solver_qmrs.h

// ---------------------------------------------------------------------
//
// Copyright (C) 1999 - 2018 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_qmrs_h
#define dealii_solver_qmrs_h

#include <deal.II/base/config.h>

#include <deal.II/base/logstream.h>
#include <deal.II/base/subscriptor.h>

#include <deal.II/lac/solver.h>
#include <deal.II/lac/solver_control.h>

#include <cmath>

DEAL_II_NAMESPACE_OPEN


template <typename VectorType = Vector<double>>
class SolverQMRS : public Solver<VectorType>
{
public:
  struct AdditionalData
  {
    explicit AdditionalData(const bool   left_preconditioning = false,
                            const double solver_tolerance     = 1.e-9,
                            const bool   breakdown_testing    = true,
                            const double breakdown_threshold  = 1.e-16)
      : left_preconditioning(left_preconditioning)
      , solver_tolerance(solver_tolerance)
      , breakdown_testing(breakdown_testing)
      , breakdown_threshold(breakdown_threshold)
    {}

    bool left_preconditioning;

    double solver_tolerance;

    bool breakdown_testing;

    double breakdown_threshold;
  };

  SolverQMRS(SolverControl &           cn,
             VectorMemory<VectorType> &mem,
             const AdditionalData &    data = AdditionalData());

  SolverQMRS(SolverControl &cn, const AdditionalData &data = AdditionalData());

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

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

protected:
  AdditionalData additional_data;

private:
  struct IterationResult
  {
    SolverControl::State state;
    double               last_residual;

    IterationResult(const SolverControl::State state,
                    const double               last_residual);
  };

  template <typename MatrixType, typename PreconditionerType>
  IterationResult
  iterate(const MatrixType &        A,
          VectorType &              x,
          const VectorType &        b,
          const PreconditionerType &preconditioner,
          VectorType &              r,
          VectorType &              u,
          VectorType &              q,
          VectorType &              t,
          VectorType &              d);

  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)
  , step(0)
{}

template <class VectorType>
SolverQMRS<VectorType>::SolverQMRS(SolverControl &       cn,
                                   const AdditionalData &data)
  : Solver<VectorType>(cn)
  , additional_data(data)
  , step(0)
{}

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 &preconditioner)
{
  LogStream::Prefix prefix("SQMR");


  // temporary vectors, allocated trough the @p VectorMemory object at the
  // start of the actual solution process and deallocated at the end.
  typename VectorMemory<VectorType>::Pointer Vr(this->memory);
  typename VectorMemory<VectorType>::Pointer Vu(this->memory);
  typename VectorMemory<VectorType>::Pointer Vq(this->memory);
  typename VectorMemory<VectorType>::Pointer Vt(this->memory);
  typename VectorMemory<VectorType>::Pointer Vd(this->memory);


  // resize the vectors, but do not set
  // the values since they'd be overwritten
  // soon anyway.
  Vr->reinit(x, true);
  Vu->reinit(x, true);
  Vq->reinit(x, true);
  Vt->reinit(x, true);
  Vd->reinit(x, true);

  step = 0;

  IterationResult state(SolverControl::failure, 0);

  do
    {
      if (step > 0)
        deallog << "Restart step " << step << std::endl;
      state = iterate(A, x, b, preconditioner, *Vr, *Vu, *Vq, *Vt, *Vd);
    }
  while (state.state == SolverControl::iterate);


  // 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,
                                VectorType &              x,
                                const VectorType &        b,
                                const PreconditionerType &preconditioner,
                                VectorType &              r,
                                VectorType &              u,
                                VectorType &              q,
                                VectorType &              t,
                                VectorType &              d)
{
  SolverControl::State state = SolverControl::iterate;

  int it = 0;

  double tau, rho, theta = 0;
  double res;

  // Compute the start residual
  A.vmult(r, x);
  r.sadd(-1., 1., b);

  // Doing the initial preconditioning
  if (additional_data.left_preconditioning)
    {
      // Left preconditioning
      preconditioner.vmult(t, r);
      q = t;
    }
  else
    {
      // Right preconditioning
      t = r;
      preconditioner.vmult(q, t);
    }

  tau = t.norm_sqr();
  res = std::sqrt(tau);

  if (this->iteration_status(step, res, x) == SolverControl::success)
    return IterationResult(SolverControl::success, res);

  rho = q * r;

  while (state == SolverControl::iterate)
    {
      step++;
      it++;
      //--------------------------------------------------------------
      // Step 1: apply the system matrix and compute one inner product
      //--------------------------------------------------------------
      A.vmult(t, q);
      const double sigma = q * t;

      // Check the breakdown criterion
      if (additional_data.breakdown_testing == true &&
          std::fabs(sigma) < additional_data.breakdown_threshold)
        return IterationResult(SolverControl::iterate, res);
      // Update the residual
      const double alpha = rho / sigma;
      r.add(-alpha, t);

      //--------------------------------------------------------------
      // Step 2: update the solution vector
      //--------------------------------------------------------------
      const double theta_old = theta;

      // Apply the preconditioner
      if (additional_data.left_preconditioning)
        {
          // Left Preconditioning
          preconditioner.vmult(t, r);
        }
      else
        {
          // Right Preconditioning
          t = r;
        }

      // Double updates
      theta            = t * t / tau;
      const double psi = 1. / (1. + theta);
      tau *= theta * psi;

      // Actual update of the solution vector
      d.sadd(psi * theta_old, psi * alpha, q);
      x += d;

      print_vectors(step, x, r, d);

      // Check for convergence
      // Compute a simple and cheap upper bound of the norm of the residual
      // vector b-Ax
      res = std::sqrt((it + 1) * tau);
      // If res lies close enough, within the desired tolerance, calculate the
      // exact residual
      if (res < additional_data.solver_tolerance)
        {
          A.vmult(u, x);
          u.sadd(-1., 1., b);
          res = u.l2_norm();
        }
      state = this->iteration_status(step, res, x);
      if ((state == SolverControl::success) ||
          (state == SolverControl::failure))
        return IterationResult(state, res);

      //--------------------------------------------------------------
      // Step 3: check breakdown criterion and update the vectors
      //--------------------------------------------------------------
      if (additional_data.breakdown_testing == true &&
          std::fabs(sigma) < additional_data.breakdown_threshold)
        return IterationResult(SolverControl::iterate, res);

      const double rho_old = rho;

      // Applying the preconditioner
      if (additional_data.left_preconditioning)
        {
          // Left preconditioning
          u = t;
        }
      else
        {
          // Right preconditioning
          preconditioner.vmult(u, t);
        }

      // Double and vector updates
      rho               = u * r;
      const double beta = rho / rho_old;
      q.sadd(beta, u);
    }
  return IterationResult(SolverControl::success, res);
}

#endif // DOXYGEN

DEAL_II_NAMESPACE_CLOSE

#endif