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// $Id: solver_gmres.h 21255 2010-06-21 17:17:23Z kanschat $
// Version: $Name$
//
// Copyright (C) 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2009, 2010 by the deal.II authors
//
// This file is subject to QPL and may not be distributed
// without copyright and license information. Please refer
// to the file deal.II/doc/license.html for the text and
// further information on this license.
//
//---------------------------- solver_gmres.h ---------------------------
#ifndef __deal2__solver_gmres_h
#define __deal2__solver_gmres_h
#include <base/config.h>
#include <base/subscriptor.h>
#include <base/logstream.h>
#include <lac/householder.h>
#include <lac/solver.h>
#include <lac/solver_control.h>
#include <lac/full_matrix.h>
#include <lac/vector.h>
#include <vector>
#include <cmath>
DEAL_II_NAMESPACE_OPEN
/*!@addtogroup Solvers */
/*@{*/
namespace internal
{
/**
* A namespace for a helper class
* to the GMRES solver.
*/
namespace SolverGMRES
{
/**
* 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 <class VECTOR>
class TmpVectors
{
public:
/**
* Constructor. Prepares an
* array of @p VECTOR of
* length @p max_size.
*/
TmpVectors(const unsigned int max_size,
VectorMemory<VECTOR> &vmem);
/**
* Delete all allocated vectors.
*/
~TmpVectors();
/**
* Get vector number
* @p i. If this vector was
* unused before, an error
* occurs.
*/
VECTOR& 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.
*/
VECTOR& operator() (const unsigned int i,
const VECTOR &temp);
private:
/**
* Pool were vectors are
* obtained from.
*/
VectorMemory<VECTOR> &mem;
/**
* Field for storing the
* vectors.
*/
std::vector<VECTOR*> data;
/**
* Offset of the first
* vector. This is for later
* when vector rotation will
* be implemented.
*/
unsigned int offset;
};
}
}
/**
* 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. Addinitonally, it allows you to choose bet 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.
*
* @author Wolfgang Bangerth, Guido Kanschat, Ralf Hartmann.
*/
template <class VECTOR = Vector<double> >
class SolverGMRES : public Solver<VECTOR>
{
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 and
* the residual of the stopping
* criterion to the default residual.
*/
AdditionalData (const unsigned int max_n_tmp_vectors = 30,
const bool right_preconditioning = false,
const bool use_default_residual = true);
/**
* Maximum number of
* temporary vectors. This
* parameter controls the
* size of the Arnoldi basis,
* which for historical
* reasons is
* #max_n_tmp_vectors-2.
*/
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;
};
/**
* Constructor.
*/
SolverGMRES (SolverControl &cn,
VectorMemory<VECTOR> &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());
/**
* Solve the linear system $Ax=b$
* for x.
*/
template<class MATRIX, class PRECONDITIONER>
void
solve (const MATRIX &A,
VECTOR &x,
const VECTOR &b,
const PRECONDITIONER &precondition);
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;
/**
* 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;
/**
* Projected system matrix
*/
FullMatrix<double> H;
/**
* Auxiliary matrix for inverting @p H
*/
FullMatrix<double> H1;
private:
/**
* No copy constructor.
*/
SolverGMRES (const SolverGMRES<VECTOR>&);
};
/**
* Implementation of the Generalized minimal residual method with flexible
* preconditioning method.
*
* This 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 also the use of a
* Krylov space method inside the preconditioner.
*
* FGMRES needs two vectors in each iteration steps yielding a total
* of <tt>2*AdditionalData::max_basis_size+1</tt> auxiliary vectors.
*
* Caveat: documentation of this class is not up to date. There are
* also a few parameters of GMRES we would like to introduce here.
*
* @author Guido Kanschat, 2003
*/
template <class VECTOR = Vector<double> >
class SolverFGMRES : public Solver<VECTOR>
{
public:
/**
* Standardized data struct to
* pipe additional data to the
* solver.
*/
struct AdditionalData
{
/**
* Constructor. By default,
* set the number of
* temporary vectors to 30,
* preconditioning from left
* and the residual of the
* stopping criterion to the
* default residual
* (cf. class documentation).
*/
AdditionalData(const unsigned int max_basis_size = 30,
const bool /*use_default_residual*/ = true)
:
max_basis_size(max_basis_size)
{}
/**
* Maximum number of
* tmp vectors.
*/
unsigned int max_basis_size;
};
/**
* Constructor.
*/
SolverFGMRES (SolverControl &cn,
VectorMemory<VECTOR> &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<class MATRIX, class PRECONDITIONER>
void
solve (const MATRIX &A,
VECTOR &x,
const VECTOR &b,
const PRECONDITIONER &precondition);
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 SolverGMRES
{
template <class VECTOR>
inline
TmpVectors<VECTOR>::
TmpVectors (const unsigned int max_size,
VectorMemory<VECTOR> &vmem)
:
mem(vmem),
data (max_size, 0),
offset(0)
{}
template <class VECTOR>
inline
TmpVectors<VECTOR>::~TmpVectors ()
{
for (typename std::vector<VECTOR*>::iterator v = data.begin();
v != data.end(); ++v)
if (*v != 0)
mem.free(*v);
}
template <class VECTOR>
inline VECTOR&
TmpVectors<VECTOR>::operator[] (const unsigned int i) const
{
Assert (i+offset<data.size(),
ExcIndexRange(i, -offset, data.size()-offset));
Assert (data[i-offset] != 0, ExcNotInitialized());
return *data[i-offset];
}
template <class VECTOR>
inline VECTOR&
TmpVectors<VECTOR>::operator() (const unsigned int i,
const VECTOR &temp)
{
Assert (i+offset<data.size(),
ExcIndexRange(i,-offset, data.size()-offset));
if (data[i-offset] == 0)
{
data[i-offset] = mem.alloc();
data[i-offset]->reinit(temp);
}
return *data[i-offset];
}
}
}
template <class VECTOR>
inline
SolverGMRES<VECTOR>::AdditionalData::
AdditionalData (const unsigned int max_n_tmp_vectors,
const bool right_preconditioning,
const bool use_default_residual)
:
max_n_tmp_vectors(max_n_tmp_vectors),
right_preconditioning(right_preconditioning),
use_default_residual(use_default_residual)
{}
template <class VECTOR>
SolverGMRES<VECTOR>::SolverGMRES (SolverControl &cn,
VectorMemory<VECTOR> &mem,
const AdditionalData &data)
:
Solver<VECTOR> (cn,mem),
additional_data(data)
{}
template <class VECTOR>
SolverGMRES<VECTOR>::SolverGMRES (SolverControl &cn,
const AdditionalData &data) :
Solver<VECTOR> (cn),
additional_data(data)
{}
template <class VECTOR>
inline
void
SolverGMRES<VECTOR>::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 VECTOR>
template<class MATRIX, class PRECONDITIONER>
void
SolverGMRES<VECTOR>::solve (const MATRIX &A,
VECTOR &x,
const VECTOR &b,
const PRECONDITIONER &precondition)
{
// this code was written a very
// long time ago by people not
// associated with deal.II. we
// don't make any guarantees to its
// optimality or that it even works
// as expected...
//TODO:[?] Check, why there are two different start residuals.
//TODO:[GK] Make sure the parameter in the constructor means maximum basis size
deallog.push("GMRES");
const unsigned int n_tmp_vectors = additional_data.max_n_tmp_vectors;
// Generate an object where basis
// vectors are stored.
internal::SolverGMRES::TmpVectors<VECTOR> 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;
// matrix used for the orthogonalization
// process later
H.reinit(n_tmp_vectors, n_tmp_vectors-1);
// some additional vectors, also used
// in the orthogonalization
dealii::Vector<double>
gamma(n_tmp_vectors),
ci (n_tmp_vectors-1),
si (n_tmp_vectors-1),
h (n_tmp_vectors-1);
unsigned int dim = 0;
SolverControl::State iteration_state = SolverControl::iterate;
// 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
VECTOR &v = tmp_vectors(0, x);
VECTOR &p = tmp_vectors(n_tmp_vectors-1, x);
// Following vectors are needed
// when not the default residuals
// are used as stopping criterion
VECTOR *r=0;
VECTOR *x_=0;
dealii::Vector<double> *gamma_=0;
if (!use_default_residual)
{
r=this->memory.alloc();
x_=this->memory.alloc();
r->reinit(x);
x_->reinit(x);
gamma_ = new dealii::Vector<double> (gamma.size());
}
///////////////////////////////////
// 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);
precondition.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)
{
iteration_state = this->control().check (
accumulated_iterations, rho);
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
precondition.vmult(*r,v);
double res = r->l2_norm();
iteration_state = this->control().check (
accumulated_iterations, res);
if (iteration_state != SolverControl::iterate)
{
this->memory.free(r);
this->memory.free(x_);
delete gamma_;
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
VECTOR& vv = tmp_vectors(inner_iteration+1, x);
if (left_precondition)
{
A.vmult(p, tmp_vectors[inner_iteration]);
precondition.vmult(vv,p);
} else {
precondition.vmult(p, tmp_vectors[inner_iteration]);
A.vmult(vv,p);
};
dim = inner_iteration+1;
/* Orthogonalization */
for (unsigned int i=0 ; i<dim ; ++i)
{
h(i) = vv * tmp_vectors[i];
vv.add(-h(i), tmp_vectors[i]);
};
/* Re-orthogonalization */
for (unsigned int i=0 ; i<dim ; ++i)
{
double htmp = vv * tmp_vectors[i];
h(i) += htmp;
vv.add(-htmp, tmp_vectors[i]);
}
const double s = vv.l2_norm();
h(inner_iteration+1) = s;
//TODO: s=0 is a lucky breakdown. Handle this somehow decently
vv *= 1./s;
/* Transformation into
triagonal structure */
givens_rotation(h,gamma,ci,si,inner_iteration);
/* append vector on matrix */
for (unsigned int i=0; i<dim; ++i)
H(i,inner_iteration) = h(i);
/* default residual */
rho = std::fabs(gamma(dim));
if (use_default_residual)
iteration_state = this->control().check (
accumulated_iterations, rho);
else
{
deallog << "default_res=" << rho << std::endl;
dealii::Vector<double> h_(dim);
*x_=x;
*gamma_=gamma;
H1.reinit(dim+1,dim);
for (unsigned int i=0; i<dim+1; ++i)
for (unsigned int j=0; j<dim; ++j)
H1(i,j) = H(i,j);
H1.backward(h_,*gamma_);
if (left_precondition)
for (unsigned int i=0 ; i<dim; ++i)
x_->add(h_(i), tmp_vectors[i]);
else
{
p = 0.;
for (unsigned int i=0; i<dim; ++i)
p.add(h_(i), tmp_vectors[i]);
precondition.vmult(*r,p);
x_->add(1.,*r);
};
A.vmult(*r,*x_);
r->sadd(-1.,1.,b);
// Now *r contains the
// unpreconditioned
// residual!!
if (left_precondition)
{
const double res=r->l2_norm();
iteration_state = this->control().check (
accumulated_iterations, res);
}
else
{
precondition.vmult(*x_, *r);
const double preconditioned_res=x_->l2_norm();
iteration_state = this->control().check (
accumulated_iterations, preconditioned_res);
}
}
};
// end of inner iteration. now
// calculate the solution from
// the temporary vectors
h.reinit(dim);
H1.reinit(dim+1,dim);
for (unsigned int i=0; i<dim+1; ++i)
for (unsigned int j=0; j<dim; ++j)
H1(i,j) = H(i,j);
H1.backward(h,gamma);
if (left_precondition)
for (unsigned int i=0 ; i<dim; ++i)
x.add(h(i), tmp_vectors[i]);
else
{
p = 0.;
for (unsigned int i=0; i<dim; ++i)
p.add(h(i), tmp_vectors[i]);
precondition.vmult(v,p);
x.add(1.,v);
};
// end of outer iteration. restart if
// no convergence and the number of
// iterations is not exceeded
}
while (iteration_state == SolverControl::iterate);
if (!use_default_residual)
{
this->memory.free(r);
this->memory.free(x_);
delete gamma_;
}
deallog.pop();
// in case of failure: throw
// exception
if (this->control().last_check() != SolverControl::success)
throw SolverControl::NoConvergence (this->control().last_step(),
this->control().last_value());
// otherwise exit as normal
}
template<class VECTOR>
double
SolverGMRES<VECTOR>::criterion ()
{
// dummy implementation. this function is
// not needed for the present implementation
// of gmres
Assert (false, ExcInternalError());
return 0;
}
//----------------------------------------------------------------------//
template <class VECTOR>
SolverFGMRES<VECTOR>::SolverFGMRES (SolverControl &cn,
VectorMemory<VECTOR> &mem,
const AdditionalData &data)
:
Solver<VECTOR> (cn, mem),
additional_data(data)
{}
template <class VECTOR>
SolverFGMRES<VECTOR>::SolverFGMRES (SolverControl &cn,
const AdditionalData &data)
:
Solver<VECTOR> (cn),
additional_data(data)
{}
template<class VECTOR>
template<class MATRIX, class PRECONDITIONER>
void
SolverFGMRES<VECTOR>::solve (
const MATRIX& A,
VECTOR& x,
const VECTOR& b,
const PRECONDITIONER& precondition)
{
deallog.push("FGMRES");
SolverControl::State iteration_state = SolverControl::iterate;
const unsigned int basis_size = additional_data.max_basis_size;
// Generate an object where basis
// vectors are stored.
typename internal::SolverGMRES::TmpVectors<VECTOR> v (basis_size, this->memory);
typename internal::SolverGMRES::TmpVectors<VECTOR> z (basis_size, this->memory);
// number of the present iteration; this
// number is not reset to zero upon a
// restart
unsigned int accumulated_iterations = 0;
// matrix used for the orthogonalization
// process later
H.reinit(basis_size+1, basis_size);
// Vectors for projected system
Vector<double> projected_rhs;
Vector<double> y;
// Iteration starts here
do
{
VECTOR* aux = this->memory.alloc();
aux->reinit(x);
A.vmult(*aux, x);
aux->sadd(-1., 1., b);
double beta = aux->l2_norm();
if (this->control().check(accumulated_iterations,beta)
== SolverControl::success)
break;
H.reinit(basis_size+1, basis_size);
double a = beta;
for (unsigned int j=0;j<basis_size;++j)
{
v(j,x).equ(1./a, *aux);
precondition.vmult(z(j,x), v[j]);
A.vmult(*aux, z[j]);
// Gram-Schmidt
for (unsigned int i=0;i<=j;++i)
{
H(i,j) = *aux * v[i];
aux->add(-H(i,j), v[i]);
}
H(j+1,j) = a = aux->l2_norm();
// Compute projected solution
if (j>0)
{
H1.reinit(j+1,j);
projected_rhs.reinit(j+1);
y.reinit(j);
projected_rhs(0) = beta;
H1.fill(H);
Householder<double> house(H1);
double res = house.least_squares(y, projected_rhs);
iteration_state = this->control().check(++accumulated_iterations, res);
if (iteration_state != SolverControl::iterate)
break;
}
}
// Update solution vector
for (unsigned int j=0;j<y.size();++j)
x.add(y(j), z[j]);
this->memory.free(aux);
} while (iteration_state == SolverControl::iterate);
deallog.pop();
// in case of failure: throw
// exception
if (this->control().last_check() != SolverControl::success)
throw SolverControl::NoConvergence (this->control().last_step(),
this->control().last_value());
}
#endif // DOXYGEN
DEAL_II_NAMESPACE_CLOSE
#endif
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