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Add restarted GMRES with deflation
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Desire NUENTSA
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unsupported/Eigen/src/IterativeSolvers/DGMRES.h
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528
unsupported/Eigen/src/IterativeSolvers/DGMRES.h
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// This file is part of Eigen, a lightweight C++ template library
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// for linear algebra.
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//
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// Copyright (C) 2012 Désiré Nuentsa-Wakam <desire.nuentsa_wakam@inria.fr>
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//
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// This Source Code Form is subject to the terms of the Mozilla
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// Public License v. 2.0. If a copy of the MPL was not distributed
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// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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#ifndef EIGEN_DGMRES_H
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#define EIGEN_DGMRES_H
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#include <Eigen/Eigenvalues>
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namespace Eigen {
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template< typename _MatrixType,
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typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> >
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class DGMRES;
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namespace internal {
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template< typename _MatrixType, typename _Preconditioner>
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struct traits<DGMRES<_MatrixType,_Preconditioner> >
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{
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typedef _MatrixType MatrixType;
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typedef _Preconditioner Preconditioner;
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};
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/** \brief Computes a permutation vector to have a sorted sequence
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* \param vec The vector to reorder.
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* \param perm gives the sorted sequence on output. Must be initialized with 0..n-1
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* \param ncut Put the ncut smallest elements at the end of the vector
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* WARNING This is an expensive sort, so should be used only
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* for small size vectors
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* TODO Use modified QuickSplit or std::nth_element to get the smallest values
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*/
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template <typename VectorType, typename IndexType>
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void sortWithPermutation (VectorType& vec, IndexType& perm, typename IndexType::Scalar& ncut)
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{
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assert(vec.size() == perm.size());
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typedef typename IndexType::Scalar Index;
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typedef typename VectorType::Scalar Scalar;
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Index n = vec.size();
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bool flag;
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for (Index k = 0; k < ncut; k++)
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{
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flag = false;
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for (Index j = 0; j < vec.size()-1; j++)
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{
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if ( vec(perm(j)) < vec(perm(j+1)) )
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{
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std::swap(perm(j),perm(j+1));
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flag = true;
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}
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if (!flag) break; // The vector is in sorted order
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}
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}
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}
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}
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/**
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* \ingroup IterativeLInearSolvers_Module
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* \brief A Restarted GMRES with deflation.
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* This class implements a modification of the GMRES solver for
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* sparse linear systems. The basis is built with modified
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* Gram-Schmidt. At each restart, a few approximated eigenvectors
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* corresponding to the smallest eigenvalues are used to build a
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* preconditioner for the next cycle. This preconditioner
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* for deflation can be combined with any other preconditioner,
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* the IncompleteLUT for instance. The preconditioner is applied
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* at right of the matrix and the combination is multiplicative.
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*
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* \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix.
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* \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
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* Typical usage :
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* \code
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* SparseMatrix<double> A;
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* VectorXd x, b;
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* //Fill A and b ...
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* DGMRES<SparseMatrix<double> > solver;
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* solver.set_restart(30); // Set restarting value
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* solver.setEigenv(1); // Set the number of eigenvalues to deflate
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* solver.compute(A);
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* x = solver.solve(b);
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* \endcode
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*
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* References :
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* [1] D. NUENTSA WAKAM and F. PACULL, Memory Efficient Hybrid
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* Algebraic Solvers for Linear Systems Arising from Compressible
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* Flows, Computers and Fluids, In Press,
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* http://dx.doi.org/10.1016/j.compfluid.2012.03.023
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* [2] K. Burrage and J. Erhel, On the performance of various
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* adaptive preconditioned GMRES strategies, 5(1998), 101-121.
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* [3] J. Erhel, K. Burrage and B. Pohl, Restarted GMRES
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* preconditioned by deflation,J. Computational and Applied
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* Mathematics, 69(1996), 303-318.
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*
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*/
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template< typename _MatrixType, typename _Preconditioner>
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class DGMRES : public IterativeSolverBase<DGMRES<_MatrixType,_Preconditioner> >
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{
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typedef IterativeSolverBase<DGMRES> Base;
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using Base::mp_matrix;
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using Base::m_error;
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using Base::m_iterations;
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using Base::m_info;
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using Base::m_isInitialized;
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using Base::m_tolerance;
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public:
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typedef _MatrixType MatrixType;
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typedef typename MatrixType::Scalar Scalar;
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typedef typename MatrixType::Index Index;
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typedef typename MatrixType::RealScalar RealScalar;
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typedef _Preconditioner Preconditioner;
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typedef Matrix<Scalar,Dynamic,Dynamic> DenseMatrix;
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typedef Matrix<Scalar,Dynamic,1> DenseVector;
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typedef std::complex<RealScalar> ComplexScalar;
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/** Default constructor. */
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DGMRES() : Base(),m_restart(30),m_neig(0),m_r(0),m_maxNeig(5),m_isDeflAllocated(false),m_isDeflInitialized(false) {}
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/** Initialize the solver with matrix \a A for further \c Ax=b solving.
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*
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* This constructor is a shortcut for the default constructor followed
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* by a call to compute().
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*
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* \warning this class stores a reference to the matrix A as well as some
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* precomputed values that depend on it. Therefore, if \a A is changed
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* this class becomes invalid. Call compute() to update it with the new
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* matrix A, or modify a copy of A.
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*/
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DGMRES(const MatrixType& A) : Base(A),m_restart(30),m_neig(0),m_r(0),m_maxNeig(5),m_isDeflAllocated(false),m_isDeflInitialized(false)
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{}
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~DGMRES() {}
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/** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A
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* \a x0 as an initial solution.
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*
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* \sa compute()
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*/
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template<typename Rhs,typename Guess>
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inline const internal::solve_retval_with_guess<DGMRES, Rhs, Guess>
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solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const
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{
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eigen_assert(m_isInitialized && "DGMRES is not initialized.");
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eigen_assert(Base::rows()==b.rows()
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&& "DGMRES::solve(): invalid number of rows of the right hand side matrix b");
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return internal::solve_retval_with_guess
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<DGMRES, Rhs, Guess>(*this, b.derived(), x0);
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}
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/** \internal */
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template<typename Rhs,typename Dest>
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void _solveWithGuess(const Rhs& b, Dest& x) const
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{
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bool failed = false;
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for(int j=0; j<b.cols(); ++j)
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{
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m_iterations = Base::maxIterations();
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m_error = Base::m_tolerance;
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typename Dest::ColXpr xj(x,j);
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dgmres(*mp_matrix, b.col(j), xj, Base::m_preconditioner);
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}
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m_info = failed ? NumericalIssue
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: m_error <= Base::m_tolerance ? Success
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: NoConvergence;
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m_isInitialized = true;
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}
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/** \internal */
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template<typename Rhs,typename Dest>
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void _solve(const Rhs& b, Dest& x) const
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{
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x = b;
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_solveWithGuess(b,x);
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}
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/**
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* Get the restart value
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*/
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int restart() { return m_restart; }
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/**
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* Set the restart value (default is 30)
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*/
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void set_restart(const int restart) { m_restart=restart; }
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/**
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* Set the number of eigenvalues to deflate at each restart
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*/
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void setEigenv(const int neig)
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{
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m_neig = neig;
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if (neig+1 > m_maxNeig) m_maxNeig = neig+1; // To allow for complex conjugates
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}
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/**
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* Get the size of the deflation subspace size
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*/
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int deflSize() {return m_r; }
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/**
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* Set the maximum size of the deflation subspace
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*/
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void setMaxEigenv(const int maxNeig) { m_maxNeig = maxNeig; }
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protected:
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// DGMRES algorithm
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template<typename Rhs, typename Dest>
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void dgmres(const MatrixType& mat,const Rhs& rhs, Dest& x, const Preconditioner& precond) const;
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// Perform one cycle of GMRES
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template<typename Dest>
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int dgmresCycle(const MatrixType& mat, const Preconditioner& precond, Dest& x, DenseVector& r0, RealScalar& beta, const RealScalar& normRhs, int& nbIts) const;
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// Compute data to use for deflation
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int dgmresComputeDeflationData(const MatrixType& mat, const Preconditioner& precond, const Index& it, Index& neig) const;
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// Apply deflation to a vector
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template<typename RhsType, typename DestType>
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int dgmresApplyDeflation(const RhsType& In, DestType& Out) const;
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// Init data for deflation
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void dgmresInitDeflation(Index& rows) const;
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mutable DenseMatrix m_V; // Krylov basis vectors
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mutable DenseMatrix m_H; // Hessenberg matrix
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mutable DenseMatrix m_Hes; // Initial hessenberg matrix wihout Givens rotations applied
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mutable Index m_restart; // Maximum size of the Krylov subspace
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mutable DenseMatrix m_U; // Vectors that form the basis of the invariant subspace
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mutable DenseMatrix m_MU; // matrix operator applied to m_U (for next cycles)
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mutable DenseMatrix m_T; /* T=U^T*M^{-1}*A*U */
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mutable PartialPivLU<DenseMatrix> m_luT; // LU factorization of m_T
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mutable int m_neig; //Number of eigenvalues to extract at each restart
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mutable int m_r; // Current number of deflated eigenvalues, size of m_U
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mutable int m_maxNeig; // Maximum number of eigenvalues to deflate
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mutable RealScalar m_lambdaN; //Modulus of the largest eigenvalue of A
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mutable bool m_isDeflAllocated;
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mutable bool m_isDeflInitialized;
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//Adaptive strategy
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mutable RealScalar m_smv; // Smaller multiple of the remaining number of steps allowed
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mutable bool m_force; // Force the use of deflation at each restart
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};
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/**
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* \brief Perform several cycles of restarted GMRES with modified Gram Schmidt,
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*
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* A right preconditioner is used combined with deflation.
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*
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*/
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template< typename _MatrixType, typename _Preconditioner>
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template<typename Rhs, typename Dest>
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void DGMRES<_MatrixType, _Preconditioner>::dgmres(const MatrixType& mat,const Rhs& rhs, Dest& x,
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const Preconditioner& precond) const
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{
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//Initialization
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int n = mat.rows();
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DenseVector r0(n);
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int nbIts = 0;
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m_H.resize(m_restart+1, m_restart);
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m_Hes.resize(m_restart, m_restart);
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m_V.resize(n,m_restart+1);
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//Initial residual vector and intial norm
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x = precond.solve(x);
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r0 = rhs - mat * x;
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RealScalar beta = r0.norm();
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RealScalar normRhs = rhs.norm();
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m_error = beta/normRhs;
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if(m_error < m_tolerance)
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m_info = Success;
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else
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m_info = NoConvergence;
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// Iterative process
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while (nbIts < m_iterations && m_info == NoConvergence)
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{
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dgmresCycle(mat, precond, x, r0, beta, normRhs, nbIts);
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// Compute the new residual vector for the restart
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if (nbIts < m_iterations && m_info == NoConvergence)
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r0 = rhs - mat * x;
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}
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}
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/**
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* \brief Perform one restart cycle of DGMRES
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* \param mat The coefficient matrix
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* \param precond The preconditioner
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* \param x the new approximated solution
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* \param r0 The initial residual vector
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* \param beta The norm of the residual computed so far
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* \param normRhs The norm of the right hand side vector
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* \param nbIts The number of iterations
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*/
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template< typename _MatrixType, typename _Preconditioner>
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template<typename Dest>
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int DGMRES<_MatrixType, _Preconditioner>::dgmresCycle(const MatrixType& mat, const Preconditioner& precond, Dest& x, DenseVector& r0, RealScalar& beta, const RealScalar& normRhs, int& nbIts) const
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{
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//Initialization
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DenseVector g(m_restart+1); // Right hand side of the least square problem
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g.setZero();
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g(0) = Scalar(beta);
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m_V.col(0) = r0/beta;
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m_info = NoConvergence;
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std::vector<JacobiRotation<Scalar> >gr(m_restart); // Givens rotations
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int it = 0; // Number of inner iterations
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int n = mat.rows();
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DenseVector tv1(n), tv2(n); //Temporary vectors
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while (m_info == NoConvergence && it < m_restart && nbIts < m_iterations)
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{
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int n = m_V.rows();
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// Apply preconditioner(s) at right
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if (m_isDeflInitialized )
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{
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dgmresApplyDeflation(m_V.col(it), tv1); // Deflation
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tv2 = precond.solve(tv1);
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}
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else
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{
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tv2 = precond.solve(m_V.col(it)); // User's selected preconditioner
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}
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tv1 = mat * tv2;
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// Orthogonalize it with the previous basis in the basis using modified Gram-Schmidt
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RealScalar coef;
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for (int i = 0; i <= it; ++i)
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{
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coef = tv1.dot(m_V.col(i));
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tv1 = tv1 - coef * m_V.col(i);
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m_H(i,it) = coef;
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m_Hes(i,it) = coef;
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}
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// Normalize the vector
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coef = tv1.norm();
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m_V.col(it+1) = tv1/coef;
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m_H(it+1, it) = coef;
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// m_Hes(it+1,it) = coef;
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// FIXME Check for happy breakdown
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// Update Hessenberg matrix with Givens rotations
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for (int i = 1; i <= it; ++i)
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{
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m_H.col(it).applyOnTheLeft(i-1,i,gr[i-1].adjoint());
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}
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// Compute the new plane rotation
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gr[it].makeGivens(m_H(it, it), m_H(it+1,it));
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// Apply the new rotation
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m_H.col(it).applyOnTheLeft(it,it+1,gr[it].adjoint());
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g.applyOnTheLeft(it,it+1, gr[it].adjoint());
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beta = std::abs(g(it+1));
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m_error = beta/normRhs;
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std::cerr << nbIts << " Relative Residual Norm " << m_error << std::endl;
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it++; nbIts++;
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if (m_error < m_tolerance)
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{
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// The method has converged
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m_info = Success;
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break;
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}
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}
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// Compute the new coefficients by solving the least square problem
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// it++;
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//FIXME Check first if the matrix is singular ... zero diagonal
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DenseVector nrs(m_restart);
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nrs = m_H.topLeftCorner(it,it).template triangularView<Upper>().solve(g.head(it));
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// Form the new solution
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if (m_isDeflInitialized)
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{
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tv1 = m_V.leftCols(it) * nrs;
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dgmresApplyDeflation(tv1, tv2);
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x = x + precond.solve(tv2);
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}
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else
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x = x + precond.solve(m_V.leftCols(it) * nrs);
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// Go for a new cycle and compute data for deflation
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if(nbIts < m_iterations && m_info == NoConvergence && m_neig > 0 && (m_r+m_neig) < m_maxNeig)
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dgmresComputeDeflationData(mat, precond, it, m_neig);
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return 0;
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}
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template< typename _MatrixType, typename _Preconditioner>
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void DGMRES<_MatrixType, _Preconditioner>::dgmresInitDeflation(Index& rows) const
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{
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m_U.resize(rows, m_maxNeig);
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m_MU.resize(rows, m_maxNeig);
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m_T.resize(m_maxNeig, m_maxNeig);
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m_lambdaN = 0.0;
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m_isDeflAllocated = true;
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}
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template< typename _MatrixType, typename _Preconditioner>
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int DGMRES<_MatrixType, _Preconditioner>::dgmresComputeDeflationData(const MatrixType& mat, const Preconditioner& precond, const Index& it, Index& neig) const
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{
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// First, find the Schur form of the Hessenberg matrix H
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RealSchur<DenseMatrix> schurofH;
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bool computeU = true;
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DenseMatrix matrixQ(it,it);
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matrixQ.setIdentity();
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schurofH.computeHessenberg(m_Hes.topLeftCorner(it,it), matrixQ, computeU);
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const DenseMatrix& T = schurofH.matrixT();
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// Extract the schur values from the diagonal of T;
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Matrix<ComplexScalar,Dynamic,1> eig(it);
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Matrix<Index,Dynamic,1>perm(it);
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int j = 0;
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while (j < it-1)
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{
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if (T(j+1,j) ==Scalar(0))
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{
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eig(j) = ComplexScalar(T(j,j),Scalar(0));
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j++;
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}
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else
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{
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eig(j) = ComplexScalar(T(j,j),T(j+1,j));
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eig(j+1) = ComplexScalar(T(j,j+1),T(j+1,j+1));
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j++;
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}
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}
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if (j < it) eig(j) = ComplexScalar(T(j,j),Scalar(0));
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// Reorder the absolute values of Schur values
|
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DenseVector 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)
|
||||
{
|
||||
//Find the maximum eigenvalue
|
||||
for (int i = 0; i < it; ++i)
|
||||
if (modulEig(i) > m_lambdaN)
|
||||
m_lambdaN = modulEig(i);
|
||||
}
|
||||
//Count the real number of extracted eigenvalues (with complex conjugates)
|
||||
int nbrEig = 0;
|
||||
while (nbrEig < neig)
|
||||
{
|
||||
if(eig(perm(it-nbrEig-1)).imag() == Scalar(0)) nbrEig++;
|
||||
else nbrEig += 2;
|
||||
}
|
||||
// Extract the smallest Schur vectors
|
||||
DenseMatrix Sr(it, nbrEig);
|
||||
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 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;
|
||||
}
|
||||
|
||||
namespace internal {
|
||||
|
||||
template<typename _MatrixType, typename _Preconditioner, typename Rhs>
|
||||
struct solve_retval<DGMRES<_MatrixType, _Preconditioner>, Rhs>
|
||||
: solve_retval_base<DGMRES<_MatrixType, _Preconditioner>, Rhs>
|
||||
{
|
||||
typedef DGMRES<_MatrixType, _Preconditioner> Dec;
|
||||
EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs)
|
||||
|
||||
template<typename Dest> void evalTo(Dest& dst) const
|
||||
{
|
||||
dec()._solve(rhs(),dst);
|
||||
}
|
||||
};
|
||||
} // end namespace internal
|
||||
|
||||
} // end namespace Eigen
|
||||
#endif
|
||||
Loading…
Reference in New Issue
Block a user