libdeal.ii-dev 8.4.2-2+b1 / usr / include / deal.II / lac / solver_bicgstab.h

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// Copyright (C) 1998 - 2015 by the deal.II authors
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#ifndef dealii__solver_bicgstab_h
#define dealii__solver_bicgstab_h


#include <deal.II/base/config.h>
#include <deal.II/base/logstream.h>
#include <deal.II/lac/solver.h>
#include <deal.II/lac/solver_control.h>
#include <cmath>
#include <deal.II/base/subscriptor.h>

DEAL_II_NAMESPACE_OPEN

/*!@addtogroup Solvers */
/*@{*/

/**
 * Bicgstab algorithm by van der Vorst.
 *
 * 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.
 *
 * The Bicgstab-method has two additional parameters: the first is a boolean,
 * deciding whether to compute the actual residual in each step (@p true) or
 * to use the length of the computed orthogonal residual (@p false). Note that
 * computing the residual causes a third matrix-vector-multiplication, though
 * no additional preconditioning, in each step. The reason for doing this is,
 * that the size of the orthogonalized residual computed during the iteration
 * may be larger by orders of magnitude than the true residual. This is due to
 * numerical instabilities related to badly conditioned matrices. Since this
 * instability results in a bad stopping criterion, the default for this
 * parameter is @p true. Whenever the user knows that the estimated residual
 * works reasonably as well, the flag should be set to @p false in order to
 * increase the performance of the solver.
 *
 * The second parameter is the size of a breakdown criterion. It is difficult
 * to find a general good criterion, so if things do not work for you, try to
 * change this value.
 *
 *
 * <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.
 *
 */
template <typename VectorType = Vector<double> >
class SolverBicgstab : public Solver<VectorType>
{
public:
  /**
   * 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. This increases the costs of the
   * algorithm, so it is should be set to false whenever the problem allows
   * it.
   *
   * Bicgstab is susceptible to breakdowns, so we need a parameter telling us,
   * which numbers are considered zero.
   */
  struct AdditionalData
  {
    /**
     * Constructor.
     *
     * The default is to perform an exact residual computation and breakdown
     * parameter 1e-10.
     */
    explicit
    AdditionalData(const bool   exact_residual = true,
                   const double breakdown      = 1.e-10)
      : exact_residual(exact_residual),
        breakdown(breakdown)
    {}
    /**
     * Flag for exact computation of residual.
     */
    bool exact_residual;
    /**
     * Breakdown threshold.
     */
    double breakdown;
  };

  /**
   * Constructor.
   */
  SolverBicgstab (SolverControl            &cn,
                  VectorMemory<VectorType> &mem,
                  const AdditionalData     &data=AdditionalData());

  /**
   * Constructor. Use an object of type GrowingVectorMemory as a default to
   * allocate memory.
   */
  SolverBicgstab (SolverControl        &cn,
                  const AdditionalData &data=AdditionalData());

  /**
   * Virtual destructor.
   */
  virtual ~SolverBicgstab ();

  /**
   * Solve primal problem only.
   */
  template<typename MatrixType, typename PreconditionerType>
  void
  solve (const MatrixType         &A,
         VectorType               &x,
         const VectorType         &b,
         const PreconditionerType &precondition);

protected:
  /**
   * Computation of the stopping criterion.
   */
  template <typename MatrixType>
  double criterion (const MatrixType &A, const VectorType &x, const VectorType &b);

  /**
   * 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;

  /**
   * Auxiliary vector.
   */
  VectorType *Vx;
  /**
   * Auxiliary vector.
   */
  VectorType *Vr;
  /**
   * Auxiliary vector.
   */
  VectorType *Vrbar;
  /**
   * Auxiliary vector.
   */
  VectorType *Vp;
  /**
   * Auxiliary vector.
   */
  VectorType *Vy;
  /**
   * Auxiliary vector.
   */
  VectorType *Vz;
  /**
   * Auxiliary vector.
   */
  VectorType *Vt;
  /**
   * Auxiliary vector.
   */
  VectorType *Vv;
  /**
   * Right hand side vector.
   */
  const VectorType *Vb;

  /**
   * Auxiliary value.
   */
  double alpha;
  /**
   * Auxiliary value.
   */
  double beta;
  /**
   * Auxiliary value.
   */
  double omega;
  /**
   * Auxiliary value.
   */
  double rho;
  /**
   * Auxiliary value.
   */
  double rhobar;

  /**
   * Current iteration step.
   */
  unsigned int step;

  /**
   * Residual.
   */
  double res;

  /**
   * Additional parameters.
   */
  AdditionalData additional_data;

private:
  /**
   * Everything before the iteration loop.
   */
  template <typename MatrixType>
  SolverControl::State start(const MatrixType &A);

  /**
   * A structure returned by the iterate() function representing what it found
   * is happening during the iteration.
   */
  struct IterationResult
  {
    bool                 breakdown;
    SolverControl::State state;
    unsigned int         last_step;
    double               last_residual;

    IterationResult (const bool breakdown,
                     const SolverControl::State state,
                     const unsigned int         last_step,
                     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);
};

/*@}*/
/*-------------------------Inline functions -------------------------------*/

#ifndef DOXYGEN


template<typename VectorType>
SolverBicgstab<VectorType>::IterationResult::IterationResult
(const bool                 breakdown,
 const SolverControl::State state,
 const unsigned int         last_step,
 const double               last_residual)
  :
  breakdown (breakdown),
  state (state),
  last_step (last_step),
  last_residual (last_residual)
{}


template<typename VectorType>
SolverBicgstab<VectorType>::SolverBicgstab (SolverControl            &cn,
                                            VectorMemory<VectorType> &mem,
                                            const AdditionalData     &data)
  :
  Solver<VectorType>(cn,mem),
  additional_data(data)
{}



template<typename VectorType>
SolverBicgstab<VectorType>::SolverBicgstab (SolverControl        &cn,
                                            const AdditionalData &data)
  :
  Solver<VectorType>(cn),
  additional_data(data)
{}



template<typename VectorType>
SolverBicgstab<VectorType>::~SolverBicgstab ()
{}



template <typename VectorType>
template <typename MatrixType>
double
SolverBicgstab<VectorType>::criterion (const MatrixType &A,
                                       const VectorType &x,
                                       const VectorType &b)
{
  A.vmult(*Vt, x);
  Vt->add(-1.,b);
  res = Vt->l2_norm();

  return res;
}



template <typename VectorType >
template <typename MatrixType>
SolverControl::State
SolverBicgstab<VectorType>::start (const MatrixType &A)
{
  A.vmult(*Vr, *Vx);
  Vr->sadd(-1.,1.,*Vb);
  res = Vr->l2_norm();

  return this->iteration_status(step, res, *Vx);
}



template<typename VectorType>
void
SolverBicgstab<VectorType>::print_vectors(const unsigned int,
                                          const VectorType &,
                                          const VectorType &,
                                          const VectorType &) const
{}



template<typename VectorType>
template<typename MatrixType, typename PreconditionerType>
typename SolverBicgstab<VectorType>::IterationResult
SolverBicgstab<VectorType>::iterate(const MatrixType         &A,
                                    const PreconditionerType &precondition)
{
//TODO:[GK] Implement "use the length of the computed orthogonal residual" in the BiCGStab method.
  SolverControl::State state = SolverControl::iterate;
  alpha = omega = rho = 1.;

  VectorType &r = *Vr;
  VectorType &rbar = *Vrbar;
  VectorType &p = *Vp;
  VectorType &y = *Vy;
  VectorType &z = *Vz;
  VectorType &t = *Vt;
  VectorType &v = *Vv;

  rbar = r;
  bool startup = true;

  do
    {
      ++step;

      rhobar = r*rbar;
      beta   = rhobar * alpha / (rho * omega);
      rho    = rhobar;
      if (startup == true)
        {
          p = r;
          startup = false;
        }
      else
        {
          p.sadd(beta, 1., r);
          p.add(-beta*omega, v);
        }

      precondition.vmult(y,p);
      A.vmult(v,y);
      rhobar = rbar * v;

      alpha = rho/rhobar;

//TODO:[?] Find better breakdown criterion

      if (std::fabs(alpha) > 1.e10)
        return IterationResult(true, state, step, res);

      res = std::sqrt(r.add_and_dot(-alpha, v, r));

      // check for early success, see the lac/bicgstab_early testcase as to
      // why this is necessary
      //
      // note: the vector *Vx we pass to the iteration_status signal here is only
      // the current approximation, not the one we will return with,
      // which will be x=*Vx + alpha*y
      if (this->iteration_status(step, res, *Vx) == SolverControl::success)
        {
          Vx->add(alpha, y);
          print_vectors(step, *Vx, r, y);
          return IterationResult(false, SolverControl::success, step, res);
        }

      precondition.vmult(z,r);
      A.vmult(t,z);
      rhobar = t*r;
      omega = rhobar/(t*t);
      Vx->add(alpha, y, omega, z);

      if (additional_data.exact_residual)
        {
          r.add(-omega, t);
          res = criterion(A, *Vx, *Vb);
        }
      else
        res = std::sqrt(r.add_and_dot(-omega, t, r));

      state = this->iteration_status(step, res, *Vx);
      print_vectors(step, *Vx, r, y);
    }
  while (state == SolverControl::iterate);
  return IterationResult(false, state, step, res);
}


template<typename VectorType>
template<typename MatrixType, typename PreconditionerType>
void
SolverBicgstab<VectorType>::solve(const MatrixType         &A,
                                  VectorType               &x,
                                  const VectorType         &b,
                                  const PreconditionerType &precondition)
{
  deallog.push("Bicgstab");
  Vr    = this->memory.alloc();
  Vr->reinit(x, true);
  Vrbar = this->memory.alloc();
  Vrbar->reinit(x, true);
  Vp    = this->memory.alloc();
  Vp->reinit(x, true);
  Vy    = this->memory.alloc();
  Vy->reinit(x, true);
  Vz    = this->memory.alloc();
  Vz->reinit(x, true);
  Vt    = this->memory.alloc();
  Vt->reinit(x, true);
  Vv    = this->memory.alloc();
  Vv->reinit(x, true);

  Vx = &x;
  Vb = &b;

  step = 0;

  IterationResult state(false,SolverControl::failure,0,0);

  // iterate while the inner iteration returns a breakdown
  do
    {
      if (step != 0)
        deallog << "Restart step " << step << std::endl;
      if (start(A) == SolverControl::success)
        {
          state.state = SolverControl::success;
          break;
        }
      state = iterate(A, precondition);
    }
  while (state.breakdown == true);

  this->memory.free(Vr);
  this->memory.free(Vrbar);
  this->memory.free(Vp);
  this->memory.free(Vy);
  this->memory.free(Vz);
  this->memory.free(Vt);
  this->memory.free(Vv);

  deallog.pop();

  // in case of failure: throw exception
  AssertThrow(state.state == SolverControl::success,
              SolverControl::NoConvergence (state.last_step,
                                            state.last_residual));
  // otherwise exit as normal
}

#endif // DOXYGEN

DEAL_II_NAMESPACE_CLOSE

#endif