Reference documentation for deal.II version Git 5533784 2018-12-09 20:47:46 -0500
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Classes | Public Member Functions | Protected Attributes | Private Member Functions | Private Attributes | List of all members
SolverQMRS< VectorType > Class Template Reference

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

Inheritance diagram for SolverQMRS< VectorType >:
[legend]

Classes

struct  AdditionalData
 
struct  IterationResult
 

Public Member Functions

 SolverQMRS (SolverControl &cn, VectorMemory< VectorType > &mem, const AdditionalData &data=AdditionalData())
 
 SolverQMRS (SolverControl &cn, const AdditionalData &data=AdditionalData())
 
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
 
- Public Member Functions inherited from Solver< VectorType >
 Solver (SolverControl &solver_control, VectorMemory< VectorType > &vector_memory)
 
 Solver (SolverControl &solver_control)
 
boost::signals2::connection connect (const std::function< SolverControl::State(const unsigned int iteration, const doublecheck_value, const VectorType &current_iterate)> &slot)
 
- Public Member Functions inherited from Subscriptor
 Subscriptor ()
 
 Subscriptor (const Subscriptor &)
 
 Subscriptor (Subscriptor &&) noexcept
 
virtual ~Subscriptor ()
 
Subscriptoroperator= (const Subscriptor &)
 
Subscriptoroperator= (Subscriptor &&) noexcept
 
void subscribe (std::atomic< bool > *const validity, const char *identifier=nullptr) const
 
void unsubscribe (std::atomic< bool > *const validity, const char *identifier=nullptr) 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)
 

Protected Attributes

AdditionalData additional_data
 
- Protected Attributes inherited from Solver< VectorType >
GrowingVectorMemory< VectorType > static_vector_memory
 
VectorMemory< VectorType > & memory
 
boost::signals2::signal< SolverControl::State(const unsigned int iteration, const double check_value, const VectorType &current_iterate), StateCombineriteration_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
 

Additional Inherited Members

- Public Types inherited from Solver< VectorType >
using vector_type = VectorType
 
- Static Public Member Functions inherited from Subscriptor
static::ExceptionBase & ExcInUse (int arg1, std::string arg2, std::string arg3)
 
static::ExceptionBase & ExcNoSubscriber (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.

Author
Guido Kanschat, 1999; Ingo Kligge 2017

Definition at line 95 of file solver_qmrs.h.

Constructor & Destructor Documentation

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

Constructor.

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.

Member Function Documentation

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.

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.

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.

Member Data Documentation

template<typename VectorType = Vector<double>>
AdditionalData SolverQMRS< VectorType >::additional_data
protected

Additional parameters.

Definition at line 200 of file solver_qmrs.h.

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

Number of the current iteration (accumulated over restarts)

Definition at line 235 of file solver_qmrs.h.


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