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// vi: set et ts=4 sw=2 sts=2:
#ifndef DUNE_ISTL_EIGENVALUE_POWERITERATION_HH
#define DUNE_ISTL_EIGENVALUE_POWERITERATION_HH
#include <cstddef> // provides std::size_t
#include <cmath> // provides std::sqrt, std::abs
#include <type_traits> // provides std::is_same
#include <iostream> // provides std::cout, std::endl
#include <limits> // provides std::numeric_limits
#include <ios> // provides std::left, std::ios::left
#include <iomanip> // provides std::setw, std::resetiosflags
#include <memory> // provides std::unique_ptr
#include <string> // provides std::string
#include <dune/common/exceptions.hh> // provides DUNE_THROW(...)
#include <dune/istl/operators.hh> // provides Dune::LinearOperator
#include <dune/istl/solvercategory.hh> // provides Dune::SolverCategory::sequential
#include <dune/istl/solvertype.hh> // provides Dune::IsDirectSolver
#include <dune/istl/operators.hh> // provides Dune::MatrixAdapter
#include <dune/istl/istlexception.hh> // provides Dune::ISTLError
#include <dune/istl/io.hh> // provides Dune::printvector(...)
#include <dune/istl/solvers.hh> // provides Dune::InverseOperatorResult
namespace Dune
{
/**
* \brief A linear operator scaling vectors by a scalar value.
* The scalar value can be changed as it is given in a
* form decomposed into an immutable and a mutable part.
*
* \author Sebastian Westerheide.
*/
template <class X, class Y = X>
class ScalingLinearOperator : public Dune::LinearOperator<X,Y>
{
public:
typedef X domain_type;
typedef Y range_type;
typedef typename X::field_type field_type;
enum {category = Dune::SolverCategory::sequential};
ScalingLinearOperator (field_type immutable_scaling,
const field_type& mutable_scaling)
: immutable_scaling_(immutable_scaling),
mutable_scaling_(mutable_scaling)
{}
virtual void apply (const X& x, Y& y) const
{
y = x;
y *= immutable_scaling_*mutable_scaling_;
}
virtual void applyscaleadd (field_type alpha, const X& x, Y& y) const
{
X temp(x);
temp *= immutable_scaling_*mutable_scaling_;
y.axpy(alpha,temp);
}
protected:
const field_type immutable_scaling_;
const field_type& mutable_scaling_;
};
/**
* \brief A linear operator representing the sum of two linear operators.
*
* \tparam OP1 Type of the first linear operator.
* \tparam OP2 Type of the second linear operator.
*
* \author Sebastian Westerheide.
*/
template <class OP1, class OP2>
class LinearOperatorSum
: public Dune::LinearOperator<typename OP1::domain_type,
typename OP1::range_type>
{
public:
typedef typename OP1::domain_type domain_type;
typedef typename OP1::range_type range_type;
typedef typename domain_type::field_type field_type;
enum {category = Dune::SolverCategory::sequential};
LinearOperatorSum (const OP1& op1, const OP2& op2)
: op1_(op1), op2_(op2)
{
static_assert(std::is_same<typename OP2::domain_type,domain_type>::value,
"Domain type of both operators doesn't match!");
static_assert(std::is_same<typename OP2::range_type,range_type>::value,
"Range type of both operators doesn't match!");
}
virtual void apply (const domain_type& x, range_type& y) const
{
op1_.apply(x,y);
op2_.applyscaleadd(1.0,x,y);
}
virtual void applyscaleadd (field_type alpha,
const domain_type& x, range_type& y) const
{
range_type temp(y);
op1_.apply(x,temp);
op2_.applyscaleadd(1.0,x,temp);
y.axpy(alpha,temp);
}
protected:
const OP1& op1_;
const OP2& op2_;
};
/**
* \brief Helper class for notifying a DUNE-ISTL linear solver about
* a change of the iteration matrix object in a unified way,
* i.e. independent from the solver's type (direct/iterative).
*
* \author Sebastian Westerheide.
*/
template <typename ISTLLinearSolver, typename BCRSMatrix>
class SolverHelper
{
public:
static void setMatrix (ISTLLinearSolver& solver,
const BCRSMatrix& matrix)
{
static const bool is_direct_solver
= Dune::IsDirectSolver<ISTLLinearSolver>::value;
SolverHelper<ISTLLinearSolver,BCRSMatrix>::
Implementation<is_direct_solver>::setMatrix(solver,matrix);
}
protected:
/**
* \brief Implementation that works together with iterative ISTL
* solvers, e.g. Dune::CGSolver or Dune::BiCGSTABSolver.
*/
template <bool is_direct_solver, typename Dummy = void>
struct Implementation
{
static void setMatrix (ISTLLinearSolver&,
const BCRSMatrix&)
{}
};
/**
* \brief Implementation that works together with direct ISTL
* solvers, e.g. Dune::SuperLU or Dune::UMFPack.
*/
template <typename Dummy>
struct Implementation<true,Dummy>
{
static void setMatrix (ISTLLinearSolver& solver,
const BCRSMatrix& matrix)
{
solver.setMatrix(matrix);
}
};
};
/**
* \brief A class template for performing some iterative eigenvalue algorithms
* based on power iteration.
*
* Given a square matrix whose eigenvalues shall be considered, this class
* template provides methods for performing the power iteration algorithm,
* the inverse iteration algorithm, the inverse iteration with shift algorithm,
* the Rayleigh quotient iteration algorithm and the TLIME iteration algorithm.
*
* \note Note that all algorithms except the power iteration algorithm require
* matrix inversion via a linear solver. When using an iterative linear
* solver, the algorithms become inexact "inner-outer" iterative methods.
* It is known that the number of inner solver iterations can increase
* steadily as the outer eigenvalue iteration proceeds. In this case, you
* should consider using a "tuned preconditioner", see e.g. [Freitag and
* Spence, 2008].
*
* \note In the current implementation, preconditioners like Dune::SeqILUn
* which are based on matrix decomposition act on the initial iteration
* matrix in each iteration, even for methods like the Rayleigh quotient
* algorithm in which the iteration matrix (m_ - mu_*I) may change in
* each iteration. This is due to the fact that those preconditioners
* currently don't support to be notified about a change of the matrix
* object.
*
* \todo The current implementation is limited to DUNE-ISTL BCRSMatrix types
* with blocklevel 2. An extension to blocklevel >= 2 might be provided
* in a future version.
*
* \tparam BCRSMatrix Type of a DUNE-ISTL BCRSMatrix whose eigenvalues
* shall be considered; is assumed to have blocklevel
* 2 with square blocks.
* \tparam BlockVector Type of the associated vectors; compatible with
* the rows of a BCRSMatrix object and its columns.
*
* \author Sebastian Westerheide.
*/
template <typename BCRSMatrix, typename BlockVector>
class PowerIteration_Algorithms
{
protected:
// Type definitions for type of iteration operator (m_ - mu_*I)
typedef typename Dune::MatrixAdapter<BCRSMatrix,BlockVector,BlockVector>
MatrixOperator;
typedef ScalingLinearOperator<BlockVector> ScalingOperator;
typedef LinearOperatorSum<MatrixOperator,ScalingOperator> OperatorSum;
public:
//! Type of underlying field
typedef typename BlockVector::field_type Real;
//! Type of iteration operator (m_ - mu_*I)
typedef OperatorSum IterationOperator;
public:
/**
* \brief Construct from required parameters.
*
* \param[in] m The square DUNE-ISTL BCRSMatrix whose
* eigenvalues shall be considered.
* \param[in] nIterationsMax The maximum number of iterations allowed.
* \param[in] verbosity_level Verbosity setting;
* >= 1: algorithms print a preamble and
* the final result,
* >= 2: algorithms print information on
* each iteration,
* >= 3: the final result output includes
* the approximated eigenvector.
*/
PowerIteration_Algorithms (const BCRSMatrix& m,
const unsigned int nIterationsMax = 1000,
const unsigned int verbosity_level = 0)
: m_(m), nIterationsMax_(nIterationsMax),
verbosity_level_(verbosity_level),
mu_(0.0),
matrixOperator_(m_),
scalingOperator_(-1.0,mu_),
itOperator_(matrixOperator_,scalingOperator_),
nIterations_(0),
title_(" PowerIteration_Algorithms: "),
blank_(title_.length(),' ')
{
// assert that BCRSMatrix type has blocklevel 2
static_assert
(BCRSMatrix::blocklevel == 2,
"Only BCRSMatrices with blocklevel 2 are supported.");
// assert that BCRSMatrix type has square blocks
static_assert
(BCRSMatrix::block_type::rows == BCRSMatrix::block_type::cols,
"Only BCRSMatrices with square blocks are supported.");
// assert that m_ is square
const int nrows = m_.M() * BCRSMatrix::block_type::rows;
const int ncols = m_.N() * BCRSMatrix::block_type::cols;
if (nrows != ncols)
DUNE_THROW(Dune::ISTLError,"Matrix is not square ("
<< nrows << "x" << ncols << ").");
}
//! disallow copying (default copy constructor does a shallow copy,
//! if copying was required a deep copy would have to be implemented
//! due to member variables which hold a dynamically allocated object)
PowerIteration_Algorithms (const PowerIteration_Algorithms&) = delete;
//! disallow copying (default assignment operator does a shallow copy,
//! if copying was required a deep copy would have to be implemented
//! due to member variables which hold a dynamically allocated object)
PowerIteration_Algorithms&
operator= (const PowerIteration_Algorithms&) = delete;
/**
* \brief Perform the power iteration algorithm to compute an approximation
* lambda of the dominant (i.e. largest magnitude) eigenvalue and
* the corresponding approximation x of an associated eigenvector.
*
* \param[in] epsilon The target residual norm.
* \param[out] lambda The approximated dominant eigenvalue.
* \param[in,out] x The associated approximated eigenvector;
* shall be initialized with an estimate
* for an eigenvector associated with the
* eigenvalue which shall be approximated.
*/
inline void applyPowerIteration (const Real& epsilon,
BlockVector& x, Real& lambda) const
{
// print verbosity information
if (verbosity_level_ > 0)
std::cout << title_
<< "Performing power iteration approximating "
<< "the dominant eigenvalue." << std::endl;
// allocate memory for auxiliary variables
BlockVector y(x);
BlockVector temp(x);
// perform power iteration
x *= (1.0 / x.two_norm());
m_.mv(x,y);
Real r_norm = std::numeric_limits<Real>::max();
nIterations_ = 0;
while (r_norm > epsilon)
{
// update and check number of iterations
if (++nIterations_ > nIterationsMax_)
DUNE_THROW(Dune::ISTLError,"Power iteration did not converge "
<< "in " << nIterationsMax_ << " iterations "
<< "(║residual║_2 = " << r_norm << ", epsilon = "
<< epsilon << ").");
// do one iteration of the power iteration algorithm
// (use that y = m_ * x)
x = y;
x *= (1.0 / y.two_norm());
// get approximated eigenvalue lambda via the Rayleigh quotient
m_.mv(x,y);
lambda = x * y;
// get norm of residual (use that y = m_ * x)
temp = y;
temp.axpy(-lambda,x);
r_norm = temp.two_norm();
// print verbosity information
if (verbosity_level_ > 1)
std::cout << blank_ << std::left
<< "iteration " << std::setw(3) << nIterations_
<< " (║residual║_2 = " << std::setw(11) << r_norm
<< "): λ = " << lambda << std::endl
<< std::resetiosflags(std::ios::left);
}
// print verbosity information
if (verbosity_level_ > 0)
{
std::cout << blank_ << "Result ("
<< "#iterations = " << nIterations_ << ", "
<< "║residual║_2 = " << r_norm << "): "
<< "λ = " << lambda << std::endl;
if (verbosity_level_ > 2)
{
// print approximated eigenvector via DUNE-ISTL I/O methods
Dune::printvector(std::cout,x,blank_+"x",blank_+"row");
}
}
}
/**
* \brief Perform the inverse iteration algorithm to compute an approximation
* lambda of the least dominant (i.e. smallest magnitude) eigenvalue
* and the corresponding approximation x of an associated eigenvector.
*
* \tparam ISTLLinearSolver Type of a DUNE-ISTL InverseOperator
* which shall be used as a linear solver.
* \tparam avoidLinSolverCrime The less accurate the linear solver is,
* the more corrupted gets the implemented
* computation of lambda and its associated
* residual. Setting this mode can help
* increasing their accuracy at the cost of
* a bit of efficiency which is beneficial
* e.g. when using a very inexact linear
* solver. Defaults to false.
*
* \param[in] epsilon The target residual norm.
* \param[in] solver The DUNE-ISTL InverseOperator which shall
* be used as a linear solver; is assumed to
* be constructed using the linear operator
* returned by getIterationOperator() (resp.
* matrix returned by getIterationMatrix()).
* \param[out] lambda The approximated least dominant eigenvalue.
* \param[in,out] x The associated approximated eigenvector;
* shall be initialized with an estimate
* for an eigenvector associated with the
* eigenvalue which shall be approximated.
*/
template <typename ISTLLinearSolver,
bool avoidLinSolverCrime = false>
inline void applyInverseIteration (const Real& epsilon,
ISTLLinearSolver& solver,
BlockVector& x, Real& lambda) const
{
constexpr Real gamma = 0.0;
applyInverseIteration(gamma,epsilon,solver,x,lambda);
}
/**
* \brief Perform the inverse iteration with shift algorithm to compute an
* approximation lambda of the eigenvalue closest to a given shift
* and the corresponding approximation x of an associated eigenvector.
*
* \tparam ISTLLinearSolver Type of a DUNE-ISTL InverseOperator
* which shall be used as a linear solver.
* \tparam avoidLinSolverCrime The less accurate the linear solver is,
* the more corrupted gets the implemented
* computation of lambda and its associated
* residual. Setting this mode can help
* increasing their accuracy at the cost of
* a bit of efficiency which is beneficial
* e.g. when using a very inexact linear
* solver. Defaults to false.
*
* \param[in] gamma The shift.
* \param[in] epsilon The target residual norm.
* \param[in] solver The DUNE-ISTL InverseOperator which shall
* be used as a linear solver; is assumed to
* be constructed using the linear operator
* returned by getIterationOperator() (resp.
* matrix returned by getIterationMatrix()).
* \param[out] lambda The approximated eigenvalue closest to gamma.
* \param[in,out] x The associated approximated eigenvector;
* shall be initialized with an estimate
* for an eigenvector associated with the
* eigenvalue which shall be approximated.
*/
template <typename ISTLLinearSolver,
bool avoidLinSolverCrime = false>
inline void applyInverseIteration (const Real& gamma,
const Real& epsilon,
ISTLLinearSolver& solver,
BlockVector& x, Real& lambda) const
{
// print verbosity information
if (verbosity_level_ > 0)
{
std::cout << title_;
if (gamma == 0.0)
std::cout << "Performing inverse iteration approximating "
<< "the least dominant eigenvalue." << std::endl;
else
std::cout << "Performing inverse iteration with shift "
<< "gamma = " << gamma << " approximating the "
<< "eigenvalue closest to gamma." << std::endl;
}
// initialize iteration operator,
// initialize iteration matrix when needed
updateShiftMu(gamma,solver);
// allocate memory for linear solver statistics
Dune::InverseOperatorResult solver_statistics;
// allocate memory for auxiliary variables
BlockVector y(x);
Real y_norm;
BlockVector temp(x);
// perform inverse iteration with shift
x *= (1.0 / x.two_norm());
Real r_norm = std::numeric_limits<Real>::max();
nIterations_ = 0;
while (r_norm > epsilon)
{
// update and check number of iterations
if (++nIterations_ > nIterationsMax_)
DUNE_THROW(Dune::ISTLError,"Inverse iteration "
<< (gamma != 0.0 ? "with shift " : "") << "did not "
<< "converge in " << nIterationsMax_ << " iterations "
<< "(║residual║_2 = " << r_norm << ", epsilon = "
<< epsilon << ").");
// do one iteration of the inverse iteration with shift algorithm,
// part 1: solve (m_ - gamma*I) * y = x for y
// (protect x from being changed)
temp = x;
solver.apply(y,temp,solver_statistics);
// get norm of y
y_norm = y.two_norm();
// compile time switch between accuracy and efficiency
if (avoidLinSolverCrime)
{
// get approximated eigenvalue lambda via the Rayleigh quotient
// (use that x_new = y / y_norm)
m_.mv(y,temp);
lambda = (y * temp) / (y_norm * y_norm);
// get norm of residual
// (use that x_new = y / y_norm, additionally use that temp = m_ * y)
temp.axpy(-lambda,y);
r_norm = temp.two_norm() / y_norm;
}
else
{
// get approximated eigenvalue lambda via the Rayleigh quotient
// (use that x_new = y / y_norm and use that (m_ - gamma*I) * y = x)
lambda = gamma + (y * x) / (y_norm * y_norm);
// get norm of residual
// (use that x_new = y / y_norm and use that (m_ - gamma*I) * y = x)
temp = x; temp.axpy(gamma-lambda,y);
r_norm = temp.two_norm() / y_norm;
}
// do one iteration of the inverse iteration with shift algorithm,
// part 2: update x
x = y;
x *= (1.0 / y_norm);
// print verbosity information
if (verbosity_level_ > 1)
std::cout << blank_ << std::left
<< "iteration " << std::setw(3) << nIterations_
<< " (║residual║_2 = " << std::setw(11) << r_norm
<< "): λ = " << lambda << std::endl
<< std::resetiosflags(std::ios::left);
}
// print verbosity information
if (verbosity_level_ > 0)
{
std::cout << blank_ << "Result ("
<< "#iterations = " << nIterations_ << ", "
<< "║residual║_2 = " << r_norm << "): "
<< "λ = " << lambda << std::endl;
if (verbosity_level_ > 2)
{
// print approximated eigenvector via DUNE-ISTL I/O methods
Dune::printvector(std::cout,x,blank_+"x",blank_+"row");
}
}
}
/**
* \brief Perform the Rayleigh quotient iteration algorithm to compute
* an approximation lambda of an eigenvalue and the corresponding
* approximation x of an associated eigenvector.
*
* \tparam ISTLLinearSolver Type of a DUNE-ISTL InverseOperator
* which shall be used as a linear solver.
* \tparam avoidLinSolverCrime The less accurate the linear solver is,
* the more corrupted gets the implemented
* computation of lambda and its associated
* residual. Setting this mode can help
* increasing their accuracy at the cost of
* a bit of efficiency which is beneficial
* e.g. when using a very inexact linear
* solver. Defaults to false.
*
* \param[in] epsilon The target residual norm.
* \param[in] solver The DUNE-ISTL InverseOperator which shall
* be used as a linear solver; is assumed to
* be constructed using the linear operator
* returned by getIterationOperator() (resp.
* matrix returned by getIterationMatrix()).
* \param[in,out] lambda The approximated eigenvalue;
* shall be initialized with an estimate for
* the eigenvalue which shall be approximated.
* \param[in,out] x The associated approximated eigenvector;
* shall be initialized with an estimate
* for an eigenvector associated with the
* eigenvalue which shall be approximated.
*/
template <typename ISTLLinearSolver,
bool avoidLinSolverCrime = false>
inline void applyRayleighQuotientIteration (const Real& epsilon,
ISTLLinearSolver& solver,
BlockVector& x, Real& lambda) const
{
// print verbosity information
if (verbosity_level_ > 0)
std::cout << title_
<< "Performing Rayleigh quotient iteration for "
<< "estimated eigenvalue " << lambda << "." << std::endl;
// allocate memory for linear solver statistics
Dune::InverseOperatorResult solver_statistics;
// allocate memory for auxiliary variables
BlockVector y(x);
Real y_norm;
Real lambda_update;
BlockVector temp(x);
// perform Rayleigh quotient iteration
x *= (1.0 / x.two_norm());
Real r_norm = std::numeric_limits<Real>::max();
nIterations_ = 0;
while (r_norm > epsilon)
{
// update and check number of iterations
if (++nIterations_ > nIterationsMax_)
DUNE_THROW(Dune::ISTLError,"Rayleigh quotient iteration did not "
<< "converge in " << nIterationsMax_ << " iterations "
<< "(║residual║_2 = " << r_norm << ", epsilon = "
<< epsilon << ").");
// update iteration operator,
// update iteration matrix when needed
updateShiftMu(lambda,solver);
// do one iteration of the Rayleigh quotient iteration algorithm,
// part 1: solve (m_ - lambda*I) * y = x for y
// (protect x from being changed)
temp = x;
solver.apply(y,temp,solver_statistics);
// get norm of y
y_norm = y.two_norm();
// compile time switch between accuracy and efficiency
if (avoidLinSolverCrime)
{
// get approximated eigenvalue lambda via the Rayleigh quotient
// (use that x_new = y / y_norm)
m_.mv(y,temp);
lambda = (y * temp) / (y_norm * y_norm);
// get norm of residual
// (use that x_new = y / y_norm, additionally use that temp = m_ * y)
temp.axpy(-lambda,y);
r_norm = temp.two_norm() / y_norm;
}
else
{
// get approximated eigenvalue lambda via the Rayleigh quotient
// (use that x_new = y / y_norm and use that (m_ - lambda_old*I) * y = x)
lambda_update = (y * x) / (y_norm * y_norm);
lambda += lambda_update;
// get norm of residual
// (use that x_new = y / y_norm and use that (m_ - lambda_old*I) * y = x)
temp = x; temp.axpy(-lambda_update,y);
r_norm = temp.two_norm() / y_norm;
}
// do one iteration of the Rayleigh quotient iteration algorithm,
// part 2: update x
x = y;
x *= (1.0 / y_norm);
// print verbosity information
if (verbosity_level_ > 1)
std::cout << blank_ << std::left
<< "iteration " << std::setw(3) << nIterations_
<< " (║residual║_2 = " << std::setw(11) << r_norm
<< "): λ = " << lambda << std::endl
<< std::resetiosflags(std::ios::left);
}
// print verbosity information
if (verbosity_level_ > 0)
{
std::cout << blank_ << "Result ("
<< "#iterations = " << nIterations_ << ", "
<< "║residual║_2 = " << r_norm << "): "
<< "λ = " << lambda << std::endl;
if (verbosity_level_ > 2)
{
// print approximated eigenvector via DUNE-ISTL I/O methods
Dune::printvector(std::cout,x,blank_+"x",blank_+"row");
}
}
}
/**
* \brief Perform the "two-level iterative method for eigenvalue calculations
* (TLIME)" iteration algorithm presented in [Szyld, 1988] to compute
* an approximation lambda of an eigenvalue and the corresponding
* approximation x of an associated eigenvector.
*
* The algorithm combines the inverse iteration with shift and the Rayleigh
* quotient iteration in order to compute an eigenvalue in a given interval
* J = (gamma - eta, gamma + eta). It guarantees that if an eigenvalue exists
* in J, the method will converge to an eigenvalue in J, while exploiting
* the cubic convergence of the Rayleigh quotient iteration, but without its
* drawback that - depending on the initial vector - it can converge to an
* arbitrary eigenvalue of the matrix. When J is free of eigenvalues, the
* method will determine this fact and converge linearly to the eigenvalue
* closest to J.
*
* \tparam ISTLLinearSolver Type of a DUNE-ISTL InverseOperator
* which shall be used as a linear solver.
* \tparam avoidLinSolverCrime The less accurate the linear solver is,
* the more corrupted gets the implemented
* computation of lambda and its associated
* residual. Setting this mode can help
* increasing their accuracy at the cost of
* a bit of efficiency which is beneficial
* e.g. when using a very inexact linear
* solver. Defaults to false.
*
* \param[in] gamma An estimate for the eigenvalue which shall
* be approximated.
* \param[in] eta Radius around gamma in which the eigenvalue
* is expected.
* \param[in] epsilon The target norm of the residual with respect
* to the Rayleigh quotient.
* \param[in] solver The DUNE-ISTL InverseOperator which shall
* be used as a linear solver; is assumed to
* be constructed using the linear operator
* returned by getIterationOperator() (resp.
* matrix returned by getIterationMatrix()).
* \param[in] delta The target relative change of the Rayleigh
* quotient, indicating that inverse iteration
* has become stationary and switching to Rayleigh
* quotient iteration is appropriate; is only
* considered if J is free of eigenvalues.
* \param[in] m The minimum number of inverse iterations before
* switching to Rayleigh quotient iteration; is
* only considered if J is free of eigenvalues.
* \param[out] extrnl If true, the interval J is free of eigenvalues;
* the approximated eigenvalue-eigenvector pair
* (lambda,x_s) then corresponds to the eigenvalue
* closest to J.
* \param[out] lambda The approximated eigenvalue.
* \param[in,out] x The associated approximated eigenvector;
* shall be initialized with an estimate
* for an eigenvector associated with the
* eigenvalue which shall be approximated.
*/
template <typename ISTLLinearSolver,
bool avoidLinSolverCrime = false>
inline void applyTLIMEIteration (const Real& gamma, const Real& eta,
const Real& epsilon,
ISTLLinearSolver& solver,
const Real& delta, const std::size_t& m,
bool& extrnl,
BlockVector& x, Real& lambda) const
{
// use same variable names as in [Szyld, 1988]
BlockVector& x_s = x;
Real& mu_s = lambda;
// print verbosity information
if (verbosity_level_ > 0)
std::cout << title_
<< "Performing TLIME iteration for "
<< "estimated eigenvalue in the "
<< "interval (" << gamma - eta << ","
<< gamma + eta << ")." << std::endl;
// allocate memory for linear solver statistics
Dune::InverseOperatorResult solver_statistics;
// allocate memory for auxiliary variables
bool doRQI;
Real mu;
BlockVector y(x_s);
Real omega;
Real mu_s_old;
Real mu_s_update;
BlockVector temp(x_s);
Real q_norm, r_norm;
// perform TLIME iteration
x_s *= (1.0 / x_s.two_norm());
extrnl = true;
doRQI = false;
r_norm = std::numeric_limits<Real>::max();
nIterations_ = 0;
while (r_norm > epsilon)
{
// update and check number of iterations
if (++nIterations_ > nIterationsMax_)
DUNE_THROW(Dune::ISTLError,"TLIME iteration did not "
<< "converge in " << nIterationsMax_
<< " iterations (║residual║_2 = " << r_norm
<< ", epsilon = " << epsilon << ").");
// set shift for next iteration according to inverse iteration
// with shift (II) resp. Rayleigh quotient iteration (RQI)
if (doRQI)
mu = mu_s;
else
mu = gamma;
// update II/RQI iteration operator,
// update II/RQI iteration matrix when needed
updateShiftMu(mu,solver);
// do one iteration of the II/RQI algorithm,
// part 1: solve (m_ - mu*I) * y = x for y
temp = x_s;
solver.apply(y,temp,solver_statistics);
// do one iteration of the II/RQI algorithm,
// part 2: compute omega
omega = (1.0 / y.two_norm());
// backup the old Rayleigh quotient
mu_s_old = mu_s;
// compile time switch between accuracy and efficiency
if (avoidLinSolverCrime)
{
// update the Rayleigh quotient mu_s, i.e. the approximated eigenvalue
// (use that x_new = y * omega)
m_.mv(y,temp);
mu_s = (y * temp) * (omega * omega);
// get norm of "the residual with respect to the shift used by II",
// use normal representation of q
// (use that x_new = y * omega, use that temp = m_ * y)
temp.axpy(-gamma,y);
q_norm = temp.two_norm() * omega;
// get norm of "the residual with respect to the Rayleigh quotient"
r_norm = q_norm*q_norm - (gamma-mu_s)*(gamma-mu_s);
// prevent that truncation errors invalidate the norm
// (we don't want to calculate sqrt of a negative number)
if (r_norm >= 0)
{
// use relation between the norms of r and q for efficiency
r_norm = std::sqrt(r_norm);
}
else
{
// use relation between r and q
// (use that x_new = y * omega, use that temp = (m_ - gamma*I) * y = q / omega)
temp.axpy(gamma-mu_s,y);
r_norm = temp.two_norm() * omega;
}
}
else
{
// update the Rayleigh quotient mu_s, i.e. the approximated eigenvalue
if (!doRQI)
{
// (use that x_new = y * omega, additionally use that (m_ - gamma*I) * y = x_s)
mu_s = gamma + (y * x_s) * (omega * omega);
}
else
{
// (use that x_new = y * omega, additionally use that (m_ - mu_s_old*I) * y = x_s)
mu_s_update = (y * x_s) * (omega * omega);
mu_s += mu_s_update;
}
// get norm of "the residual with respect to the shift used by II"
if (!doRQI)
{
// use special representation of q in the II case
// (use that x_new = y * omega, additionally use that (m_ - gamma*I) * y = x_s)
q_norm = omega;
}
else
{
// use special representation of q in the RQI case
// (use that x_new = y * omega, additionally use that (m_ - mu_s_old*I) * y = x_s)
temp = x_s; temp.axpy(mu_s-gamma,y);
q_norm = temp.two_norm() * omega;
}
// get norm of "the residual with respect to the Rayleigh quotient"
// don't use efficient relation between the norms of r and q, as
// this relation seems to yield a less accurate r_norm in the case
// where linear solver crime is admitted
if (!doRQI)
{
// (use that x_new = y * omega and use that (m_ - gamma*I) * y = x_s)
temp = x_s; temp.axpy(gamma-lambda,y);
r_norm = temp.two_norm() * omega;
}
else
{
// (use that x_new = y * omega and use that (m_ - mu_s_old*I) * y = x_s)
temp = x_s; temp.axpy(-mu_s_update,y);
r_norm = temp.two_norm() * omega;
}
}
// do one iteration of the II/RQI algorithm,
// part 3: update x
x_s = y; x_s *= omega;
// // for relative residual norm mode, scale with mu_s^{-1}
// r_norm /= std::abs(mu_s);
// print verbosity information
if (verbosity_level_ > 1)
std::cout << blank_ << "iteration "
<< std::left << std::setw(3) << nIterations_
<< " (" << (doRQI ? "RQI," : "II, ")
<< " " << (doRQI ? "—>" : " ") << " "
<< "║r║_2 = " << std::setw(11) << r_norm
<< ", " << (doRQI ? " " : "—>") << " "
<< "║q║_2 = " << std::setw(11) << q_norm
<< "): λ = " << lambda << std::endl
<< std::resetiosflags(std::ios::left);
// check if the eigenvalue closest to gamma lies in J
if (!doRQI && q_norm < eta)
{
// J is not free of eigenvalues
extrnl = false;
// by theory we know now that mu_s also lies in J
assert(std::abs(mu_s-gamma) < eta);
// switch to RQI
doRQI = true;
}
// revert to II if J is not free of eigenvalues but
// at some point mu_s falls back again outside J
if (!extrnl && doRQI && std::abs(mu_s-gamma) >= eta)
doRQI = false;
// if eigenvalue closest to gamma does not lie in J use RQI
// solely to accelerate the convergence to this eigenvalue
// when II has become stationary
if (extrnl && !doRQI)
{
// switch to RQI if the relative change of the Rayleigh
// quotient indicates that II has become stationary
if (nIterations_ >= m &&
std::abs(mu_s - mu_s_old) / std::abs(mu_s) < delta)
doRQI = true;
}
}
// // compute final residual and lambda again (paranoia....)
// m_.mv(x_s,temp);
// mu_s = x_s * temp;
// temp.axpy(-mu_s,x_s);
// r_norm = temp.two_norm();
// // r_norm /= std::abs(mu_s);
// print verbosity information
if (verbosity_level_ > 0)
{
if (extrnl)
std::cout << blank_ << "Interval "
<< "(" << gamma - eta << "," << gamma + eta
<< ") is free of eigenvalues, approximating "
<< "the closest eigenvalue." << std::endl;
std::cout << blank_ << "Result ("
<< "#iterations = " << nIterations_ << ", "
<< "║residual║_2 = " << r_norm << "): "
<< "λ = " << lambda << std::endl;
if (verbosity_level_ > 2)
{
// print approximated eigenvector via DUNE-ISTL I/O methods
Dune::printvector(std::cout,x,blank_+"x",blank_+"row");
}
}
}
/**
* \brief Return the iteration operator (m_ - mu_*I).
*
* The linear operator returned by this method shall be used
* to create the linear solver object. For linear solvers or
* preconditioners which require that the matrix is provided
* explicitly use getIterationMatrix() instead/additionally.
*/
inline IterationOperator& getIterationOperator ()
{
// return iteration operator
return itOperator_;
}
/**
* \brief Return the iteration matrix (m_ - mu_*I), provided
* on demand when needed (e.g. for direct solvers or
* preconditioning).
*
* The matrix returned by this method shall be used to create
* the linear solver object if it requires that the matrix is
* provided explicitly. For linear solvers which operate
* completely matrix free use getIterationOperator() instead.
*
* \note Calling this method creates a new DUNE-ISTL
* BCRSMatrix object which requires as much memory as
* the matrix whose eigenvalues shall be considered.
*/
inline const BCRSMatrix& getIterationMatrix () const
{
// create iteration matrix on demand
if (!itMatrix_)
itMatrix_ = std::unique_ptr<BCRSMatrix>(new BCRSMatrix(m_));
// return iteration matrix
return *itMatrix_;
}
/**
* \brief Return the number of iterations in last application
* of an algorithm.
*/
inline unsigned int getIterationCount () const
{
if (nIterations_ == 0)
DUNE_THROW(Dune::ISTLError,"No algorithm applied, yet.");
return nIterations_;
}
protected:
/**
* \brief Update shift mu_, i.e. update iteration operator/matrix
* (m_ - mu_*I).
*
* \note Does nothing if new shift equals the old one.
*
* \tparam ISTLLinearSolver Type of a DUNE-ISTL InverseOperator
* which is used as a linear solver.
*
* \param[in] mu The new shift.
* \param[in] solver The DUNE-ISTL InverseOperator which is used
* as a linear solver.
*
*/
template <typename ISTLLinearSolver>
inline void updateShiftMu (const Real& mu,
ISTLLinearSolver& solver) const
{
// do nothing if new shift equals the old one
if (mu == mu_) return;
// update shift mu_, i.e. update iteration operator
mu_ = mu;
// update iteration matrix when needed
if (itMatrix_)
{
// iterate over entries in iteration matrix diagonal
constexpr int rowBlockSize = BCRSMatrix::block_type::rows;
constexpr int colBlockSize = BCRSMatrix::block_type::cols;
for (typename BCRSMatrix::size_type i = 0;
i < itMatrix_->M()*rowBlockSize; ++i)
{
// access m_[i,i] where i is the flat index of a row/column
const Real& m_entry = m_
[i/rowBlockSize][i/colBlockSize][i%rowBlockSize][i%colBlockSize];
// access *itMatrix[i,i] where i is the flat index of a row/column
Real& entry = (*itMatrix_)
[i/rowBlockSize][i/colBlockSize][i%rowBlockSize][i%colBlockSize];
// change current entry in iteration matrix diagonal
entry = m_entry - mu_;
}
// notify linear solver about change of the iteration matrix object
SolverHelper<ISTLLinearSolver,BCRSMatrix>::setMatrix
(solver,*itMatrix_);
}
}
protected:
// parameters related to iterative eigenvalue algorithms
const BCRSMatrix& m_;
const unsigned int nIterationsMax_;
// verbosity setting
const unsigned int verbosity_level_;
// shift mu_ used by iteration operator/matrix (m_ - mu_*I)
mutable Real mu_;
// iteration operator (m_ - mu_*I), passing shift mu_ by reference
const MatrixOperator matrixOperator_;
const ScalingOperator scalingOperator_;
OperatorSum itOperator_;
// iteration matrix (m_ - mu_*I), provided on demand when needed
// (e.g. for preconditioning)
mutable std::unique_ptr<BCRSMatrix> itMatrix_;
// memory for storing temporary variables (mutable as they shall
// just be effectless auxiliary variables of the const apply*(...)
// methods)
mutable unsigned int nIterations_;
// constants for printing verbosity information
const std::string title_;
const std::string blank_;
};
} // namespace Dune
#endif // DUNE_ISTL_EIGENVALUE_POWERITERATION_HH
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