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//
// Copyright (C) 1999 - 2015 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 at
// the top level of the deal.II distribution.
//
// ---------------------------------------------------------------------
#ifndef dealii__solver_qmrs_h
#define dealii__solver_qmrs_h
#include <deal.II/base/config.h>
#include <deal.II/lac/solver.h>
#include <deal.II/lac/solver_control.h>
#include <deal.II/base/logstream.h>
#include <cmath>
#include <deal.II/base/subscriptor.h>
#include <cmath>
DEAL_II_NAMESPACE_OPEN
/*!@addtogroup Solvers */
/*@{*/
/**
* Quasi-minimal residual method for symmetric matrices.
*
* The QMRS method is supposed to solve symmetric indefinite linear systems
* with symmetric, not necessarily definite preconditioners. This version of
* QMRS is adapted from Freund/Nachtigal: Software for simplified Lanczos and
* QMR algorithms, Appl. Num. Math. 19 (1995), pp. 319-341
*
* This version is for right preconditioning only, since then only the
* preconditioner is used: left preconditioning seems to require the inverse.
*
* 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 @p
* 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 @p 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.
*
* However, since the QMRS method does not need additional data, the
* respective structure is empty and does not offer any functionality. The
* constructor has a default argument, so you may call it without the
* additional parameter.
*
*
* <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.
*
*
* @author Guido Kanschat, 1999
*/
template <typename VectorType = Vector<double> >
class SolverQMRS : public Solver<VectorType>
{
public:
/**
* Standardized data struct to pipe additional data to the solver.
*
* There are two possibilities to compute the residual: one is an estimate
* using the computed value @p tau. The other is exact computation using
* another matrix vector multiplication.
*
* QMRS, is susceptible to breakdowns, so we need a parameter telling us,
* which numbers are considered zero. The proper breakdown criterion is very
* unclear, so experiments may be necessary here.
*/
struct AdditionalData
{
/**
* Constructor.
*
* The default is no exact residual computation and breakdown parameter
* 1e-16.
*/
explicit
AdditionalData(bool exact_residual = false,
double breakdown=1.e-16) :
exact_residual(exact_residual),
breakdown(breakdown)
{}
/**
* Flag for exact computation of residual.
*/
bool exact_residual;
/**
* Breakdown threshold.
*/
double breakdown;
};
/**
* Constructor.
*/
SolverQMRS (SolverControl &cn,
VectorMemory<VectorType> &mem,
const AdditionalData &data=AdditionalData());
/**
* Constructor. Use an object of type GrowingVectorMemory as a default to
* allocate memory.
*/
SolverQMRS (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 &precondition);
/**
* 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.
*/
virtual void print_vectors (const unsigned int step,
const VectorType &x,
const VectorType &r,
const VectorType &d) const;
protected:
/**
* Implementation of the computation of the norm of the residual.
*/
virtual double criterion();
/**
* Temporary vectors, allocated through the @p VectorMemory object at the
* start of the actual solution process and deallocated at the end.
*/
VectorType *Vv;
VectorType *Vp;
VectorType *Vq;
VectorType *Vt;
VectorType *Vd;
/**
* Iteration vector.
*/
VectorType *Vx;
/**
* RHS vector.
*/
const VectorType *Vb;
/**
* Within the iteration loop, the square of the residual vector is stored in
* this variable. The function @p criterion uses this variable to compute
* the convergence value, which in this class is the norm of the residual
* vector and thus the square root of the @p res2 value.
*/
double res2;
/**
* Additional parameters.
*/
AdditionalData additional_data;
private:
/**
* A structure returned by the iterate() function representing what it found
* is happening during the iteration.
*/
struct IterationResult
{
SolverControl::State state;
double last_residual;
IterationResult (const SolverControl::State state,
const double last_residual);
};
/**
* The iteration loop itself. The function returns a structure indicating
* what happened in this function.
*/
template<typename MatrixType, typename PreconditionerType>
IterationResult
iterate (const MatrixType &A,
const PreconditionerType &precondition);
/**
* Number of the current iteration (accumulated over restarts)
*/
unsigned int step;
};
/*@}*/
/*------------------------- Implementation ----------------------------*/
#ifndef DOXYGEN
template<class VectorType>
SolverQMRS<VectorType>::IterationResult::IterationResult (const SolverControl::State state,
const double last_residual)
:
state (state),
last_residual (last_residual)
{}
template<class VectorType>
SolverQMRS<VectorType>::SolverQMRS (SolverControl &cn,
VectorMemory<VectorType> &mem,
const AdditionalData &data)
:
Solver<VectorType>(cn,mem),
additional_data(data)
{}
template<class VectorType>
SolverQMRS<VectorType>::SolverQMRS(SolverControl &cn,
const AdditionalData &data)
:
Solver<VectorType>(cn),
additional_data(data)
{}
template<class VectorType>
double
SolverQMRS<VectorType>::criterion()
{
return std::sqrt(res2);
}
template<class VectorType>
void
SolverQMRS<VectorType>::print_vectors(const unsigned int,
const VectorType &,
const VectorType &,
const VectorType &) const
{}
template<class VectorType>
template<typename MatrixType, typename PreconditionerType>
void
SolverQMRS<VectorType>::solve (const MatrixType &A,
VectorType &x,
const VectorType &b,
const PreconditionerType &precondition)
{
deallog.push("QMRS");
// Memory allocation
Vv = this->memory.alloc();
Vp = this->memory.alloc();
Vq = this->memory.alloc();
Vt = this->memory.alloc();
Vd = this->memory.alloc();
Vx = &x;
Vb = &b;
// resize the vectors, but do not set
// the values since they'd be overwritten
// soon anyway.
Vv->reinit(x, true);
Vp->reinit(x, true);
Vq->reinit(x, true);
Vt->reinit(x, true);
step = 0;
IterationResult state (SolverControl::failure,0);
do
{
if (step > 0)
deallog << "Restart step " << step << std::endl;
state = iterate(A, precondition);
}
while (state.state == SolverControl::iterate);
// Deallocate Memory
this->memory.free(Vv);
this->memory.free(Vp);
this->memory.free(Vq);
this->memory.free(Vt);
this->memory.free(Vd);
// Output
deallog.pop();
// in case of failure: throw exception
AssertThrow(state.state == SolverControl::success,
SolverControl::NoConvergence (step,
state.last_residual));
// otherwise exit as normal
}
template<class VectorType>
template<typename MatrixType, typename PreconditionerType>
typename SolverQMRS<VectorType>::IterationResult
SolverQMRS<VectorType>::iterate(const MatrixType &A,
const PreconditionerType &precondition)
{
/* Remark: the matrix A in the article is the preconditioned matrix.
* Therefore, we have to precondition x before we compute the first residual.
* In step 1 we replace p by q to avoid one preconditioning step.
* There are still two steps left, making this algorithm expensive.
*/
SolverControl::State state = SolverControl::iterate;
// define some aliases for simpler access
VectorType &v = *Vv;
VectorType &p = *Vp;
VectorType &q = *Vq;
VectorType &t = *Vt;
VectorType &d = *Vd;
VectorType &x = *Vx;
const VectorType &b = *Vb;
int it=0;
double tau, rho, theta=0, sigma, alpha, psi, theta_old, rho_old, beta;
double res;
d.reinit(x);
// Apply right preconditioning to x
precondition.vmult(q,x);
// Preconditioned residual
A.vmult(v,q);
v.sadd(-1.,1.,b);
res = v.l2_norm();
if (this->iteration_status(step, res, x) == SolverControl::success)
return IterationResult(SolverControl::success, res);
p = v;
precondition.vmult(q,p);
tau = v.norm_sqr();
rho = q*v;
while (state == SolverControl::iterate)
{
step++;
it++;
// Step 1
A.vmult(t,q);
// Step 2
sigma = q*t;
//TODO:[?] Find a really good breakdown criterion. The absolute one detects breakdown instead of convergence
if (std::fabs(sigma/rho) < additional_data.breakdown)
return IterationResult(SolverControl::iterate, std::fabs(sigma/rho));
// Step 3
alpha = rho/sigma;
v.add(-alpha,t);
// Step 4
theta_old = theta;
theta = v*v/tau;
psi = 1./(1.+theta);
tau *= theta*psi;
d.sadd(psi*theta_old, psi*alpha, p);
x.add(d);
print_vectors(step,x,v,d);
// Step 5
if (additional_data.exact_residual)
{
A.vmult(q,x);
q.sadd(-1.,1.,b);
res = q.l2_norm();
}
else
res = std::sqrt((it+1)*tau);
state = this->iteration_status(step,res,x);
if ((state == SolverControl::success)
|| (state == SolverControl::failure))
return IterationResult(state, res);
// Step 6
if (std::fabs(rho) < additional_data.breakdown)
return IterationResult(SolverControl::iterate, std::fabs(rho));
// Step 7
rho_old = rho;
precondition.vmult(q,v);
rho = q*v;
beta = rho/rho_old;
p.sadd(beta,v);
precondition.vmult(q,p);
}
return IterationResult(SolverControl::success, std::fabs(rho));
}
#endif // DOXYGEN
DEAL_II_NAMESPACE_CLOSE
#endif
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