooFewTmpVectors (int arg1)

static ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBaseExcNoSubscriber (std::string arg1, std::string arg2)

## Protected Member Functions

virtual double criterion ()

void givens_rotation (Vector< double > &h, Vector< double > &b, Vector< double > &ci, Vector< double > &si, int col) const

## Static Protected Member Functions

static double modified_gram_schmidt (const internal::SolverGMRESImplementation::TmpVectors< VectorType > &orthogonal_vectors, const unsigned int dim, const unsigned int accumulated_iterations, VectorType &vv, Vector< double > &h, bool &re_orthogonalize, const boost::signals2::signal< void(int)> &re_orthogonalize_signal=boost::signals2::signal< void(int)>())

static void compute_eigs_and_cond (const FullMatrix< double > &H_orig, const unsigned int dim, const boost::signals2::signal< void(const std::vector< std::complex< double >> &)> &eigenvalues_signal, const boost::signals2::signal< void(const FullMatrix< double > &)> &hessenberg_signal, const boost::signals2::signal< void(double)> &cond_signal)

## Protected Attributes

boost::signals2::signal< void(double)> condition_number_signal

boost::signals2::signal< void(double)> all_condition_numbers_signal

boost::signals2::signal< void(const std::vector< std::complex< double >> &)> eigenvalues_signal

boost::signals2::signal< void(const std::vector< std::complex< double >> &)> all_eigenvalues_signal

boost::signals2::signal< void(const FullMatrix< double > &)> hessenberg_signal

boost::signals2::signal< void(const FullMatrix< double > &)> all_hessenberg_signal

boost::signals2::signal< void(const internal::SolverGMRESImplementation::TmpVectors< VectorType > &)> krylov_space_signal

boost::signals2::signal< void(int)> re_orthogonalize_signal

FullMatrix< doubleH

FullMatrix< doubleH1

GrowingVectorMemory< VectorTypestatic_vector_memory

VectorMemory< VectorType > & memory

boost::signals2::signal< SolverControl::State(const unsigned int iteration, const double check_value, const VectorType &current_iterate), StateCombineriteration_status

## Detailed Description

### template<class VectorType = Vector<double>> class SolverGMRES< VectorType >

Implementation of the Restarted Preconditioned Direct Generalized Minimal Residual Method. The stopping criterion is the norm of the residual.

The AdditionalData structure contains the number of temporary vectors used. The size of the Arnoldi basis is this number minus three. Additionally, it allows you to choose between right or left preconditioning. The default is left preconditioning. Finally it includes a flag indicating whether or not the default residual is used as stopping criterion.

### Left versus right preconditioning

AdditionalData allows you to choose between left and right preconditioning. As expected, this switches between solving for the systems P-1A and AP-1, respectively.

A second consequence is the type of residual which is used to measure convergence. With left preconditioning, this is the preconditioned residual, while with right preconditioning, it is the residual of the unpreconditioned system.

Optionally, this behavior can be overridden by using the flag AdditionalData::use_default_residual. A true value refers to the behavior described in the previous paragraph, while false reverts it. Be aware though that additional residuals have to be computed in this case, impeding the overall performance of the solver.

### The size of the Arnoldi basis

The maximal basis size is controlled by AdditionalData::max_n_tmp_vectors, and it is this number minus 2. If the number of iteration steps exceeds this number, all basis vectors are discarded and the iteration starts anew from the approximation obtained so far.

Note that the minimizing property of GMRes only pertains to the Krylov space spanned by the Arnoldi basis. Therefore, restarted GMRes is not minimizing anymore. The choice of the basis length is a trade- off between memory consumption and convergence speed, since a longer basis means minimization over a larger space.

For the requirements on matrices and vectors in order to work with this class, see the documentation of the Solver base class.

### Observing the progress of linear solver iterations

The solve() function of this class uses the mechanism described in the Solver base class to determine convergence. This mechanism can also be used to observe the progress of the iteration.

### Eigenvalue and condition number estimates

This class can estimate eigenvalues and condition number during the solution process. This is done by creating the Hessenberg matrix during the inner iterations. The eigenvalues are estimated as the eigenvalues of the Hessenberg matrix and the condition number is estimated as the ratio of the largest and smallest singular value of the Hessenberg matrix. The estimates can be obtained by connecting a function as a slot using connect_condition_number_slot and connect_eigenvalues_slot. These slots will then be called from the solver with the estimates as argument.

Definition at line 175 of file solver_gmres.h.

## ◆ vector_type

template<class VectorType = Vector<double>>
 using SolverBase< VectorType >::vector_type = VectorType
inherited

An alias for the underlying vector type

Definition at line 347 of file solver.h.

## ◆ SolverGMRES() [1/3]

template<class VectorType = Vector<double>>
 SolverGMRES< VectorType >::SolverGMRES ( SolverControl & cn, VectorMemory< VectorType > & mem, const AdditionalData & data = AdditionalData() )

Constructor.

## ◆ SolverGMRES() [2/3]

template<class VectorType = Vector<double>>
 SolverGMRES< VectorType >::SolverGMRES ( SolverControl & cn, const AdditionalData & data = AdditionalData() )

Constructor. Use an object of type GrowingVectorMemory as a default to allocate memory.

## ◆ SolverGMRES() [3/3]

template<class VectorType = Vector<double>>
 SolverGMRES< VectorType >::SolverGMRES ( const SolverGMRES< VectorType > & )
delete

The copy constructor is deleted.

## ◆ solve()

template<class VectorType = Vector<double>>
template<typename MatrixType , typename PreconditionerType >
 void SolverGMRES< VectorType >::solve ( const MatrixType & A, VectorType & x, const VectorType & b, const PreconditionerType & preconditioner )

Solve the linear system $$Ax=b$$ for x.

## ◆ connect_condition_number_slot()

template<class VectorType = Vector<double>>
 boost::signals2::connection SolverGMRES< VectorType >::connect_condition_number_slot ( const std::function< void(double)> & slot, const bool every_iteration = false )

Connect a slot to retrieve the estimated condition number. Called on each outer iteration if every_iteration=true, otherwise called once when iterations are ended (i.e., either because convergence has been achieved, or because divergence has been detected).

## ◆ connect_eigenvalues_slot()

template<class VectorType = Vector<double>>
 boost::signals2::connection SolverGMRES< VectorType >::connect_eigenvalues_slot ( )

Connect a slot to retrieve the estimated eigenvalues. Called on each outer iteration if every_iteration=true, otherwise called once when iterations are ended (i.e., either because convergence has been achieved, or because divergence has been detected).

## ◆ connect_hessenberg_slot()

template<class VectorType = Vector<double>>
 boost::signals2::connection SolverGMRES< VectorType >::connect_hessenberg_slot ( const std::function< void(const FullMatrix< double > &)> & slot, const bool every_iteration = true )

Connect a slot to retrieve the Hessenberg matrix obtained by the projection of the initial matrix on the Krylov basis. Called on each outer iteration if every_iteration=true, otherwise called once when iterations are ended (i.e., either because convergence has been achieved, or because divergence has been detected).

## ◆ connect_krylov_space_slot()

template<class VectorType = Vector<double>>
 boost::signals2::connection SolverGMRES< VectorType >::connect_krylov_space_slot ( const std::function< void(const internal::SolverGMRESImplementation::TmpVectors< VectorType > &)> & slot )

Connect a slot to retrieve the basis vectors of the Krylov space generated by the Arnoldi algorithm. Called at once when iterations are completed (i.e., either because convergence has been achieved, or because divergence has been detected).

## ◆ connect_re_orthogonalization_slot()

template<class VectorType = Vector<double>>
 boost::signals2::connection SolverGMRES< VectorType >::connect_re_orthogonalization_slot ( const std::function< void(int)> & slot )

Connect a slot to retrieve a notification when the vectors are re-orthogonalized.

## ◆ criterion()

template<class VectorType = Vector<double>>
 virtual double SolverGMRES< VectorType >::criterion ( )
protectedvirtual

Implementation of the computation of the norm of the residual.

## ◆ givens_rotation()

template<class VectorType = Vector<double>>
 void SolverGMRES< VectorType >::givens_rotation ( Vector< double > & h, Vector< double > & b, Vector< double > & ci, Vector< double > & si, int col ) const
protected

Transformation of an upper Hessenberg matrix into tridiagonal structure by givens rotation of the last column

## ◆ modified_gram_schmidt()

template<class VectorType = Vector<double>>
 static double SolverGMRES< VectorType >::modified_gram_schmidt ( const internal::SolverGMRESImplementation::TmpVectors< VectorType > & orthogonal_vectors, const unsigned int dim, const unsigned int accumulated_iterations, VectorType & vv, Vector< double > & h, bool & re_orthogonalize, const boost::signals2::signal< void(int)> & re_orthogonalize_signal = boost::signals2::signal< void(int)>() )
staticprotected

Orthogonalize the vector vv against the dim (orthogonal) vectors given by the first argument using the modified Gram-Schmidt algorithm. The factors used for orthogonalization are stored in h. The boolean re_orthogonalize specifies whether the modified Gram-Schmidt algorithm should be applied twice. The algorithm checks loss of orthogonality in the procedure every fifth step and sets the flag to true in that case. All subsequent iterations use re-orthogonalization. Calls the signal re_orthogonalize_signal if it is connected.

## ◆ compute_eigs_and_cond()

template<class VectorType = Vector<double>>
 static void SolverGMRES< VectorType >::compute_eigs_and_cond ( const FullMatrix< double > & H_orig, const unsigned int dim )
staticprotected

Estimates the eigenvalues from the Hessenberg matrix, H_orig, generated during the inner iterations. Uses these estimate to compute the condition number. Calls the signals eigenvalues_signal and cond_signal with these estimates as arguments.

## ◆ connect()

template<class VectorType >
 boost::signals2::connection SolverBase< VectorType >::connect ( const std::function< SolverControl::State(const unsigned int iteration, const double check_value, const VectorType ¤t_iterate)> & slot )
inlineinherited

Connect a function object that will be called periodically within iterative solvers. This function is used to attach monitors to iterative solvers, either to determine when convergence has happened, or simply to observe the progress of an iteration. See the documentation of this class for more information.

Parameters
 slot A function object specified here will, with each call, receive the number of the current iteration, the value that is used to check for convergence (typically the residual of the current iterate with respect to the linear system to be solved) and the currently best available guess for the current iterate. Note that some solvers do not update the approximate solution in every iteration but only after convergence or failure has been determined (GMRES is an example); in such cases, the vector passed as the last argument to the signal is simply the best approximate at the time the signal is called, but not the vector that will be returned if the signal's return value indicates that the iteration should be terminated. The function object must return a SolverControl::State value that indicates whether the iteration should continue, has failed, or has succeeded. The results of all connected functions will then be combined to determine what should happen with the iteration.
Returns
A connection object that represents the connection from the signal to the function object. It can be used to disconnect the function object again from the signal. See the documentation of the BOOST Signals2 library for more information on connection management.

Definition at line 554 of file solver.h.

## ◆ subscribe()

 void Subscriptor::subscribe ( std::atomic< bool > *const validity, const std::string & identifier = "" ) const
inherited

Subscribes a user of the object by storing the pointer validity. The subscriber may be identified by text supplied as identifier.

Definition at line 136 of file subscriptor.cc.

## ◆ unsubscribe()

 void Subscriptor::unsubscribe ( std::atomic< bool > *const validity, const std::string & identifier = "" ) const
inherited

Unsubscribes a user from the object.

Note
The identifier and the validity pointer must be the same as the one supplied to subscribe().

Definition at line 156 of file subscriptor.cc.

## ◆ n_subscriptions()

 unsigned int Subscriptor::n_subscriptions ( ) const
inlineinherited

Return the present number of subscriptions to this object. This allows to use this class for reference counted lifetime determination where the last one to unsubscribe also deletes the object.

Definition at line 301 of file subscriptor.h.

## ◆ list_subscribers() [1/2]

template<typename StreamType >
 void Subscriptor::list_subscribers ( StreamType & stream ) const
inlineinherited

List the subscribers to the input stream.

Definition at line 318 of file subscriptor.h.

## ◆ list_subscribers() [2/2]

 void Subscriptor::list_subscribers ( ) const
inherited

List the subscribers to deallog.

Definition at line 204 of file subscriptor.cc.

## ◆ serialize()

template<class Archive >
 void Subscriptor::serialize ( Archive & ar, const unsigned int version )
inlineinherited

Read or write the data of this object to or from a stream for the purpose of serialization using the BOOST serialization library.

This function does not actually serialize any of the member variables of this class. The reason is that what this class stores is only who subscribes to this object, but who does so at the time of storing the contents of this object does not necessarily have anything to do with who subscribes to the object when it is restored. Consequently, we do not want to overwrite the subscribers at the time of restoring, and then there is no reason to write the subscribers out in the first place.

Definition at line 310 of file subscriptor.h.

## Member Data Documentation

template<class VectorType = Vector<double>>
protected

Includes the maximum number of tmp vectors.

Definition at line 317 of file solver_gmres.h.

## ◆ condition_number_signal

template<class VectorType = Vector<double>>
 boost::signals2::signal SolverGMRES< VectorType >::condition_number_signal
protected

Signal used to retrieve the estimated condition number. Called once when all iterations are ended.

Definition at line 323 of file solver_gmres.h.

## ◆ all_condition_numbers_signal

template<class VectorType = Vector<double>>
 boost::signals2::signal SolverGMRES< VectorType >::all_condition_numbers_signal
protected

Signal used to retrieve the estimated condition numbers. Called on each outer iteration.

Definition at line 329 of file solver_gmres.h.

## ◆ eigenvalues_signal

template<class VectorType = Vector<double>>
 boost::signals2::signal> &)> SolverGMRES< VectorType >::eigenvalues_signal
protected

Signal used to retrieve the estimated eigenvalues. Called once when all iterations are ended.

Definition at line 336 of file solver_gmres.h.

## ◆ all_eigenvalues_signal

template<class VectorType = Vector<double>>
 boost::signals2::signal> &)> SolverGMRES< VectorType >::all_eigenvalues_signal
protected

Signal used to retrieve the estimated eigenvalues. Called on each outer iteration.

Definition at line 343 of file solver_gmres.h.

## ◆ hessenberg_signal

template<class VectorType = Vector<double>>
 boost::signals2::signal &)> SolverGMRES< VectorType >::hessenberg_signal
protected

Signal used to retrieve the Hessenberg matrix. Called once when all iterations are ended.

Definition at line 349 of file solver_gmres.h.

## ◆ all_hessenberg_signal

template<class VectorType = Vector<double>>
 boost::signals2::signal &)> SolverGMRES< VectorType >::all_hessenberg_signal
protected

Signal used to retrieve the Hessenberg matrix. Called on each outer iteration.

Definition at line 356 of file solver_gmres.h.

## ◆ krylov_space_signal

template<class VectorType = Vector<double>>
 boost::signals2::signal &)> SolverGMRES< VectorType >::krylov_space_signal
protected

Signal used to retrieve the Krylov space basis vectors. Called once when all iterations are ended.

Definition at line 364 of file solver_gmres.h.

## ◆ re_orthogonalize_signal

template<class VectorType = Vector<double>>
 boost::signals2::signal SolverGMRES< VectorType >::re_orthogonalize_signal
protected

Signal used to retrieve a notification when the vectors are re-orthogonalized.

Definition at line 370 of file solver_gmres.h.

## ◆ H

template<class VectorType = Vector<double>>
 FullMatrix SolverGMRES< VectorType >::H
protected

Projected system matrix

Definition at line 430 of file solver_gmres.h.

## ◆ H1

template<class VectorType = Vector<double>>
 FullMatrix SolverGMRES< VectorType >::H1
protected

Auxiliary matrix for inverting H

Definition at line 435 of file solver_gmres.h.

## ◆ static_vector_memory

template<class VectorType = Vector<double>>
 GrowingVectorMemory SolverBase< VectorType >::static_vector_memory
mutableprotectedinherited

A static vector memory object to be used whenever no such object has been given to the constructor.

Definition at line 415 of file solver.h.

## ◆ memory

template<class VectorType = Vector<double>>
 VectorMemory& SolverBase< VectorType >::memory
protectedinherited

A reference to an object that provides memory for auxiliary vectors.

Definition at line 420 of file solver.h.

## ◆ iteration_status

template<class VectorType = Vector<double>>
 boost::signals2::signal< SolverControl::State(const unsigned int iteration, const double check_value, const VectorType & current_iterate), StateCombiner> SolverBase< VectorType >::iteration_status
protectedinherited

A signal that iterative solvers can execute at the end of every iteration (or in an otherwise periodic fashion) to find out whether we should continue iterating or not. The signal may call one or more slots that each will make this determination by themselves, and the result over all slots (function calls) will be determined by the StateCombiner object.

The arguments passed to the signal are (i) the number of the current iteration; (ii) the value that is used to determine convergence (oftentimes the residual, but in other cases other quantities may be used as long as they converge to zero as the iterate approaches the solution of the linear system); and (iii) a vector that corresponds to the current best guess for the solution at the point where the signal is called. Note that some solvers do not update the approximate solution in every iteration but only after convergence or failure has been determined (GMRES is an example); in such cases, the vector passed as the last argument to the signal is simply the best approximate at the time the signal is called, but not the vector that will be returned if the signal's return value indicates that the iteration should be terminated.

Definition at line 471 of file solver.h.

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