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solver_gmres.h
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solver_gmres.h
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// ---------------------------------------------------------------------
//
// Copyright (C) 1998 - 2021 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE.md at
// the top level directory of deal.II.
//
// ---------------------------------------------------------------------
#ifndef dealii_solver_gmres_h
#define dealii_solver_gmres_h
#include <deal.II/base/config.h>
#include <deal.II/base/logstream.h>
#include <deal.II/base/subscriptor.h>
#include <deal.II/lac/full_matrix.h>
#include <deal.II/lac/householder.h>
#include <deal.II/lac/lapack_full_matrix.h>
#include <deal.II/lac/solver.h>
#include <deal.II/lac/solver_control.h>
#include <deal.II/lac/vector.h>
#include <algorithm>
#include <cmath>
#include <limits>
#include <memory>
#include <vector>
DEAL_II_NAMESPACE_OPEN
/**
* @addtogroup Solvers
* @{
*/
namespace internal
{
/**
* A namespace for a helper class to the GMRES solver.
*/
namespace SolverGMRESImplementation
{
/**
* Class to hold temporary vectors. This class automatically allocates a
* new vector, once it is needed.
*
* A future version should also be able to shift through vectors
* automatically, avoiding restart.
*/
template <typename VectorType>
class TmpVectors
{
public:
/**
* Constructor. Prepares an array of @p VectorType of length @p
* max_size.
*/
TmpVectors(const unsigned int max_size, VectorMemory<VectorType> &vmem);
/**
* Destructor. Delete all allocated vectors.
*/
~TmpVectors() = default;
/**
* Get vector number @p i. If this vector was unused before, an error
* occurs.
*/
VectorType &
operator[](const unsigned int i) const;
/**
* Get vector number @p i. Allocate it if necessary.
*
* If a vector must be allocated, @p temp is used to reinit it to the
* proper dimensions.
*/
VectorType &
operator()(const unsigned int i, const VectorType &temp);
/**
* Return size of data vector. It is used in the solver to store
* the Arnoldi vectors.
*/
unsigned int
size() const;
private:
/**
* Pool where vectors are obtained from.
*/
VectorMemory<VectorType> &mem;
/**
* Field for storing the vectors.
*/
std::vector<typename VectorMemory<VectorType>::Pointer> data;
};
} // namespace SolverGMRESImplementation
} // namespace internal
/**
* 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.
*
*
* <h3>Left versus right preconditioning</h3>
*
* @p AdditionalData allows you to choose between left and right
* preconditioning. As expected, this switches between solving for the systems
* <i>P<sup>-1</sup>A</i> and <i>AP<sup>-1</sup></i>, respectively.
*
* A second consequence is the type of residual which is used to measure
* convergence. With left preconditioning, this is the <b>preconditioned</b>
* 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 <tt>true</tt> value refers to the
* behavior described in the previous paragraph, while <tt>false</tt> reverts
* it. Be aware though that additional residuals have to be computed in this
* case, impeding the overall performance of the solver.
*
*
* <h3>The size of the Arnoldi basis</h3>
*
* 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
* <b>not</b> 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.
*
*
* <h3>Observing the progress of linear solver iterations</h3>
*
* 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.
*
*
* <h3>Eigenvalue and condition number estimates</h3>
*
* 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 @p
* connect_condition_number_slot and @p connect_eigenvalues_slot. These slots
* will then be called from the solver with the estimates as argument.
*/
template <class VectorType = Vector<double>>
class SolverGMRES : public SolverBase<VectorType>
{
public:
/**
* Standardized data struct to pipe additional data to the solver.
*/
struct AdditionalData
{
/**
* Constructor. By default, set the number of temporary vectors to 30,
* i.e. do a restart every 28 iterations. Also set preconditioning from
* left, the residual of the stopping criterion to the default residual,
* and re-orthogonalization only if necessary. Also, the batched mode with
* reduced functionality to track information is disabled by default.
*/
explicit AdditionalData(const unsigned int max_n_tmp_vectors = 30,
const bool right_preconditioning = false,
const bool use_default_residual = true,
const bool force_re_orthogonalization = false,
const bool batched_mode = false);
/**
* Maximum number of temporary vectors. This parameter controls the size
* of the Arnoldi basis, which for historical reasons is
* #max_n_tmp_vectors-2. SolverGMRES assumes that there are at least three
* temporary vectors, so this value must be greater than or equal to three.
*/
unsigned int max_n_tmp_vectors;
/**
* Flag for right preconditioning.
*
* @note Change between left and right preconditioning will also change
* the way residuals are evaluated. See the corresponding section in the
* SolverGMRES.
*/
bool right_preconditioning;
/**
* Flag for the default residual that is used to measure convergence.
*/
bool use_default_residual;
/**
* Flag to force re-orthogonalization of orthonormal basis in every step.
* If set to false, the solver automatically checks for loss of
* orthogonality every 5 iterations and enables re-orthogonalization only
* if necessary.
*/
bool force_re_orthogonalization;
/**
* Flag to control whether a reduced mode of the solver should be
* run. This is necessary when running (several) SolverGMRES instances
* involving very small and cheap linear systems where the feedback from
* all signals, eigenvalue computations, and log stream are disabled.
*/
bool batched_mode;
};
/**
* Constructor.
*/
SolverGMRES(SolverControl & cn,
VectorMemory<VectorType> &mem,
const AdditionalData & data = AdditionalData());
/**
* Constructor. Use an object of type GrowingVectorMemory as a default to
* allocate memory.
*/
SolverGMRES(SolverControl &cn, const AdditionalData &data = AdditionalData());
/**
* The copy constructor is deleted.
*/
SolverGMRES(const SolverGMRES<VectorType> &) = delete;
/**
* Solve the linear system $Ax=b$ for x.
*/
template <typename MatrixType, typename PreconditionerType>
void
solve(const MatrixType & A,
VectorType & x,
const VectorType & b,
const PreconditionerType &preconditioner);
/**
* 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).
*/
boost::signals2::connection
connect_condition_number_slot(const std::function<void(double)> &slot,
const bool every_iteration = false);
/**
* 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).
*/
boost::signals2::connection
connect_eigenvalues_slot(
const std::function<void(const std::vector<std::complex<double>> &)> &slot,
const bool every_iteration = false);
/**
* 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).
*/
boost::signals2::connection
connect_hessenberg_slot(
const std::function<void(const FullMatrix<double> &)> &slot,
const bool every_iteration = true);
/**
* 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).
*/
boost::signals2::connection
connect_krylov_space_slot(
const std::function<
void(const internal::SolverGMRESImplementation::TmpVectors<VectorType> &)>
&slot);
/**
* Connect a slot to retrieve a notification when the vectors are
* re-orthogonalized.
*/
boost::signals2::connection
connect_re_orthogonalization_slot(const std::function<void(int)> &slot);
DeclException1(ExcTooFewTmpVectors,
int,
<< "The number of temporary vectors you gave (" << arg1
<< ") is too small. It should be at least 10 for "
<< "any results, and much more for reasonable ones.");
protected:
/**
* Includes the maximum number of tmp vectors.
*/
AdditionalData additional_data;
/**
* Signal used to retrieve the estimated condition number. Called once when
* all iterations are ended.
*/
boost::signals2::signal<void(double)> condition_number_signal;
/**
* Signal used to retrieve the estimated condition numbers. Called on each
* outer iteration.
*/
boost::signals2::signal<void(double)> all_condition_numbers_signal;
/**
* Signal used to retrieve the estimated eigenvalues. Called once when all
* iterations are ended.
*/
boost::signals2::signal<void(const std::vector<std::complex<double>> &)>
eigenvalues_signal;
/**
* Signal used to retrieve the estimated eigenvalues. Called on each outer
* iteration.
*/
boost::signals2::signal<void(const std::vector<std::complex<double>> &)>
all_eigenvalues_signal;
/**
* Signal used to retrieve the Hessenberg matrix. Called once when
* all iterations are ended.
*/
boost::signals2::signal<void(const FullMatrix<double> &)> hessenberg_signal;
/**
* Signal used to retrieve the Hessenberg matrix. Called on each outer
* iteration.
*/
boost::signals2::signal<void(const FullMatrix<double> &)>
all_hessenberg_signal;
/**
* Signal used to retrieve the Krylov space basis vectors. Called once
* when all iterations are ended.
*/
boost::signals2::signal<void(
const internal::SolverGMRESImplementation::TmpVectors<VectorType> &)>
krylov_space_signal;
/**
* Signal used to retrieve a notification
* when the vectors are re-orthogonalized.
*/
boost::signals2::signal<void(int)> re_orthogonalize_signal;
/**
* A reference to the underlying SolverControl object. In the regular case,
* this is not needed, as the signal from the base class is used, but the
* batched variant cannot use those mechanisms due to the high costs.
*/
SolverControl &solver_control;
/**
* Implementation of the computation of the norm of the residual.
*/
virtual double
criterion();
/**
* Transformation of an upper Hessenberg matrix into tridiagonal structure
* by givens rotation of the last column
*/
void
givens_rotation(Vector<double> &h,
Vector<double> &b,
Vector<double> &ci,
Vector<double> &si,
int col) const;
/**
* Orthogonalize the vector @p vv against the @p dim (orthogonal) vectors
* given by the first argument using the modified Gram-Schmidt algorithm.
* The factors used for orthogonalization are stored in @p h. The boolean @p
* 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.
*/
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)>());
/**
* 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.
*/
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);
/**
* Projected system matrix
*/
FullMatrix<double> H;
/**
* Auxiliary vector for orthogonalization
*/
Vector<double> gamma;
/**
* Auxiliary vector for orthogonalization
*/
Vector<double> ci;
/**
* Auxiliary vector for orthogonalization
*/
Vector<double> si;
/**
* Auxiliary vector for orthogonalization
*/
Vector<double> h;
};
/**
* Implementation of the Generalized minimal residual method with flexible
* preconditioning (flexible GMRES or FGMRES).
*
* This flexible version of the GMRES method allows for the use of a different
* preconditioner in each iteration step. Therefore, it is also more robust
* with respect to inaccurate evaluation of the preconditioner. An important
* application is the use of a Krylov space method inside the
* preconditioner. As opposed to SolverGMRES which allows one to choose
* between left and right preconditioning, this solver always applies the
* preconditioner from the right.
*
* FGMRES needs two vectors in each iteration steps yielding a total of
* <tt>2*SolverFGMRES::%AdditionalData::%max_basis_size+1</tt> auxiliary
* vectors. Otherwise, FGMRES requires roughly the same number of operations
* per iteration compared to GMRES, except one application of the
* preconditioner less at each restart and at the end of solve().
*
* For more details see @cite Saad1991.
*/
template <class VectorType = Vector<double>>
class SolverFGMRES : public SolverBase<VectorType>
{
public:
/**
* Standardized data struct to pipe additional data to the solver.
*/
struct AdditionalData
{
/**
* Constructor. By default, set the maximum basis size to 30.
*/
explicit AdditionalData(const unsigned int max_basis_size = 30)
: max_basis_size(max_basis_size)
{}
/**
* Maximum basis size.
*/
unsigned int max_basis_size;
};
/**
* Constructor.
*/
SolverFGMRES(SolverControl & cn,
VectorMemory<VectorType> &mem,
const AdditionalData & data = AdditionalData());
/**
* Constructor. Use an object of type GrowingVectorMemory as a default to
* allocate memory.
*/
SolverFGMRES(SolverControl & cn,
const AdditionalData &data = AdditionalData());
/**
* Solve the linear system $Ax=b$ for x.
*/
template <typename MatrixType, typename PreconditionerType>
void
solve(const MatrixType & A,
VectorType & x,
const VectorType & b,
const PreconditionerType &preconditioner);
private:
/**
* Additional flags.
*/
AdditionalData additional_data;
/**
* Projected system matrix
*/
FullMatrix<double> H;
/**
* Auxiliary matrix for inverting @p H
*/
FullMatrix<double> H1;
};
/** @} */
/* --------------------- Inline and template functions ------------------- */
#ifndef DOXYGEN
namespace internal
{
namespace SolverGMRESImplementation
{
template <class VectorType>
inline TmpVectors<VectorType>::TmpVectors(const unsigned int max_size,
VectorMemory<VectorType> &vmem)
: mem(vmem)
, data(max_size)
{}
template <class VectorType>
inline VectorType &
TmpVectors<VectorType>::operator[](const unsigned int i) const
{
AssertIndexRange(i, data.size());
Assert(data[i] != nullptr, ExcNotInitialized());
return *data[i];
}
template <class VectorType>
inline VectorType &
TmpVectors<VectorType>::operator()(const unsigned int i,
const VectorType & temp)
{
AssertIndexRange(i, data.size());
if (data[i] == nullptr)
{
data[i] = std::move(typename VectorMemory<VectorType>::Pointer(mem));
data[i]->reinit(temp, true);
}
return *data[i];
}
template <class VectorType>
unsigned int
TmpVectors<VectorType>::size() const
{
return (data.size() > 0 ? data.size() - 1 : 0);
}
// A comparator for better printing eigenvalues
inline bool
complex_less_pred(const std::complex<double> &x,
const std::complex<double> &y)
{
return x.real() < y.real() ||
(x.real() == y.real() && x.imag() < y.imag());
}
// A function to solve the (upper) triangular system after Givens
// rotations on a matrix that has possibly unused rows and columns
inline void
solve_triangular(const unsigned int dim,
const FullMatrix<double> &H,
const Vector<double> & rhs,
Vector<double> & solution)
{
for (int i = dim - 1; i >= 0; --i)
{
double s = rhs(i);
for (unsigned int j = i + 1; j < dim; ++j)
s -= solution(j) * H(i, j);
solution(i) = s / H(i, i);
AssertIsFinite(solution(i));
}
}
} // namespace SolverGMRESImplementation
} // namespace internal
template <class VectorType>
inline SolverGMRES<VectorType>::AdditionalData::AdditionalData(
const unsigned int max_n_tmp_vectors,
const bool right_preconditioning,
const bool use_default_residual,
const bool force_re_orthogonalization,
const bool batched_mode)
: max_n_tmp_vectors(max_n_tmp_vectors)
, right_preconditioning(right_preconditioning)
, use_default_residual(use_default_residual)
, force_re_orthogonalization(force_re_orthogonalization)
, batched_mode(batched_mode)
{
Assert(3 <= max_n_tmp_vectors,
ExcMessage("SolverGMRES needs at least three "
"temporary vectors."));
}
template <class VectorType>
SolverGMRES<VectorType>::SolverGMRES(SolverControl & cn,
VectorMemory<VectorType> &mem,
const AdditionalData & data)
: SolverBase<VectorType>(cn, mem)
, additional_data(data)
, solver_control(cn)
{}
template <class VectorType>
SolverGMRES<VectorType>::SolverGMRES(SolverControl & cn,
const AdditionalData &data)
: SolverBase<VectorType>(cn)
, additional_data(data)
, solver_control(cn)
{}
template <class VectorType>
inline void
SolverGMRES<VectorType>::givens_rotation(Vector<double> &h,
Vector<double> &b,
Vector<double> &ci,
Vector<double> &si,
int col) const
{
for (int i = 0; i < col; ++i)
{
const double s = si(i);
const double c = ci(i);
const double dummy = h(i);
h(i) = c * dummy + s * h(i + 1);
h(i + 1) = -s * dummy + c * h(i + 1);
};
const double r = 1. / std::sqrt(h(col) * h(col) + h(col + 1) * h(col + 1));
si(col) = h(col + 1) * r;
ci(col) = h(col) * r;
h(col) = ci(col) * h(col) + si(col) * h(col + 1);
b(col + 1) = -si(col) * b(col);
b(col) *= ci(col);
}
template <class VectorType>
inline 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 & reorthogonalize,
const boost::signals2::signal<void(int)> &reorthogonalize_signal)
{
Assert(dim > 0, ExcInternalError());
const unsigned int inner_iteration = dim - 1;
// need initial norm for detection of re-orthogonalization, see below
double norm_vv_start = 0;
const bool consider_reorthogonalize =
(reorthogonalize == false) && (inner_iteration % 5 == 4);
if (consider_reorthogonalize)
norm_vv_start = vv.l2_norm();
// Orthogonalization
h(0) = vv * orthogonal_vectors[0];
for (unsigned int i = 1; i < dim; ++i)
h(i) = vv.add_and_dot(-h(i - 1),
orthogonal_vectors[i - 1],
orthogonal_vectors[i]);
double norm_vv =
std::sqrt(vv.add_and_dot(-h(dim - 1), orthogonal_vectors[dim - 1], vv));
// Re-orthogonalization if loss of orthogonality detected. For the test, use
// a strategy discussed in C. T. Kelley, Iterative Methods for Linear and
// Nonlinear Equations, SIAM, Philadelphia, 1995: Compare the norm of vv
// after orthogonalization with its norm when starting the
// orthogonalization. If vv became very small (here: less than the square
// root of the machine precision times 10), it is almost in the span of the
// previous vectors, which indicates loss of precision.
if (consider_reorthogonalize)
{
if (norm_vv >
10. * norm_vv_start *
std::sqrt(
std::numeric_limits<typename VectorType::value_type>::epsilon()))
return norm_vv;
else
{
reorthogonalize = true;
if (!reorthogonalize_signal.empty())
reorthogonalize_signal(accumulated_iterations);
}
}
if (reorthogonalize == true)
{
double htmp = vv * orthogonal_vectors[0];
h(0) += htmp;
for (unsigned int i = 1; i < dim; ++i)
{
htmp = vv.add_and_dot(-htmp,
orthogonal_vectors[i - 1],
orthogonal_vectors[i]);
h(i) += htmp;
}
norm_vv =
std::sqrt(vv.add_and_dot(-htmp, orthogonal_vectors[dim - 1], vv));
}
return norm_vv;
}
template <class VectorType>
inline void
SolverGMRES<VectorType>::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)
{
// Avoid copying the Hessenberg matrix if it isn't needed.
if ((!eigenvalues_signal.empty() || !hessenberg_signal.empty() ||
!cond_signal.empty()) &&
dim > 0)
{
LAPACKFullMatrix<double> mat(dim, dim);
for (unsigned int i = 0; i < dim; ++i)
for (unsigned int j = 0; j < dim; ++j)
mat(i, j) = H_orig(i, j);
hessenberg_signal(H_orig);
// Avoid computing eigenvalues if they are not needed.
if (!eigenvalues_signal.empty())
{
// Copy mat so that we can compute svd below. Necessary since
// compute_eigenvalues will leave mat in state
// LAPACKSupport::unusable.
LAPACKFullMatrix<double> mat_eig(mat);
mat_eig.compute_eigenvalues();
std::vector<std::complex<double>> eigenvalues(dim);
for (unsigned int i = 0; i < mat_eig.n(); ++i)
eigenvalues[i] = mat_eig.eigenvalue(i);
// Sort eigenvalues for nicer output.
std::sort(eigenvalues.begin(),
eigenvalues.end(),
internal::SolverGMRESImplementation::complex_less_pred);
eigenvalues_signal(eigenvalues);
}
// Calculate condition number, avoid calculating the svd if a slot
// isn't connected. Need at least a 2-by-2 matrix to do the estimate.
if (!cond_signal.empty() && (mat.n() > 1))
{
mat.compute_svd();
double condition_number =
mat.singular_value(0) / mat.singular_value(mat.n() - 1);
cond_signal(condition_number);
}
}
}
template <class VectorType>
template <typename MatrixType, typename PreconditionerType>
void
SolverGMRES<VectorType>::solve(const MatrixType & A,
VectorType & x,
const VectorType & b,
const PreconditionerType &preconditioner)
{
// TODO:[?] Check, why there are two different start residuals.
// TODO:[GK] Make sure the parameter in the constructor means maximum basis
// size
std::unique_ptr<LogStream::Prefix> prefix;
if (!additional_data.batched_mode)
prefix = std::make_unique<LogStream::Prefix>("GMRES");
// extra call to std::max to placate static analyzers: coverity rightfully
// complains that data.max_n_tmp_vectors - 2 may overflow
const unsigned int n_tmp_vectors =
std::max(additional_data.max_n_tmp_vectors, 3u);
// Generate an object where basis vectors are stored.
internal::SolverGMRESImplementation::TmpVectors<VectorType> tmp_vectors(
n_tmp_vectors, this->memory);
// number of the present iteration; this
// number is not reset to zero upon a
// restart
unsigned int accumulated_iterations = 0;
const bool do_eigenvalues =
!additional_data.batched_mode &&
(!condition_number_signal.empty() ||
!all_condition_numbers_signal.empty() || !eigenvalues_signal.empty() ||
!all_eigenvalues_signal.empty() || !hessenberg_signal.empty() ||
!all_hessenberg_signal.empty());
// for eigenvalue computation, need to collect the Hessenberg matrix (before
// applying Givens rotations)
FullMatrix<double> H_orig;
if (do_eigenvalues)
H_orig.reinit(n_tmp_vectors, n_tmp_vectors - 1);
// matrix used for the orthogonalization process later
H.reinit(n_tmp_vectors, n_tmp_vectors - 1);
// some additional vectors, also used in the orthogonalization
gamma.reinit(n_tmp_vectors);
ci.reinit(n_tmp_vectors - 1);
si.reinit(n_tmp_vectors - 1);
h.reinit(n_tmp_vectors - 1);
unsigned int dim = 0;
SolverControl::State iteration_state = SolverControl::iterate;
double last_res = std::numeric_limits<double>::lowest();
// switch to determine whether we want a left or a right preconditioner. at
// present, left is default, but both ways are implemented
const bool left_precondition = !additional_data.right_preconditioning;
// Per default the left preconditioned GMRes uses the preconditioned
// residual and the right preconditioned GMRes uses the unpreconditioned
// residual as stopping criterion.
const bool use_default_residual = additional_data.use_default_residual;
// define two aliases
VectorType &v = tmp_vectors(0, x);
VectorType &p = tmp_vectors(n_tmp_vectors - 1, x);
// Following vectors are needed when we are not using the default residuals
// as stopping criterion
typename VectorMemory<VectorType>::Pointer r;
typename VectorMemory<VectorType>::Pointer x_;
std::unique_ptr<dealii::Vector<double>> gamma_;
if (!use_default_residual)
{
r = std::move(typename VectorMemory<VectorType>::Pointer(this->memory));
x_ = std::move(typename VectorMemory<VectorType>::Pointer(this->memory));
r->reinit(x);
x_->reinit(x);
gamma_ = std::make_unique<dealii::Vector<double>>(gamma.size());
}
bool re_orthogonalize = additional_data.force_re_orthogonalization;
///////////////////////////////////////////////////////////////////////////
// outer iteration: loop until we either reach convergence or the maximum
// number of iterations is exceeded. each cycle of this loop amounts to one
// restart
do
{
// reset this vector to the right size
h.reinit(n_tmp_vectors - 1);
if (left_precondition)
{
A.vmult(p, x);
p.sadd(-1., 1., b);
preconditioner.vmult(v, p);
}
else
{
A.vmult(v, x);
v.sadd(-1., 1., b);
};
double rho = v.l2_norm();
// check the residual here as well since it may be that we got the exact
// (or an almost exact) solution vector at the outset. if we wouldn't
// check here, the next scaling operation would produce garbage
if (use_default_residual)
{
last_res = rho;
if (additional_data.batched_mode)
iteration_state = solver_control.check(accumulated_iterations, rho);
else
iteration_state =
this->iteration_status(accumulated_iterations, rho, x);
if (iteration_state != SolverControl::iterate)
break;
}
else
{
deallog << "default_res=" << rho << std::endl;
if (left_precondition)
{
A.vmult(*r, x);
r->sadd(-1., 1., b);
}
else
preconditioner.vmult(*r, v);
double res = r->l2_norm();
last_res = res;
if (additional_data.batched_mode)
iteration_state = solver_control.check(accumulated_iterations, rho);
else
iteration_state =
this->iteration_status(accumulated_iterations, res, x);
if (iteration_state != SolverControl::iterate)
break;
}
gamma(0) = rho;
v *= 1. / rho;
// inner iteration doing at most as many steps as there are temporary
// vectors. the number of steps actually been done is propagated outside
// through the @p dim variable
for (unsigned int inner_iteration = 0;
((inner_iteration < n_tmp_vectors - 2) &&
(iteration_state == SolverControl::iterate));
++inner_iteration)
{
++accumulated_iterations;
// yet another alias