DGMRES.h

// 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/.
// SPDX-License-Identifier: MPL-2.0
 
#ifndef EIGEN_DGMRES_H
#define EIGEN_DGMRES_H
 
#include "../../Eigenvalues"
 
// IWYU pragma: private
#include "./InternalHeaderCheck.h"
 
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_> > {
  using MatrixType = MatrixType_;
  using Preconditioner = Preconditioner_;
};

template <typename VectorType, typename IndexType>
void sortWithPermutation(VectorType& vec, IndexType& perm, typename IndexType::Scalar& ncut) {
  eigen_assert(vec.size() == perm.size());
  for (Index k = 0; k < ncut; k++) {
    bool 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
    }
  }
}
 
}  // namespace internal
template <typename MatrixType_, typename Preconditioner_>
class DGMRES : public IterativeSolverBase<DGMRES<MatrixType_, Preconditioner_> > {
 protected:
  using Base = IterativeSolverBase<DGMRES>;
  using Base::m_error;
  using Base::m_info;
  using Base::m_isInitialized;
  using Base::m_iterations;
  using Base::m_tolerance;
  using Base::matrix;
 
 public:
  using Base::_solve_impl;
  using Base::_solve_with_guess_impl;
  using MatrixType = MatrixType_;
  using Scalar = typename MatrixType::Scalar;
  using StorageIndex = typename MatrixType::StorageIndex;
  using RealScalar = typename MatrixType::RealScalar;
  using ComplexScalar = internal::make_complex_t<Scalar>;
  using Preconditioner = Preconditioner_;
  using DenseMatrix = Matrix<Scalar, Dynamic, Dynamic>;
  using DenseRealMatrix = Matrix<RealScalar, Dynamic, Dynamic>;
  using DenseVector = Matrix<Scalar, Dynamic, 1>;
  using DenseRealVector = Matrix<RealScalar, Dynamic, 1>;
  using ComplexVector = Matrix<ComplexScalar, Dynamic, 1>;

  DGMRES()
      : Base(), m_restart(30), m_neig(0), m_r(0), m_maxNeig(5), m_isDeflAllocated(false), m_isDeflInitialized(false) {}

  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) {}

  template <typename Rhs, typename Dest>
  void _solve_vector_with_guess_impl(const Rhs& b, Dest& x) const {
    EIGEN_STATIC_ASSERT(Rhs::ColsAtCompileTime == 1 || Dest::ColsAtCompileTime == 1,
                        YOU_TRIED_CALLING_A_VECTOR_METHOD_ON_A_MATRIX);
 
    m_iterations = Base::maxIterations();
    m_error = Base::m_tolerance;
 
    dgmres(matrix(), b, x, Base::m_preconditioner);
  }

  Index restart() const { return m_restart; }

  void set_restart(const Index restart) { m_restart = restart; }

  void setEigenv(const Index neig) {
    m_neig = neig;
    if (neig + 1 > m_maxNeig) m_maxNeig = neig + 1;  // To allow for complex conjugates
  }

  Index deflSize() const { return m_r; }

  void setMaxEigenv(const Index 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>
  Index dgmresCycle(const MatrixType& mat, const Preconditioner& precond, Dest& x, DenseVector& r0, RealScalar& beta,
                    const RealScalar& normRhs, Index& nbIts) const;
  // Compute data to use for deflation
  Index dgmresComputeDeflationData(const MatrixType& mat, const Preconditioner& precond, const Index& it,
                                   StorageIndex& neig) const;
  // Apply deflation to a vector
  template <typename RhsType, typename DestType>
  Index 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 without 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 Index m_r;                        // Current number of deflated eigenvalues, size of m_U
  mutable Index 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
};

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 {
  const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
 
  RealScalar normRhs = rhs.norm();
  if (normRhs <= considerAsZero) {
    x.setZero();
    m_error = 0;
    m_iterations = 0;
    return;
  }
 
  // Initialization
  m_isDeflInitialized = false;
  Index n = mat.rows();
  DenseVector r0(n);
  Index 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 initial norm
  if (x.squaredNorm() == 0) x = precond.solve(rhs);
  r0.noalias() = rhs - mat * x;
  RealScalar beta = r0.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.noalias() = rhs - mat * x;
      beta = r0.norm();
    }
  }
  // m_iterations carried the iteration cap for the loops above; report the number actually performed.
  m_iterations = nbIts;
}

template <typename MatrixType_, typename Preconditioner_>
template <typename Dest>
Index DGMRES<MatrixType_, Preconditioner_>::dgmresCycle(const MatrixType& mat, const Preconditioner& precond, Dest& x,
                                                        DenseVector& r0, RealScalar& beta, const RealScalar& normRhs,
                                                        Index& 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
  Index it = 0;                                        // Number of inner iterations
  Index 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.noalias() = mat * tv2;
 
    // Orthogonalize it with the previous basis in the basis using modified Gram-Schmidt
    Scalar coef;
    for (Index 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 == 0 is an Arnoldi happy breakdown: the new
    // direction lies in span(V[0..it]), so skip the division (which would
    // poison m_V.col(it+1) with NaN) and fall through to the termination
    // check below.
    coef = tv1.norm();
    const bool happy_breakdown = numext::is_exactly_zero(coef);
    if (!happy_breakdown) {
      m_V.col(it + 1) = tv1 / coef;
    }
    m_H(it + 1, it) = coef;
 
    // Update Hessenberg matrix with Givens rotations
    for (Index i = 1; i <= it; ++i) {
      m_H.col(it).applyOnTheLeft(i - 1, i, gr[i - 1].adjoint());
    }
 
    // If the rotated diagonal is also zero, the reduced triangular system
    // becomes singular and the back-substitution below would produce Inf/NaN.
    // Stop with NumericalIssue instead of polluting x.
    if (happy_breakdown && numext::is_exactly_zero(m_H(it, it))) {
      m_info = NumericalIssue;
      break;
    }
 
    // 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 = numext::abs(g(it + 1));
    m_error = beta / normRhs;
    it++;
    nbIts++;
 
    if (m_error < m_tolerance || happy_breakdown) {
      // Happy breakdown: residual on the current subspace is exactly zero, so
      // the it-dim triangular system yields the exact solution.
      m_info = Success;
      break;
    }
  }
 
  // Compute the new coefficients by solving the least square problem.
  DenseVector nrs = m_H.topLeftCorner(it, it).template triangularView<Upper>().solve(g.head(it));
 
  // Form the new solution
  if (m_isDeflInitialized) {
    tv1.noalias() = 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 {
  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) = ComplexScalar(T(j, j), RealScalar(0));
      j++;
    } else {
      eig(j) = ComplexScalar(T(j, j), T(j + 1, j));
      eig(j + 1) = ComplexScalar(T(j, j + 1), T(j + 1, j + 1));
      j++;
    }
  }
  if (j < it - 1) eig(j) = ComplexScalar(T(j, j), RealScalar(0));
  return eig;
}
 
template <typename MatrixType_, typename Preconditioner_>
Index 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
  std::conditional_t<NumTraits<Scalar>::IsComplex, ComplexSchur<DenseMatrix>, RealSchur<DenseMatrix> > schurofH;
  bool computeU = true;
  DenseMatrix matrixQ(it, it);
  matrixQ.setIdentity();
  schurofH.computeFromHessenberg(m_Hes.topLeftCorner(it, it), matrixQ, computeU);
 
  ComplexVector eig = this->schurValues(schurofH);
  Matrix<StorageIndex, Dynamic, 1> perm(it);
 
  // Reorder the absolute values of Schur values
  DenseRealVector modulEig(it);
  for (Index j = 0; j < it; ++j) modulEig(j) = numext::abs(eig(j));
  perm.setLinSpaced(it, 0, internal::convert_index<StorageIndex>(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)
  Index 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 (Index 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 = m_V.leftCols(it) * Sr;
  if (m_r) {
    // Orthogonalize X against m_U using modified Gram-Schmidt
    for (Index j = 0; j < nbrEig; j++)
      for (Index 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 MX = M^-1 * A * X
  Index m = m_V.rows();
  if (!m_isDeflAllocated) dgmresInitDeflation(m);
  DenseMatrix MX(m, nbrEig);
  DenseVector tv1(m);
  for (Index j = 0; j < nbrEig; j++) {
    tv1.noalias() = 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).noalias() = X.transpose() * MX;
  if (m_r) {
    m_T.block(0, m_r, m_r, nbrEig).noalias() = m_U.leftCols(m_r).transpose() * MX;
    m_T.block(m_r, 0, nbrEig, m_r).noalias() = X.transpose() * m_MU.leftCols(m_r);
  }
 
  // Save X into m_U and m_MX in m_MU
  for (Index j = 0; j < nbrEig; j++) m_U.col(m_r + j) = X.col(j);
  for (Index 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>
Index 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