// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2012 Désiré Nuentsa-Wakam <desire.nuentsa_wakam@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
#ifndef EIGEN_DGMRES_H
#define EIGEN_DGMRES_H
#include <Eigen/Eigenvalues>
namespace Eigen {
template< typename _MatrixType,
typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> >
class DGMRES;
namespace internal {
template< typename _MatrixType, typename _Preconditioner>
struct traits<DGMRES<_MatrixType,_Preconditioner> >
{
typedef _MatrixType MatrixType;
typedef _Preconditioner Preconditioner;
};
/** \brief Computes a permutation vector to have a sorted sequence
* \param vec The vector to reorder.
* \param perm gives the sorted sequence on output. Must be initialized with 0..n-1
* \param ncut Put the ncut smallest elements at the end of the vector
* WARNING This is an expensive sort, so should be used only
* for small size vectors
* TODO Use modified QuickSplit or std::nth_element to get the smallest values
*/
template <typename VectorType, typename IndexType>
void sortWithPermutation (VectorType& vec, IndexType& perm, typename IndexType::Scalar& ncut)
{
eigen_assert(vec.size() == perm.size());
typedef typename IndexType::Scalar Index;
bool flag;
for (Index k = 0; k < ncut; k++)
{
flag = false;
for (Index j = 0; j < vec.size()-1; j++)
{
if ( vec(perm(j)) < vec(perm(j+1)) )
{
std::swap(perm(j),perm(j+1));
flag = true;
}
if (!flag) break; // The vector is in sorted order
}
}
}
}
/**
* \ingroup IterativeLInearSolvers_Module
* \brief A Restarted GMRES with deflation.
* This class implements a modification of the GMRES solver for
* sparse linear systems. The basis is built with modified
* Gram-Schmidt. At each restart, a few approximated eigenvectors
* corresponding to the smallest eigenvalues are used to build a
* preconditioner for the next cycle. This preconditioner
* for deflation can be combined with any other preconditioner,
* the IncompleteLUT for instance. The preconditioner is applied
* at right of the matrix and the combination is multiplicative.
*
* \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix.
* \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
* Typical usage :
* \code
* SparseMatrix<double> A;
* VectorXd x, b;
* //Fill A and b ...
* DGMRES<SparseMatrix<double> > solver;
* solver.set_restart(30); // Set restarting value
* solver.setEigenv(1); // Set the number of eigenvalues to deflate
* solver.compute(A);
* x = solver.solve(b);
* \endcode
*
* DGMRES can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink.
*
* References :
* [1] D. NUENTSA WAKAM and F. PACULL, Memory Efficient Hybrid
* Algebraic Solvers for Linear Systems Arising from Compressible
* Flows, Computers and Fluids, In Press,
* http://dx.doi.org/10.1016/j.compfluid.2012.03.023
* [2] K. Burrage and J. Erhel, On the performance of various
* adaptive preconditioned GMRES strategies, 5(1998), 101-121.
* [3] J. Erhel, K. Burrage and B. Pohl, Restarted GMRES
* preconditioned by deflation,J. Computational and Applied
* Mathematics, 69(1996), 303-318.
*
*/
template< typename _MatrixType, typename _Preconditioner>
class DGMRES : public IterativeSolverBase<DGMRES<_MatrixType,_Preconditioner> >
{
typedef IterativeSolverBase<DGMRES> Base;
using Base::matrix;
using Base::m_error;
using Base::m_iterations;
using Base::m_info;
using Base::m_isInitialized;
using Base::m_tolerance;
public:
using Base::_solve_impl;
typedef _MatrixType MatrixType;
typedef typename MatrixType::Scalar Scalar;
typedef typename MatrixType::Index Index;
typedef typename MatrixType::StorageIndex StorageIndex;
typedef typename MatrixType::RealScalar RealScalar;
typedef _Preconditioner Preconditioner;
typedef Matrix<Scalar,Dynamic,Dynamic> DenseMatrix;
typedef Matrix<RealScalar,Dynamic,Dynamic> DenseRealMatrix;
typedef Matrix<Scalar,Dynamic,1> DenseVector;
typedef Matrix<RealScalar,Dynamic,1> DenseRealVector;
typedef Matrix<std::complex<RealScalar>, Dynamic, 1> ComplexVector;
/** Default constructor. */
DGMRES() : Base(),m_restart(30),m_neig(0),m_r(0),m_maxNeig(5),m_isDeflAllocated(false),m_isDeflInitialized(false) {}
/** Initialize the solver with matrix \a A for further \c Ax=b solving.
*
* This constructor is a shortcut for the default constructor followed
* by a call to compute().
*
* \warning this class stores a reference to the matrix A as well as some
* precomputed values that depend on it. Therefore, if \a A is changed
* this class becomes invalid. Call compute() to update it with the new
* matrix A, or modify a copy of A.
*/
template<typename MatrixDerived>
explicit DGMRES(const EigenBase<MatrixDerived>& A) : Base(A.derived()), m_restart(30),m_neig(0),m_r(0),m_maxNeig(5),m_isDeflAllocated(false),m_isDeflInitialized(false) {}
~DGMRES() {}
/** \internal */
template<typename Rhs,typename Dest>
void _solve_with_guess_impl(const Rhs& b, Dest& x) const
{
bool failed = false;
for(int j=0; j<b.cols(); ++j)
{
m_iterations = Base::maxIterations();
m_error = Base::m_tolerance;
typename Dest::ColXpr xj(x,j);
dgmres(matrix(), b.col(j), xj, Base::m_preconditioner);
}
m_info = failed ? NumericalIssue
: m_error <= Base::m_tolerance ? Success
: NoConvergence;
m_isInitialized = true;
}
/** \internal */
template<typename Rhs,typename Dest>
void _solve_impl(const Rhs& b, MatrixBase<Dest>& x) const
{
x = b;
_solve_with_guess_impl(b,x.derived());
}
/**
* Get the restart value
*/
int restart() { return m_restart; }
/**
* Set the restart value (default is 30)
*/
void set_restart(const int restart) { m_restart=restart; }
/**
* Set the number of eigenvalues to deflate at each restart
*/
void setEigenv(const int neig)
{
m_neig = neig;
if (neig+1 > m_maxNeig) m_maxNeig = neig+1; // To allow for complex conjugates
}
/**
* Get the size of the deflation subspace size
*/
int deflSize() {return m_r; }
/**
* Set the maximum size of the deflation subspace
*/
void setMaxEigenv(const int maxNeig) { m_maxNeig = maxNeig; }
protected:
// DGMRES algorithm
template<typename Rhs, typename Dest>
void dgmres(const MatrixType& mat,const Rhs& rhs, Dest& x, const Preconditioner& precond) const;
// Perform one cycle of GMRES
template<typename Dest>
int dgmresCycle(const MatrixType& mat, const Preconditioner& precond, Dest& x, DenseVector& r0, RealScalar& beta, const RealScalar& normRhs, int& nbIts) const;
// Compute data to use for deflation
int dgmresComputeDeflationData(const MatrixType& mat, const Preconditioner& precond, const Index& it, StorageIndex& neig) const;
// Apply deflation to a vector
template<typename RhsType, typename DestType>
int dgmresApplyDeflation(const RhsType& In, DestType& Out) const;
ComplexVector schurValues(const ComplexSchur<DenseMatrix>& schurofH) const;
ComplexVector schurValues(const RealSchur<DenseMatrix>& schurofH) const;
// Init data for deflation
void dgmresInitDeflation(Index& rows) const;
mutable DenseMatrix m_V; // Krylov basis vectors
mutable DenseMatrix m_H; // Hessenberg matrix
mutable DenseMatrix m_Hes; // Initial hessenberg matrix wihout Givens rotations applied
mutable Index m_restart; // Maximum size of the Krylov subspace
mutable DenseMatrix m_U; // Vectors that form the basis of the invariant subspace
mutable DenseMatrix m_MU; // matrix operator applied to m_U (for next cycles)
mutable DenseMatrix m_T; /* T=U^T*M^{-1}*A*U */
mutable PartialPivLU<DenseMatrix> m_luT; // LU factorization of m_T
mutable StorageIndex m_neig; //Number of eigenvalues to extract at each restart
mutable int m_r; // Current number of deflated eigenvalues, size of m_U
mutable int m_maxNeig; // Maximum number of eigenvalues to deflate
mutable RealScalar m_lambdaN; //Modulus of the largest eigenvalue of A
mutable bool m_isDeflAllocated;
mutable bool m_isDeflInitialized;
//Adaptive strategy
mutable RealScalar m_smv; // Smaller multiple of the remaining number of steps allowed
mutable bool m_force; // Force the use of deflation at each restart
};
/**
* \brief Perform several cycles of restarted GMRES with modified Gram Schmidt,
*
* A right preconditioner is used combined with deflation.
*
*/
template< typename _MatrixType, typename _Preconditioner>
template<typename Rhs, typename Dest>
void DGMRES<_MatrixType, _Preconditioner>::dgmres(const MatrixType& mat,const Rhs& rhs, Dest& x,
const Preconditioner& precond) const
{
//Initialization
int n = mat.rows();
DenseVector r0(n);
int nbIts = 0;
m_H.resize(m_restart+1, m_restart);
m_Hes.resize(m_restart, m_restart);
m_V.resize(n,m_restart+1);
//Initial residual vector and intial norm
x = precond.solve(x);
r0 = rhs - mat * x;
RealScalar beta = r0.norm();
RealScalar normRhs = rhs.norm();
m_error = beta/normRhs;
if(m_error < m_tolerance)
m_info = Success;
else
m_info = NoConvergence;
// Iterative process
while (nbIts < m_iterations && m_info == NoConvergence)
{
dgmresCycle(mat, precond, x, r0, beta, normRhs, nbIts);
// Compute the new residual vector for the restart
if (nbIts < m_iterations && m_info == NoConvergence)
r0 = rhs - mat * x;
}
}
/**
* \brief Perform one restart cycle of DGMRES
* \param mat The coefficient matrix
* \param precond The preconditioner
* \param x the new approximated solution
* \param r0 The initial residual vector
* \param beta The norm of the residual computed so far
* \param normRhs The norm of the right hand side vector
* \param nbIts The number of iterations
*/
template< typename _MatrixType, typename _Preconditioner>
template<typename Dest>
int DGMRES<_MatrixType, _Preconditioner>::dgmresCycle(const MatrixType& mat, const Preconditioner& precond, Dest& x, DenseVector& r0, RealScalar& beta, const RealScalar& normRhs, int& nbIts) const
{
//Initialization
DenseVector g(m_restart+1); // Right hand side of the least square problem
g.setZero();
g(0) = Scalar(beta);
m_V.col(0) = r0/beta;
m_info = NoConvergence;
std::vector<JacobiRotation<Scalar> >gr(m_restart); // Givens rotations
int it = 0; // Number of inner iterations
int n = mat.rows();
DenseVector tv1(n), tv2(n); //Temporary vectors
while (m_info == NoConvergence && it < m_restart && nbIts < m_iterations)
{
// Apply preconditioner(s) at right
if (m_isDeflInitialized )
{
dgmresApplyDeflation(m_V.col(it), tv1); // Deflation
tv2 = precond.solve(tv1);
}
else
{
tv2 = precond.solve(m_V.col(it)); // User's selected preconditioner
}
tv1 = mat * tv2;
// Orthogonalize it with the previous basis in the basis using modified Gram-Schmidt
Scalar coef;
for (int i = 0; i <= it; ++i)
{
coef = tv1.dot(m_V.col(i));
tv1 = tv1 - coef * m_V.col(i);
m_H(i,it) = coef;
m_Hes(i,it) = coef;
}
// Normalize the vector
coef = tv1.norm();
m_V.col(it+1) = tv1/coef;
m_H(it+1, it) = coef;
// m_Hes(it+1,it) = coef;
// FIXME Check for happy breakdown
// Update Hessenberg matrix with Givens rotations
for (int i = 1; i <= it; ++i)
{
m_H.col(it).applyOnTheLeft(i-1,i,gr[i-1].adjoint());
}
// Compute the new plane rotation
gr[it].makeGivens(m_H(it, it), m_H(it+1,it));
// Apply the new rotation
m_H.col(it).applyOnTheLeft(it,it+1,gr[it].adjoint());
g.applyOnTheLeft(it,it+1, gr[it].adjoint());
beta = std::abs(g(it+1));
m_error = beta/normRhs;
// std::cerr << nbIts << " Relative Residual Norm " << m_error << std::endl;
it++; nbIts++;
if (m_error < m_tolerance)
{
// The method has converged
m_info = Success;
break;
}
}
// Compute the new coefficients by solving the least square problem
// it++;
//FIXME Check first if the matrix is singular ... zero diagonal
DenseVector nrs(m_restart);
nrs = m_H.topLeftCorner(it,it).template triangularView<Upper>().solve(g.head(it));
// Form the new solution
if (m_isDeflInitialized)
{
tv1 = m_V.leftCols(it) * nrs;
dgmresApplyDeflation(tv1, tv2);
x = x + precond.solve(tv2);
}
else
x = x + precond.solve(m_V.leftCols(it) * nrs);
// Go for a new cycle and compute data for deflation
if(nbIts < m_iterations && m_info == NoConvergence && m_neig > 0 && (m_r+m_neig) < m_maxNeig)
dgmresComputeDeflationData(mat, precond, it, m_neig);
return 0;
}
template< typename _MatrixType, typename _Preconditioner>
void DGMRES<_MatrixType, _Preconditioner>::dgmresInitDeflation(Index& rows) const
{
m_U.resize(rows, m_maxNeig);
m_MU.resize(rows, m_maxNeig);
m_T.resize(m_maxNeig, m_maxNeig);
m_lambdaN = 0.0;
m_isDeflAllocated = true;
}
template< typename _MatrixType, typename _Preconditioner>
inline typename DGMRES<_MatrixType, _Preconditioner>::ComplexVector DGMRES<_MatrixType, _Preconditioner>::schurValues(const ComplexSchur<DenseMatrix>& schurofH) const
{
return schurofH.matrixT().diagonal();
}
template< typename _MatrixType, typename _Preconditioner>
inline typename DGMRES<_MatrixType, _Preconditioner>::ComplexVector DGMRES<_MatrixType, _Preconditioner>::schurValues(const RealSchur<DenseMatrix>& schurofH) const
{
typedef typename MatrixType::Index Index;
const DenseMatrix& T = schurofH.matrixT();
Index it = T.rows();
ComplexVector eig(it);
Index j = 0;
while (j < it-1)
{
if (T(j+1,j) ==Scalar(0))
{
eig(j) = std::complex<RealScalar>(T(j,j),RealScalar(0));
j++;
}
else
{
eig(j) = std::complex<RealScalar>(T(j,j),T(j+1,j));
eig(j+1) = std::complex<RealScalar>(T(j,j+1),T(j+1,j+1));
j++;
}
}
if (j < it-1) eig(j) = std::complex<RealScalar>(T(j,j),RealScalar(0));
return eig;
}
template< typename _MatrixType, typename _Preconditioner>
int DGMRES<_MatrixType, _Preconditioner>::dgmresComputeDeflationData(const MatrixType& mat, const Preconditioner& precond, const Index& it, StorageIndex& neig) const
{
// First, find the Schur form of the Hessenberg matrix H
typename internal::conditional<NumTraits<Scalar>::IsComplex, ComplexSchur<DenseMatrix>, RealSchur<DenseMatrix> >::type schurofH;
bool computeU = true;
DenseMatrix matrixQ(it,it);
matrixQ.setIdentity();
schurofH.computeFromHessenberg(m_Hes.topLeftCorner(it,it), matrixQ, computeU);
ComplexVector eig(it);
Matrix<StorageIndex,Dynamic,1>perm(it);
eig = this->schurValues(schurofH);
// Reorder the absolute values of Schur values
DenseRealVector modulEig(it);
for (int j=0; j<it; ++j) modulEig(j) = std::abs(eig(j));
perm.setLinSpaced(it,0,it-1);
internal::sortWithPermutation(modulEig, perm, neig);
if (!m_lambdaN)
{
m_lambdaN = (std::max)(modulEig.maxCoeff(), m_lambdaN);
}
//Count the real number of extracted eigenvalues (with complex conjugates)
int nbrEig = 0;
while (nbrEig < neig)
{
if(eig(perm(it-nbrEig-1)).imag() == RealScalar(0)) nbrEig++;
else nbrEig += 2;
}
// Extract the Schur vectors corresponding to the smallest Ritz values
DenseMatrix Sr(it, nbrEig);
Sr.setZero();
for (int j = 0; j < nbrEig; j++)
{
Sr.col(j) = schurofH.matrixU().col(perm(it-j-1));
}
// Form the Schur vectors of the initial matrix using the Krylov basis
DenseMatrix X;
X = m_V.leftCols(it) * Sr;
if (m_r)
{
// Orthogonalize X against m_U using modified Gram-Schmidt
for (int j = 0; j < nbrEig; j++)
for (int k =0; k < m_r; k++)
X.col(j) = X.col(j) - (m_U.col(k).dot(X.col(j)))*m_U.col(k);
}
// Compute m_MX = A * M^-1 * X
Index m = m_V.rows();
if (!m_isDeflAllocated)
dgmresInitDeflation(m);
DenseMatrix MX(m, nbrEig);
DenseVector tv1(m);
for (int j = 0; j < nbrEig; j++)
{
tv1 = mat * X.col(j);
MX.col(j) = precond.solve(tv1);
}
//Update m_T = [U'MU U'MX; X'MU X'MX]
m_T.block(m_r, m_r, nbrEig, nbrEig) = X.transpose() * MX;
if(m_r)
{
m_T.block(0, m_r, m_r, nbrEig) = m_U.leftCols(m_r).transpose() * MX;
m_T.block(m_r, 0, nbrEig, m_r) = X.transpose() * m_MU.leftCols(m_r);
}
// Save X into m_U and m_MX in m_MU
for (int j = 0; j < nbrEig; j++) m_U.col(m_r+j) = X.col(j);
for (int j = 0; j < nbrEig; j++) m_MU.col(m_r+j) = MX.col(j);
// Increase the size of the invariant subspace
m_r += nbrEig;
// Factorize m_T into m_luT
m_luT.compute(m_T.topLeftCorner(m_r, m_r));
//FIXME CHeck if the factorization was correctly done (nonsingular matrix)
m_isDeflInitialized = true;
return 0;
}
template<typename _MatrixType, typename _Preconditioner>
template<typename RhsType, typename DestType>
int DGMRES<_MatrixType, _Preconditioner>::dgmresApplyDeflation(const RhsType &x, DestType &y) const
{
DenseVector x1 = m_U.leftCols(m_r).transpose() * x;
y = x + m_U.leftCols(m_r) * ( m_lambdaN * m_luT.solve(x1) - x1);
return 0;
}
} // end namespace Eigen
#endif