Reference documentation for deal.II version GIT 3801df8983 2022-05-22 23:30:02+00:00
SolverCG< VectorType > Class Template Reference

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

Inheritance diagram for SolverCG< VectorType >:
[legend]

## Public Types

using size_type = types::global_dof_index

using vector_type = Vector< double >

## Public Member Functions

virtual ~SolverCG () override=default

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

boost::signals2::connection connect_coefficients_slot (const std::function< void(typename VectorType::value_type, typename VectorType::value_type)> &slot)

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< double > &)> &slot, const bool every_iteration=false)

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

## Protected Member Functions

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

## Static Protected Member Functions

static void compute_eigs_and_cond (const std::vector< typename VectorType::value_type > &diagonal, const std::vector< typename VectorType::value_type > &offdiagonal, const boost::signals2::signal< void(const std::vector< double > &)> &eigenvalues_signal, const boost::signals2::signal< void(double)> &cond_signal)

## Protected Attributes

boost::signals2::signal< void(typename VectorType::value_type, typename VectorType::value_type)> coefficients_signal

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

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

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

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

bool determine_beta_by_flexible_formula

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

## 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

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

using map_value_type = decltype(counter_map)::value_type

using map_iterator = decltype(counter_map)::iterator

std::atomic< unsigned int > counter

std::map< std::string, unsigned int > counter_map

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

const std::type_info * object_info

static std::mutex mutex

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

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

void check_no_subscribers () const noexcept

## Detailed Description

### template<typename VectorType = Vector<double>> class SolverCG< VectorType >

This class implements the preconditioned Conjugate Gradients (CG) method that can be used to solve linear systems with a symmetric positive definite matrix. This class is used first in step-3 and step-4, but is used in many other tutorial programs as well. Like all other solver classes, it can work on any kind of vector and matrix as long as they satisfy certain requirements (for the requirements on matrices and vectors in order to work with this class, see the documentation of the Solver base class). The type of the solution vector must be passed as template argument, and defaults to Vector<double>.

Note
The CG method requires a symmetric preconditioner (i.e., for example, SOR is not a possible choice). There is a variant of the solver, SolverFlexibleCG, that allows to use a variable preconditioner or a preconditioner with some slight non-symmetry (like weighted Schwarz methods), by using a different formula for the step length in the computation of the next search direction.

### Eigenvalue computation

The cg-method performs an orthogonal projection of the original preconditioned linear system to another system of smaller dimension. Furthermore, the projected matrix T is tri-diagonal. Since the projection is orthogonal, the eigenvalues of T approximate those of the original preconditioned matrix PA. In fact, after n steps, where n is the dimension of the original system, the eigenvalues of both matrices are equal. But, even for small numbers of iteration steps, the condition number of T is a good estimate for the one of PA.

After m steps the matrix T_m can be written in terms of the coefficients alpha and beta as the tri-diagonal matrix with diagonal elements 1/alpha_0, 1/alpha_1 + beta_0/alpha_0, ..., 1/alpha_{m-1}+beta_{m-2}/alpha_{m-2} and off-diagonal elements sqrt(beta_0)/alpha_0, ..., sqrt(beta_{m-2})/alpha_{m-2}. The eigenvalues of this matrix can be computed by postprocessing.

Y. Saad: "Iterative methods for Sparse Linear Systems", section 6.7.3 for details.

The coefficients, eigenvalues and condition number (computed as the ratio of the largest over smallest eigenvalue) can be obtained by connecting a function as a slot to the solver using one of the functions connect_coefficients_slot, connect_eigenvalues_slot and connect_condition_number_slot. These slots will then be called from the solver with the estimates as argument.

### 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.

#### Optimized operations with specific MatrixType argument

This class enables to embed the vector updates into the matrix-vector product in case the MatrixType and PreconditionerType support such a mode of operation. To this end, the VectorType needs to be LinearAlgebra::distributed::Vector, the class MatrixType needs to provide a function with the signature

void MatrixType::vmult(
VectorType &,
const VectorType &,
const std::function<void(const unsigned int, const unsigned int)> &,
const std::function<void(const unsigned int, const unsigned int)> &) const

where the two given functions run before and after the matrix-vector product, respectively, and the PreconditionerType needs to provide a function either the signature

Number PreconditionerType::apply(unsigned int index, const Number src) const
auto apply(F &&fn, Tuple &&t) -> decltype(apply_impl(std::forward< F >(fn), std::forward< Tuple >(t), std::make_index_sequence< std::tuple_size< typename std::remove_reference< Tuple >::type >::value >()))
Definition: tuple.h:36

to apply the action of the preconditioner on a single element (effectively being a diagonal preconditioner), or the signature

void PreconditionerType::apply_to_subrange(unsigned int start_range,
unsigned int end_range,
const Number* src_ptr_to_subrange,
Number* dst_ptr_to_subrange)

where the pointers src_ptr_to_subrange and dst_ptr_to_subrange point to the location in the vector where the operation should be applied to. If both functions are given, the more optimized apply path is selected. The functions passed to MatrixType::vmult take as arguments a sub-range among the locally owned elements of the vector, defined as half-open intervals. The intervals are designed to be scheduled close to the time the matrix-vector product touches those entries in the src and dst vectors, respectively, with the requirement that

• the matrix-vector product may only access an entry in src or dst once the operation_before_matrix_vector_product has been run on that vector entry;
• operation_after_matrix_vector_product may run on a range of entries [i,j) once the matrix-vector product does not access the entries [i,j) in src and dst any more.

The motivation for this function is to increase data locality and hence cache usage. For the example of a class similar to the one in the step-37 tutorial program, the implementation is

void
const std::function<void(const unsigned int, const unsigned int)>
&operation_before_matrix_vector_product,
const std::function<void(const unsigned int, const unsigned int)>
&operation_after_matrix_vector_product) const
{
data.cell_loop(&LaplaceOperator::local_apply,
this,
dst,
src,
operation_before_matrix_vector_product,
operation_after_matrix_vector_product);
}

In terms of the SolverCG implementation, the operation before the loop will run the updates on the vectors according to a variant presented in Algorithm 2.2 of [Chronopoulos1989] (but for a preconditioner), whereas the operation after the loop performs a total of 7 reductions in parallel.

Definition at line 177 of file solver_cg.h.

## ◆ size_type

template<typename VectorType = Vector<double>>
 using SolverCG< VectorType >::size_type = types::global_dof_index

Declare type for container size.

Definition at line 183 of file solver_cg.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.

## ◆ SolverCG() [1/2]

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

Constructor.

## ◆ SolverCG() [2/2]

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

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

## ◆ ~SolverCG()

template<typename VectorType = Vector<double>>
 virtual SolverCG< VectorType >::~SolverCG ( )
overridevirtualdefault

Virtual destructor.

## ◆ solve()

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

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

## ◆ connect_coefficients_slot()

template<typename VectorType = Vector<double>>
 boost::signals2::connection SolverCG< VectorType >::connect_coefficients_slot ( const std::function< void(typename VectorType::value_type, typename VectorType::value_type)> & slot )

Connect a slot to retrieve the CG coefficients. The slot will be called with alpha as the first argument and with beta as the second argument, where alpha and beta follow the notation in Y. Saad: "Iterative methods for Sparse Linear Systems", section 6.7. Called once per iteration

## ◆ connect_condition_number_slot()

template<typename VectorType = Vector<double>>
 boost::signals2::connection SolverCG< 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 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<typename VectorType = Vector<double>>
 boost::signals2::connection SolverCG< VectorType >::connect_eigenvalues_slot ( const std::function< void(const std::vector< double > &)> & slot, const bool every_iteration = false )

Connect a slot to retrieve the estimated eigenvalues. Called on each 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).

## ◆ print_vectors()

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

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 graphical output of the convergence history.

## ◆ compute_eigs_and_cond()

template<typename VectorType = Vector<double>>
 static void SolverCG< VectorType >::compute_eigs_and_cond ( const std::vector< typename VectorType::value_type > & diagonal, const std::vector< typename VectorType::value_type > & offdiagonal, const boost::signals2::signal< void(const std::vector< double > &)> & eigenvalues_signal, const boost::signals2::signal< void(double)> & cond_signal )
staticprotected

Estimates the eigenvalues from diagonal and offdiagonal. Uses these estimate to compute the condition number. Calls the signals eigenvalues_signal and cond_signal with these estimates as arguments.

## ◆ connect()

 boost::signals2::connection SolverBase< Vector< double > >::connect ( const std::function< SolverControl::State(const unsigned int iteration, const double check_value, const VectorType ¤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 281 of file solver_cg.h.

## ◆ coefficients_signal

template<typename VectorType = Vector<double>>
 boost::signals2::signal SolverCG< VectorType >::coefficients_signal
protected

Signal used to retrieve the CG coefficients. Called on each iteration.

Definition at line 288 of file solver_cg.h.

## ◆ condition_number_signal

template<typename VectorType = Vector<double>>
 boost::signals2::signal SolverCG< VectorType >::condition_number_signal
protected

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

Definition at line 294 of file solver_cg.h.

## ◆ all_condition_numbers_signal

template<typename VectorType = Vector<double>>
 boost::signals2::signal SolverCG< VectorType >::all_condition_numbers_signal
protected

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

Definition at line 300 of file solver_cg.h.

## ◆ eigenvalues_signal

template<typename VectorType = Vector<double>>
 boost::signals2::signal &)> SolverCG< VectorType >::eigenvalues_signal
protected

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

Definition at line 306 of file solver_cg.h.

## ◆ all_eigenvalues_signal

template<typename VectorType = Vector<double>>
 boost::signals2::signal &)> SolverCG< VectorType >::all_eigenvalues_signal
protected

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

Definition at line 313 of file solver_cg.h.

## ◆ determine_beta_by_flexible_formula

template<typename VectorType = Vector<double>>
 bool SolverCG< VectorType >::determine_beta_by_flexible_formula
protected

Flag to indicate whether the classical Fletcher–Reeves update formula for the parameter $$\beta_k$$ (standard CG algorithm, minimal storage needs) or the flexible conjugate gradient method with Polak-Ribiere formula for $$\beta_k$$ should be used. This base class implementation of SolverCG will always use the former method, whereas the derived class SolverFlexibleCG will use the latter.

Definition at line 323 of file solver_cg.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: