eigen.h

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//
// Copyright (C) 2000 - 2020 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.
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#ifndef dealii_eigen_h
#define dealii_eigen_h


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

#include <deal.II/lac/linear_operator.h>
#include <deal.II/lac/precondition.h>
#include <deal.II/lac/solver.h>
#include <deal.II/lac/solver_cg.h>
#include <deal.II/lac/solver_control.h>
#include <deal.II/lac/solver_gmres.h>
#include <deal.II/lac/solver_minres.h>
#include <deal.II/lac/vector_memory.h>

#include <cmath>

DEAL_II_NAMESPACE_OPEN



template <typename VectorType = Vector<double>>
class EigenPower : private SolverBase<VectorType>
{
public:
  using size_type = types::global_dof_index;

  struct AdditionalData
  {
    double shift;
    AdditionalData(const double shift = 0.)
      : shift(shift)
    {}
  };

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


  template <typename MatrixType>
  void
  solve(double &value, const MatrixType &A, VectorType &x);

protected:
  AdditionalData additional_data;
};

template <typename VectorType = Vector<double>>
class EigenInverse : private SolverBase<VectorType>
{
public:
  using size_type = types::global_dof_index;

  struct AdditionalData
  {
    double relaxation;

    unsigned int start_adaption;
    bool use_residual;
    AdditionalData(double       relaxation     = 1.,
                   unsigned int start_adaption = 6,
                   bool         use_residual   = true)
      : relaxation(relaxation)
      , start_adaption(start_adaption)
      , use_residual(use_residual)
    {}
  };

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

  template <typename MatrixType>
  void
  solve(double &value, const MatrixType &A, VectorType &x);

protected:
  AdditionalData additional_data;
};

//---------------------------------------------------------------------------


template <class VectorType>
EigenPower<VectorType>::EigenPower(SolverControl &           cn,
                                   VectorMemory<VectorType> &mem,
                                   const AdditionalData &    data)
  : SolverBase<VectorType>(cn, mem)
  , additional_data(data)
{}



template <class VectorType>
template <typename MatrixType>
void
EigenPower<VectorType>::solve(double &value, const MatrixType &A, VectorType &x)
{
  SolverControl::State conv = SolverControl::iterate;

  LogStream::Prefix prefix("Power method");

  typename VectorMemory<VectorType>::Pointer Vy(this->memory);
  VectorType &                               y = *Vy;
  y.reinit(x);
  typename VectorMemory<VectorType>::Pointer Vr(this->memory);
  VectorType &                               r = *Vr;
  r.reinit(x);

  double length     = x.l2_norm();
  double old_length = 0.;
  x *= 1. / length;

  A.vmult(y, x);

  // Main loop
  int iter = 0;
  for (; conv == SolverControl::iterate; iter++)
    {
      y.add(additional_data.shift, x);

      // Compute absolute value of eigenvalue
      old_length = length;
      length     = y.l2_norm();

      // do a little trick to compute the sign
      // with not too much effect of round-off errors.
      double    entry  = 0.;
      size_type i      = 0;
      double    thresh = length / x.size();
      do
        {
          Assert(i < x.size(), ExcInternalError());
          entry = y(i++);
        }
      while (std::fabs(entry) < thresh);

      --i;

      // Compute unshifted eigenvalue
      value = (entry * x(i) < 0.) ? -length : length;
      value -= additional_data.shift;

      // Update normalized eigenvector
      x.equ(1 / length, y);

      // Compute residual
      A.vmult(y, x);

      // Check the change of the eigenvalue
      // Brrr, this is not really a good criterion
      conv = this->iteration_status(iter,
                                    std::fabs(1. / length - 1. / old_length),
                                    x);
    }

  // in case of failure: throw exception
  AssertThrow(conv == SolverControl::success,
              SolverControl::NoConvergence(
                iter, std::fabs(1. / length - 1. / old_length)));

  // otherwise exit as normal
}

//---------------------------------------------------------------------------

template <class VectorType>
EigenInverse<VectorType>::EigenInverse(SolverControl &           cn,
                                       VectorMemory<VectorType> &mem,
                                       const AdditionalData &    data)
  : SolverBase<VectorType>(cn, mem)
  , additional_data(data)
{}



template <class VectorType>
template <typename MatrixType>
void
EigenInverse<VectorType>::solve(double &          value,
                                const MatrixType &A,
                                VectorType &      x)
{
  LogStream::Prefix prefix("Wielandt");

  SolverControl::State conv = SolverControl::iterate;

  // Prepare matrix for solver
  auto   A_op          = linear_operator(A);
  double current_shift = -value;
  auto   A_s           = A_op + current_shift * identity_operator(A_op);

  // Define solver
  ReductionControl        inner_control(5000, 1.e-16, 1.e-5, false, false);
  PreconditionIdentity    prec;
  SolverGMRES<VectorType> solver(inner_control, this->memory);

  // Next step for recomputing the shift
  unsigned int goal = additional_data.start_adaption;

  // Auxiliary vector
  typename VectorMemory<VectorType>::Pointer Vy(this->memory);
  VectorType &                               y = *Vy;
  y.reinit(x);
  typename VectorMemory<VectorType>::Pointer Vr(this->memory);
  VectorType &                               r = *Vr;
  r.reinit(x);

  double length    = x.l2_norm();
  double old_value = value;

  x *= 1. / length;

  // Main loop
  double    res  = -std::numeric_limits<double>::max();
  size_type iter = 0;
  for (; conv == SolverControl::iterate; iter++)
    {
      solver.solve(A_s, y, x, prec);

      // Compute absolute value of eigenvalue
      length = y.l2_norm();

      // do a little trick to compute the sign
      // with not too much effect of round-off errors.
      double    entry  = 0.;
      size_type i      = 0;
      double    thresh = length / x.size();
      do
        {
          Assert(i < x.size(), ExcInternalError());
          entry = y(i++);
        }
      while (std::fabs(entry) < thresh);

      --i;

      // Compute unshifted eigenvalue
      value = (entry * x(i) < 0. ? -1. : 1.) / length - current_shift;

      if (iter == goal)
        {
          const auto & relaxation = additional_data.relaxation;
          const double new_shift =
            relaxation * (-value) + (1. - relaxation) * current_shift;

          A_s           = A_op + new_shift * identity_operator(A_op);
          current_shift = new_shift;

          ++goal;
        }

      // Update normalized eigenvector
      x.equ(1. / length, y);
      // Compute residual
      if (additional_data.use_residual)
        {
          y.equ(value, x);
          A.vmult(r, x);
          r.sadd(-1., value, x);
          res = r.l2_norm();
          // Check the residual
          conv = this->iteration_status(iter, res, x);
        }
      else
        {
          res  = std::fabs(1. / value - 1. / old_value);
          conv = this->iteration_status(iter, res, x);
        }
      old_value = value;
    }

  // in case of failure: throw
  // exception
  AssertThrow(conv == SolverControl::success,
              SolverControl::NoConvergence(iter, res));
  // otherwise exit as normal
}

DEAL_II_NAMESPACE_CLOSE

#endif