Reference documentation for deal.II version 9.4.0
SolverQMRS< VectorType > Class Template Reference

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

Inheritance diagram for SolverQMRS< VectorType >:
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## Classes

struct  IterationResult

## Public Types

using vector_type = Vector< double >

## Public Member Functions

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

virtual void print_vectors (const unsigned int step, const VectorType &x, const VectorType &r, const VectorType &d) const

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

## Protected Attributes

GrowingVectorMemory< Vector< double > > static_vector_memory

VectorMemory< Vector< double > > & memory

boost::signals2::signal< SolverControl::State(const unsigned int iteration, const double check_value, const Vector< double > &current_iterate), StateCombiner > iteration_status

## Private Member Functions

template<typename MatrixType , typename PreconditionerType >
IterationResult iterate (const MatrixType &A, VectorType &x, const VectorType &b, const PreconditionerType &preconditioner, VectorType &r, VectorType &u, VectorType &q, VectorType &t, VectorType &d)

## Private Attributes

unsigned int step

## Subscriptor functionality

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

std::atomic< unsigned intcounter

std::map< std::string, unsigned intcounter_map

std::vector< std::atomic< bool > * > validity_pointers

const std::type_info * object_info

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

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

void check_no_subscribers () const noexcept

using map_value_type = decltype(counter_map)::value_type

using map_iterator = decltype(counter_map)::iterator

static std::mutex mutex

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

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

## Detailed Description

template<typename VectorType = Vector<double>>
class SolverQMRS< VectorType >

### Quasi-minimal method for symmetric matrices (SQMR)

The SQMR (symmetric quasi-minimal residual) method is supposed to solve symmetric indefinite linear systems with symmetric, not necessarily definite preconditioners. It is a variant of the original quasi-minimal residual method (QMR) and produces the same iterative solution. This version of SQMR is adapted from the respective symmetric QMR-from-BiCG algorithm given by both Freund/Nachtigal: A new Krylov-subspace method for symmetric indefinite linear systems, NASA STI/Recon Technical Report N, 95 (1994) and Freund/Nachtigal: Software for simplified Lanczos and QMR algorithms, Appl. Num. Math. 19 (1995), pp. 319-341 and provides both right and left (but not split) preconditioning.

### Trade off of stability to simplicity

Note, that the QMR implementation that the given algorithm is based on is derived from classical BiCG. It can be shown (Freund/Szeto: A transpose-free quasi-minimal residual squared algorithm for non-Hermitian linear systems, Advances in Computer Methods for Partial Differential Equations VII (IMACS, New Brunswick, NJ, 1992) pp. 258-264) that the QMR iterates can be generated from the BiCG iteration through one additional vector and some scalar updates. Possible breakdowns (or precisely, divisions by zero) of BiCG therefore obviously transfer to this simple no-look-ahead algorithm.

In return the algorithm is cheap compared to classical QMR or BiCGStab, using only one matrix-vector product with the system matrix and one application of the preconditioner per iteration respectively.

The residual used for measuring convergence is only approximately calculated by an upper bound. If this value comes below a threshold prescribed within the AdditionalData struct, then the exact residual of the current QMR iterate will be calculated using another multiplication with the system matrix. By experience (according to Freund and Nachtigal) this technique is useful for a threshold that is ten times the solving tolerance, and in that case will be only used in the last one or two steps of the complete iteration.

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

Like all other solver classes, this class has a local structure called AdditionalData which is used to pass additional parameters to the solver, like damping parameters or the number of temporary vectors. We use this additional structure instead of passing these values directly to the constructor because this makes the use of the SolverSelector and other classes much easier and guarantees that these will continue to work even if number or type of the additional parameters for a certain solver changes.

### 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.

Definition at line 92 of file solver_qmrs.h.

## ◆ vector_type

 using SolverBase< Vector< double > >::vector_type = Vector< double >
inherited

An alias for the underlying vector type

Definition at line 346 of file solver.h.

## ◆ SolverQMRS() [1/2]

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

Constructor.

## ◆ SolverQMRS() [2/2]

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

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

## ◆ solve()

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

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

## ◆ print_vectors()

template<typename VectorType = Vector<double>>
 virtual void SolverQMRS< VectorType >::print_vectors ( const unsigned int step, const VectorType & x, const VectorType & r, const VectorType & d ) const
virtual

Interface for derived class. This function gets the current iteration vector, the residual and the update vector in each step. It can be used for a graphical output of the convergence history.

## ◆ iterate()

template<typename VectorType = Vector<double>>
template<typename MatrixType , typename PreconditionerType >
 IterationResult SolverQMRS< VectorType >::iterate ( const MatrixType & A, VectorType & x, const VectorType & b, const PreconditionerType & preconditioner, VectorType & r, VectorType & u, VectorType & q, VectorType & t, VectorType & d )
private

The iteration loop itself. The function returns a structure indicating what happened in this function.

## ◆ connect()

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

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.

## Member Data Documentation

template<typename VectorType = Vector<double>>
protected

Definition at line 197 of file solver_qmrs.h.

## ◆ step

template<typename VectorType = Vector<double>>
 unsigned int SolverQMRS< VectorType >::step
private

Number of the current iteration (accumulated over restarts)

Definition at line 232 of file solver_qmrs.h.

## ◆ static_vector_memory

 GrowingVectorMemory > SolverBase< Vector< double > >::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 414 of file solver.h.

## ◆ memory

 VectorMemory >& SolverBase< Vector< double > >::memory
protectedinherited

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

Definition at line 419 of file solver.h.

## ◆ iteration_status

 boost::signals2::signal< SolverControl::State(const unsigned int iteration, const double check_value, const Vector< double > & current_iterate), StateCombiner> SolverBase< Vector< double > >::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 470 of file solver.h.

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