gmm_solver_Schwarz_additive.h

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/**@file gmm_solver_Schwarz_additive.h
   @author  Yves Renard <[email protected]>
   @author  Michel Fournie <[email protected]>
   @date October 13, 2002.
*/
 
#ifndef GMM_SOLVERS_SCHWARZ_ADDITIVE_H__
#define GMM_SOLVERS_SCHWARZ_ADDITIVE_H__ 
 
#include "gmm_kernel.h"
#include "gmm_superlu_interface.h"
#include "gmm_solver_cg.h"
#include "gmm_solver_gmres.h"
#include "gmm_solver_bicgstab.h"
#include "gmm_solver_qmr.h"
 
namespace gmm {
      
  /* ******************************************************************** */
  /*            Additive Schwarz interfaced local solvers                 */
  /* ******************************************************************** */
 
  struct using_cg {};
  struct using_gmres {};
  struct using_bicgstab {};
  struct using_qmr {};
 
  template <typename P, typename local_solver, typename Matrix>
  struct actual_precond {
    typedef P APrecond;
    static const APrecond &transform(const P &PP) { return PP; }
  };
 
  template <typename Matrix1, typename Precond, typename Vector> 
  void AS_local_solve(using_cg, const Matrix1 &A, Vector &x, const Vector &b,
                 const Precond &P, iteration &iter)
  { cg(A, x, b, P, iter); }
 
  template <typename Matrix1, typename Precond, typename Vector> 
  void AS_local_solve(using_gmres, const Matrix1 &A, Vector &x,
                      const Vector &b, const Precond &P, iteration &iter)
  { gmres(A, x, b, P, 100, iter); }
  
  template <typename Matrix1, typename Precond, typename Vector> 
  void AS_local_solve(using_bicgstab, const Matrix1 &A, Vector &x,
                      const Vector &b, const Precond &P, iteration &iter)
  { bicgstab(A, x, b, P, iter); }
 
  template <typename Matrix1, typename Precond, typename Vector> 
  void AS_local_solve(using_qmr, const Matrix1 &A, Vector &x,
                      const Vector &b, const Precond &P, iteration &iter)
  { qmr(A, x, b, P, iter); }
 
#if defined(GMM_USES_SUPERLU)
  struct using_superlu {};
 
  template <typename P, typename Matrix>
  struct actual_precond<P, using_superlu, Matrix> {
    typedef typename linalg_traits<Matrix>::value_type value_type;
    typedef SuperLU_factor<value_type> APrecond;
    template <typename PR>
    static APrecond transform(const PR &) { return APrecond(); }
    static const APrecond &transform(const APrecond &PP) { return PP; }
  };
 
  template <typename Matrix1, typename Precond, typename Vector> 
  void AS_local_solve(using_superlu, const Matrix1 &, Vector &x,
                      const Vector &b, const Precond &P, iteration &iter)
  { P.solve(x, b); iter.set_iteration(1); }
#endif
 
  /* ******************************************************************** */
  /*            Additive Schwarz Linear system                            */
  /* ******************************************************************** */
 
  template <typename Matrix1, typename Matrix2, typename Precond,
            typename local_solver>
  struct add_schwarz_mat{
    typedef typename linalg_traits<Matrix1>::value_type value_type;
 
    const Matrix1 *A;
    const std::vector<Matrix2> *vB;
    std::vector<Matrix2> vAloc;
    mutable iteration iter;
    double residual;
    mutable size_type itebilan;
    mutable std::vector<std::vector<value_type> > gi, fi;
    std::vector<typename actual_precond<Precond, local_solver,
                                        Matrix1>::APrecond> precond1;
 
    void init(const Matrix1 &A_, const std::vector<Matrix2> &vB_,
              iteration iter_, const Precond &P, double residual_);
 
    add_schwarz_mat(void) {}
    add_schwarz_mat(const Matrix1 &A_, const std::vector<Matrix2> &vB_,
                iteration iter_, const Precond &P, double residual_)
    { init(A_, vB_, iter_, P, residual_); }
  };
 
  template <typename Matrix1, typename Matrix2, typename Precond,
            typename local_solver>
  void add_schwarz_mat<Matrix1, Matrix2, Precond, local_solver>::init(
       const Matrix1 &A_, const std::vector<Matrix2> &vB_,
       iteration iter_, const Precond &P, double residual_) {
 
    vB = &vB_; A = &A_; iter = iter_;
    residual = residual_;
    
    size_type nb_sub = vB->size();
    vAloc.resize(nb_sub);
    gi.resize(nb_sub); fi.resize(nb_sub);
    precond1.resize(nb_sub);
    std::fill(precond1.begin(), precond1.end(),
              actual_precond<Precond, local_solver, Matrix1>::transform(P));
    itebilan = 0;
    
    if (iter.get_noisy()) cout << "Init pour sub dom ";
#ifdef GMM_USES_MPI
    int size,tranche,borne_sup,borne_inf,rank,tag1=11,tag2=12,tag3=13,sizepr = 0;
    //    int tab[4];
    double t_ref,t_final;
    MPI_Status status;
    t_ref=MPI_Wtime();
    MPI_Comm_rank(MPI_COMM_WORLD, &rank);
    MPI_Comm_size(MPI_COMM_WORLD, &size);
    tranche=nb_sub/size;
    borne_inf=rank*tranche;
    borne_sup=(rank+1)*tranche;
    // if (rank==size-1) borne_sup = nb_sub;
 
    cout << "Nombre de sous domaines " << borne_sup - borne_inf << endl;
 
    int sizeA = mat_nrows(*A);
    gmm::csr_matrix<value_type> Acsr(sizeA, sizeA), Acsrtemp(sizeA, sizeA);
    gmm::copy(gmm::eff_matrix(*A), Acsr);
    int next = (rank + 1) % size;
    int previous = (rank + size - 1) % size;
    //communication of local information on ring pattern
    //Each process receive  Nproc-1 contributions 
 
    for (int nproc = 0; nproc < size; ++nproc) {
       for (size_type i = size_type(borne_inf); i < size_type(borne_sup); ++i) {
//      for (size_type i = 0; i < nb_sub/size; ++i) {
//      for (size_type i = 0; i < nb_sub; ++i) {
        // size_type i=(rank+size*(j-1)+nb_sub)%nb_sub;
 
        cout << "Sous domaines " << i << " : " << mat_ncols((*vB)[i]) << endl;
#else
        for (size_type i = 0; i < nb_sub; ++i) {
#endif
          
          if (iter.get_noisy()) cout << i << " " << std::flush;
          Matrix2 Maux(mat_ncols((*vB)[i]), mat_nrows((*vB)[i]));
          
#ifdef GMM_USES_MPI
          Matrix2 Maux2(mat_ncols((*vB)[i]), mat_ncols((*vB)[i]));
          if (nproc == 0) {
            gmm::resize(vAloc[i], mat_ncols((*vB)[i]), mat_ncols((*vB)[i]));
            gmm::clear(vAloc[i]);
          }
          gmm::mult(gmm::transposed((*vB)[i]), Acsr, Maux);
          gmm::mult(Maux, (*vB)[i], Maux2);
          gmm::add(Maux2, vAloc[i]);
#else
          gmm::resize(vAloc[i], mat_ncols((*vB)[i]), mat_ncols((*vB)[i]));
          gmm::mult(gmm::transposed((*vB)[i]), *A, Maux);
          gmm::mult(Maux, (*vB)[i], vAloc[i]);
#endif
 
#ifdef GMM_USES_MPI
          if (nproc == size - 1 ) {
#endif
            precond1[i].build_with(vAloc[i]);
            gmm::resize(fi[i], mat_ncols((*vB)[i]));
            gmm::resize(gi[i], mat_ncols((*vB)[i]));
#ifdef GMM_USES_MPI
          }
#else
        }
#endif
#ifdef GMM_USES_MPI
     }
      if (nproc != size - 1) {
        MPI_Sendrecv(&(Acsr.jc[0]), sizeA+1, MPI_INT, next, tag2,
                     &(Acsrtemp.jc[0]), sizeA+1, MPI_INT, previous, tag2,
                     MPI_COMM_WORLD, &status);
        if (Acsrtemp.jc[sizeA] > size_type(sizepr)) {
          sizepr = Acsrtemp.jc[sizeA];
          gmm::resize(Acsrtemp.pr, sizepr);
          gmm::resize(Acsrtemp.ir, sizepr);
        }
        MPI_Sendrecv(&(Acsr.ir[0]), Acsr.jc[sizeA], MPI_INT, next, tag1,
                     &(Acsrtemp.ir[0]), Acsrtemp.jc[sizeA], MPI_INT, previous, tag1,
                     MPI_COMM_WORLD, &status);
        
        MPI_Sendrecv(&(Acsr.pr[0]), Acsr.jc[sizeA], mpi_type(value_type()), next, tag3, 
                     &(Acsrtemp.pr[0]), Acsrtemp.jc[sizeA], mpi_type(value_type()), previous, tag3,
                     MPI_COMM_WORLD, &status);
        gmm::copy(Acsrtemp, Acsr);
      }
    }
      t_final=MPI_Wtime();
    cout<<"temps boucle precond "<< t_final-t_ref<<endl;
#endif
    if (iter.get_noisy()) cout << "\n";
  }
  
  template <typename Matrix1, typename Matrix2, typename Precond,
            typename Vector2, typename Vector3, typename local_solver>
  void mult(const add_schwarz_mat<Matrix1, Matrix2, Precond, local_solver> &M,
            const Vector2 &p, Vector3 &q) {
    size_type itebilan = 0;
#ifdef GMM_USES_MPI
    static double tmult_tot = 0.0;
    double t_ref = MPI_Wtime();
#endif
    // cout << "tmult AS begin " << endl;
    mult(*(M.A), p, q);
#ifdef GMM_USES_MPI
    tmult_tot += MPI_Wtime()-t_ref;
    cout << "tmult_tot = " << tmult_tot << endl;
#endif
    std::vector<double> qbis(gmm::vect_size(q));
    std::vector<double> qter(gmm::vect_size(q));
#ifdef GMM_USES_MPI
    //    MPI_Status status;
    //    MPI_Request request,request1;
    //    int tag=111;
    int size,tranche,borne_sup,borne_inf,rank;
    size_type nb_sub=M.fi.size();
    MPI_Comm_rank(MPI_COMM_WORLD, &rank);
    MPI_Comm_size(MPI_COMM_WORLD, &size);
    tranche=nb_sub/size;
    borne_inf=rank*tranche;
    borne_sup=(rank+1)*tranche;
    // if (rank==size-1) borne_sup=nb_sub;
    //    int next = (rank + 1) % size;
    //    int previous = (rank + size - 1) % size;
    t_ref = MPI_Wtime();
     for (size_type i = size_type(borne_inf); i < size_type(borne_sup); ++i)
//        for (size_type i = 0; i < nb_sub/size; ++i)
      // for (size_type j = 0; j < nb_sub; ++j)
#else
    for (size_type i = 0; i < M.fi.size(); ++i)
#endif
      {
#ifdef GMM_USES_MPI
        // size_type i=j; // (rank+size*(j-1)+nb_sub)%nb_sub;
#endif
        gmm::mult(gmm::transposed((*(M.vB))[i]), q, M.fi[i]);
       M.iter.init();
       AS_local_solve(local_solver(), (M.vAloc)[i], (M.gi)[i],
                      (M.fi)[i],(M.precond1)[i],M.iter);
       itebilan = std::max(itebilan, M.iter.get_iteration());
       }
 
#ifdef GMM_USES_MPI
    cout << "First  AS loop time " <<  MPI_Wtime() - t_ref << endl;
#endif
 
    gmm::clear(q);
#ifdef GMM_USES_MPI
    t_ref = MPI_Wtime();
    // for (size_type j = 0; j < nb_sub; ++j)
    for (size_type i = size_type(borne_inf); i < size_type(borne_sup); ++i)
 
#else
      for (size_type i = 0; i < M.gi.size(); ++i)
#endif
        {
 
#ifdef GMM_USES_MPI
          // size_type i=j; // (rank+size*(j-1)+nb_sub)%nb_sub;
//        gmm::mult((*(M.vB))[i], M.gi[i], qbis,qbis);
          gmm::mult((*(M.vB))[i], M.gi[i], qter);
          add(qter,qbis,qbis);
#else
          gmm::mult((*(M.vB))[i], M.gi[i], q, q);
#endif
        }
#ifdef GMM_USES_MPI
     //WARNING this add only if you use the ring pattern below
  // need to do this below if using a n explicit ring pattern communication
 
//      add(qbis,q,q);
    cout << "Second AS loop time " <<  MPI_Wtime() - t_ref << endl;
#endif
 
 
#ifdef GMM_USES_MPI
    //    int tag1=11;
    static double t_tot = 0.0;
    double t_final;
    t_ref=MPI_Wtime();
//     int next = (rank + 1) % size;
//     int previous = (rank + size - 1) % size;
    //communication of local information on ring pattern
    //Each process receive  Nproc-1 contributions 
 
//     if (size > 1) {
//     for (int nproc = 0; nproc < size-1; ++nproc) 
//       {
 
//      MPI_Sendrecv(&(qbis[0]), gmm::vect_size(q), MPI_DOUBLE, next, tag1,
//                 &(qter[0]), gmm::vect_size(q),MPI_DOUBLE,previous,tag1,
//                 MPI_COMM_WORLD,&status);
//      gmm::copy(qter, qbis);
//      add(qbis,q,q);
//       }
//     }
    MPI_Allreduce(&(qbis[0]), &(q[0]),gmm::vect_size(q), MPI_DOUBLE,
                  MPI_SUM,MPI_COMM_WORLD);
    t_final=MPI_Wtime();
    t_tot += t_final-t_ref;
     cout<<"["<< rank<<"] temps reduce Resol "<< t_final-t_ref << " t_tot = " << t_tot << endl;
#endif 
 
    if (M.iter.get_noisy() > 0) cout << "itebloc = " << itebilan << endl;
    M.itebilan += itebilan;
    M.iter.set_resmax((M.iter.get_resmax() + M.residual) * 0.5);
  }
 
  template <typename Matrix1, typename Matrix2, typename Precond,
            typename Vector2, typename Vector3, typename local_solver>
  void mult(const add_schwarz_mat<Matrix1, Matrix2, Precond, local_solver> &M,
            const Vector2 &p, const Vector3 &q) {
    mult(M, p, const_cast<Vector3 &>(q));
  }
 
  template <typename Matrix1, typename Matrix2, typename Precond,
            typename Vector2, typename Vector3, typename Vector4,
            typename local_solver>
  void mult(const add_schwarz_mat<Matrix1, Matrix2, Precond, local_solver> &M,
            const Vector2 &p, const Vector3 &p2, Vector4 &q)
  { mult(M, p, q); add(p2, q); }
 
  template <typename Matrix1, typename Matrix2, typename Precond,
            typename Vector2, typename Vector3, typename Vector4,
            typename local_solver>
  void mult(const add_schwarz_mat<Matrix1, Matrix2, Precond, local_solver> &M,
            const Vector2 &p, const Vector3 &p2, const Vector4 &q)
  { mult(M, p, const_cast<Vector4 &>(q)); add(p2, q); }
 
  /* ******************************************************************** */
  /*            Additive Schwarz interfaced global solvers                */
  /* ******************************************************************** */
 
  template <typename ASM_type, typename Vect>
  void AS_global_solve(using_cg, const ASM_type &ASM, Vect &x,
                       const Vect &b, iteration &iter)
  { cg(ASM, x, b, *(ASM.A), identity_matrix(), iter); }
 
  template <typename ASM_type, typename Vect>
  void AS_global_solve(using_gmres, const ASM_type &ASM, Vect &x,
                       const Vect &b, iteration &iter)
  { gmres(ASM, x, b, identity_matrix(), 100, iter); }
 
  template <typename ASM_type, typename Vect>
  void AS_global_solve(using_bicgstab, const ASM_type &ASM, Vect &x,
                       const Vect &b, iteration &iter)
  { bicgstab(ASM, x, b, identity_matrix(), iter); }
 
  template <typename ASM_type, typename Vect>
  void AS_global_solve(using_qmr,const ASM_type &ASM, Vect &x,
                       const Vect &b, iteration &iter)
  { qmr(ASM, x, b, identity_matrix(), iter); }
 
#if defined(GMM_USES_SUPERLU)
  template <typename ASM_type, typename Vect>
  void AS_global_solve(using_superlu, const ASM_type &, Vect &,
                       const Vect &, iteration &) {
    GMM_ASSERT1(false, "You cannot use SuperLU as "
                "global solver in additive Schwarz meethod");
  }
#endif
  
  /* ******************************************************************** */
  /*                Linear Additive Schwarz method                        */
  /* ******************************************************************** */
  /* ref : Domain decomposition algorithms for the p-version finite       */
  /*       element method for elliptic problems, Luca F. Pavarino,        */
  /*       PhD thesis, Courant Institute of Mathematical Sciences, 1992.  */
  /* ******************************************************************** */
 
  /** Function to call if the ASM matrix is precomputed for successive solve
   * with the same system.
   */
  template <typename Matrix1, typename Matrix2,
            typename Vector2, typename Vector3, typename Precond,
            typename local_solver, typename global_solver>
  void additive_schwarz(
    add_schwarz_mat<Matrix1, Matrix2, Precond, local_solver> &ASM, Vector3 &u,
    const Vector2 &f, iteration &iter, const global_solver&) {
 
    typedef typename linalg_traits<Matrix1>::value_type value_type;
 
    size_type nb_sub = ASM.vB->size(), nb_dof = gmm::vect_size(f);
    ASM.itebilan = 0;
    std::vector<value_type> g(nb_dof);
    std::vector<value_type> gbis(nb_dof);
#ifdef GMM_USES_MPI
    double t_init=MPI_Wtime();
    int size,tranche,borne_sup,borne_inf,rank;
    MPI_Comm_rank(MPI_COMM_WORLD, &rank);
    MPI_Comm_size(MPI_COMM_WORLD, &size);
    tranche=nb_sub/size;
    borne_inf=rank*tranche;
    borne_sup=(rank+1)*tranche;
    // if (rank==size-1) borne_sup=nb_sub*size;
    for (size_type i = size_type(borne_inf); i < size_type(borne_sup); ++i)
//     for (size_type i = 0; i < nb_sub/size; ++i)
      // for (size_type j = 0; j < nb_sub; ++j)
      // for (size_type i = rank; i < nb_sub; i+=size)
#else
    for (size_type i = 0; i < nb_sub; ++i)
#endif
    {
 
#ifdef GMM_USES_MPI
      // size_type i=j; // (rank+size*(j-1)+nb_sub)%nb_sub;
#endif
      gmm::mult(gmm::transposed((*(ASM.vB))[i]), f, ASM.fi[i]);
      ASM.iter.init();
      AS_local_solve(local_solver(), ASM.vAloc[i], ASM.gi[i], ASM.fi[i],
                     ASM.precond1[i], ASM.iter);
      ASM.itebilan = std::max(ASM.itebilan, ASM.iter.get_iteration());
#ifdef GMM_USES_MPI
    gmm::mult((*(ASM.vB))[i], ASM.gi[i], gbis,gbis);
#else   
    gmm::mult((*(ASM.vB))[i], ASM.gi[i], g, g);
#endif
    }
#ifdef GMM_USES_MPI
    cout<<"temps boucle init "<< MPI_Wtime()-t_init<<endl;
    double t_ref,t_final;
    t_ref=MPI_Wtime();
    MPI_Allreduce(&(gbis[0]), &(g[0]),gmm::vect_size(g), MPI_DOUBLE,
                  MPI_SUM,MPI_COMM_WORLD);
    t_final=MPI_Wtime();
    cout<<"temps reduce init "<< t_final-t_ref<<endl;
#endif
#ifdef GMM_USES_MPI
    t_ref=MPI_Wtime();
    cout<<"begin global AS"<<endl;
#endif
    AS_global_solve(global_solver(), ASM, u, g, iter);
#ifdef GMM_USES_MPI
    t_final=MPI_Wtime();
    cout<<"temps AS Global Solve "<< t_final-t_ref<<endl;
#endif
    if (iter.get_noisy())
      cout << "Total number of internal iterations : " << ASM.itebilan << endl;
  }
 
  /** Global function. Compute the ASM matrix and call the previous function.
   *  The ASM matrix represent the preconditionned linear system.
   */
  template <typename Matrix1, typename Matrix2,
            typename Vector2, typename Vector3, typename Precond,
            typename local_solver, typename global_solver>
  void additive_schwarz(const Matrix1 &A, Vector3 &u,
                                  const Vector2 &f, const Precond &P,
                                  const std::vector<Matrix2> &vB,
                                  iteration &iter, local_solver,
                                  global_solver) {
    iter.set_rhsnorm(vect_norm2(f));
    if (iter.get_rhsnorm() == 0.0) { gmm::clear(u); return; }
    iteration iter2 = iter; iter2.reduce_noisy();
    iter2.set_maxiter(size_type(-1));
    add_schwarz_mat<Matrix1, Matrix2, Precond, local_solver>
      ASM(A, vB, iter2, P, iter.get_resmax());
    additive_schwarz(ASM, u, f, iter, global_solver());
  }
 
  /* ******************************************************************** */
  /*            Sequential Non-Linear Additive Schwarz method             */
  /* ******************************************************************** */
  /* ref : Nonlinearly Preconditionned Inexact Newton Algorithms,         */
  /*       Xiao-Chuan Cai, David E. Keyes,                                */
  /*       SIAM J. Sci. Comp. 24: p183-200.  l                             */
  /* ******************************************************************** */
 
  template <typename Matrixt, typename MatrixBi> 
  class NewtonAS_struct {
    
  public :
    typedef Matrixt tangent_matrix_type;
    typedef MatrixBi B_matrix_type;
    typedef typename linalg_traits<Matrixt>::value_type value_type;
    typedef std::vector<value_type> Vector;
    
    virtual size_type size(void) = 0;
    virtual const std::vector<MatrixBi> &get_vB() = 0;
    
    virtual void compute_F(Vector &f, Vector &x) = 0;
    virtual void compute_tangent_matrix(Matrixt &M, Vector &x) = 0;
    // compute Bi^T grad(F(X)) Bi
    virtual void compute_sub_tangent_matrix(Matrixt &Mloc, Vector &x,
                                            size_type i) = 0;
    // compute Bi^T F(X)
    virtual void compute_sub_F(Vector &fi, Vector &x, size_type i) = 0;
 
    virtual ~NewtonAS_struct() {}
  };
 
  template <typename Matrixt, typename MatrixBi> 
  struct AS_exact_gradient {
    const std::vector<MatrixBi> &vB;
    std::vector<Matrixt> vM;
    std::vector<Matrixt> vMloc;
 
    void init(void) {
      for (size_type i = 0; i < vB.size(); ++i) {
        Matrixt aux(gmm::mat_ncols(vB[i]), gmm::mat_ncols(vM[i]));
        gmm::resize(vMloc[i], gmm::mat_ncols(vB[i]), gmm::mat_ncols(vB[i]));
        gmm::mult(gmm::transposed(vB[i]), vM[i], aux);
        gmm::mult(aux, vB[i], vMloc[i]);
      }
    }
    AS_exact_gradient(const std::vector<MatrixBi> &vB_) : vB(vB_) {
      vM.resize(vB.size()); vMloc.resize(vB.size());
      for (size_type i = 0; i < vB.size(); ++i) {
        gmm::resize(vM[i], gmm::mat_nrows(vB[i]), gmm::mat_nrows(vB[i]));
      }
    }
  };
 
  template <typename Matrixt, typename MatrixBi,
            typename Vector2, typename Vector3>
  void mult(const AS_exact_gradient<Matrixt, MatrixBi> &M,
            const Vector2 &p, Vector3 &q) {
    gmm::clear(q);
    typedef typename gmm::linalg_traits<Vector3>::value_type T;
    std::vector<T> v(gmm::vect_size(p)), w, x;
    for (size_type i = 0; i < M.vB.size(); ++i) {
      w.resize(gmm::mat_ncols(M.vB[i]));
      x.resize(gmm::mat_ncols(M.vB[i]));
      gmm::mult(M.vM[i], p, v);
      gmm::mult(gmm::transposed(M.vB[i]), v, w);
      double rcond;
      SuperLU_solve(M.vMloc[i], x, w, rcond);
      // gmm::iteration iter(1E-10, 0, 100000);
      //gmm::gmres(M.vMloc[i], x, w, gmm::identity_matrix(), 50, iter);
      gmm::mult_add(M.vB[i], x, q);
    }
  }
 
  template <typename Matrixt, typename MatrixBi,
            typename Vector2, typename Vector3>
  void mult(const AS_exact_gradient<Matrixt, MatrixBi> &M,
            const Vector2 &p, const Vector3 &q) {
    mult(M, p, const_cast<Vector3 &>(q));
  }
 
  template <typename Matrixt, typename MatrixBi,
            typename Vector2, typename Vector3, typename Vector4>
  void mult(const AS_exact_gradient<Matrixt, MatrixBi> &M,
            const Vector2 &p, const Vector3 &p2, Vector4 &q)
  { mult(M, p, q); add(p2, q); }
 
  template <typename Matrixt, typename MatrixBi,
            typename Vector2, typename Vector3, typename Vector4>
  void mult(const AS_exact_gradient<Matrixt, MatrixBi> &M,
            const Vector2 &p, const Vector3 &p2, const Vector4 &q)
  { mult(M, p, const_cast<Vector4 &>(q)); add(p2, q); }
 
  struct S_default_newton_line_search {
    
    double conv_alpha, conv_r;
    size_t it, itmax, glob_it;
 
    double alpha, alpha_old, alpha_mult, first_res, alpha_max_ratio;
    double alpha_min_ratio, alpha_min;
    size_type count, count_pat;
    bool max_ratio_reached;
    double alpha_max_ratio_reached, r_max_ratio_reached;
    size_type it_max_ratio_reached;
 
    
    double converged_value(void) { return conv_alpha; };
    double converged_residual(void) { return conv_r; };
 
    virtual void init_search(double r, size_t git, double = 0.0) {
      alpha_min_ratio = 0.9;
      alpha_min = 1e-10;
      alpha_max_ratio = 10.0;
      alpha_mult = 0.25;
      itmax = size_type(-1);
      glob_it = git; if (git <= 1) count_pat = 0;
      conv_alpha = alpha = alpha_old = 1.;
      conv_r = first_res = r; it = 0;
      count = 0;
      max_ratio_reached = false;
    }
    virtual double next_try(void) {
      alpha_old = alpha;
      if (alpha >= 0.4) alpha *= 0.5; else alpha *= alpha_mult; ++it;
      return alpha_old;
    }
    virtual bool is_converged(double r, double = 0.0) {
      // cout << "r = " << r << " alpha = " << alpha / alpha_mult << " count_pat = " << count_pat << endl;
      if (!max_ratio_reached && r < first_res * alpha_max_ratio) {
        alpha_max_ratio_reached = alpha_old; r_max_ratio_reached = r;
        it_max_ratio_reached = it; max_ratio_reached = true; 
      }
      if (max_ratio_reached && r < r_max_ratio_reached * 0.5
          && r > first_res * 1.1 && it <= it_max_ratio_reached+1) {
        alpha_max_ratio_reached = alpha_old; r_max_ratio_reached = r;
        it_max_ratio_reached = it;
      }
      if (count == 0 || r < conv_r)
        { conv_r = r; conv_alpha = alpha_old; count = 1; }
      if (conv_r < first_res) ++count;
 
      if (r < first_res *  alpha_min_ratio)
        { count_pat = 0; return true; }      
      if (count >= 5 || (alpha < alpha_min && max_ratio_reached)) {
        if (conv_r < first_res * 0.99) count_pat = 0;
        if (/*gmm::random() * 50. < -log(conv_alpha)-4.0 ||*/ count_pat >= 3)
          { conv_r=r_max_ratio_reached; conv_alpha=alpha_max_ratio_reached; }
        if (conv_r >= first_res * 0.9999) count_pat++;
        return true;
      }
      return false;
    }
    S_default_newton_line_search(void) { count_pat = 0; }
  };
 
 
  
  template <typename Matrixt, typename MatrixBi, typename Vector,
            typename Precond, typename local_solver, typename global_solver>
  void Newton_additive_Schwarz(NewtonAS_struct<Matrixt, MatrixBi> &NS,
                               const Vector &u_,
                               iteration &iter, const Precond &P,
                               local_solver, global_solver) {
    Vector &u = const_cast<Vector &>(u_);
    typedef typename linalg_traits<Vector>::value_type value_type;
    typedef typename number_traits<value_type>::magnitude_type mtype;
    typedef actual_precond<Precond, local_solver, Matrixt> chgt_precond;
    
    double residual = iter.get_resmax();
 
    S_default_newton_line_search internal_ls;
    S_default_newton_line_search external_ls;
 
    typename chgt_precond::APrecond PP = chgt_precond::transform(P);
    iter.set_rhsnorm(mtype(1));
    iteration iternc(iter);
    iternc.reduce_noisy(); iternc.set_maxiter(size_type(-1));
    iteration iter2(iternc);
    iteration iter3(iter2); iter3.reduce_noisy();
    iteration iter4(iter3);
    iternc.set_name("Local Newton");
    iter2.set_name("Linear System for Global Newton");
    iternc.set_resmax(residual/100.0);
    iter3.set_resmax(residual/10000.0);
    iter2.set_resmax(residual/1000.0);
    iter4.set_resmax(residual/1000.0);
    std::vector<value_type> rhs(NS.size()), x(NS.size()), d(NS.size());
    std::vector<value_type> xi, xii, fi, di;
 
    std::vector< std::vector<value_type> > vx(NS.get_vB().size());
    for (size_type i = 0; i < NS.get_vB().size(); ++i) // for exact gradient
      vx[i].resize(NS.size()); // for exact gradient
 
    Matrixt Mloc, M(NS.size(), NS.size());
    NS.compute_F(rhs, u);
    mtype act_res=gmm::vect_norm2(rhs), act_res_new(0), precond_res = act_res;
    mtype alpha;
    
    while(!iter.finished(std::min(act_res, precond_res))) {
      for (int SOR_step = 0;  SOR_step >= 0; --SOR_step) {
        gmm::clear(rhs);
        for (size_type isd = 0; isd < NS.get_vB().size(); ++isd) {
          const MatrixBi &Bi = (NS.get_vB())[isd];
          size_type si = mat_ncols(Bi);
          gmm::resize(Mloc, si, si);
          xi.resize(si); xii.resize(si); fi.resize(si); di.resize(si);
          
          iternc.init();
          iternc.set_maxiter(30); // ?
          if (iternc.get_noisy())
            cout << "Non-linear local problem " << isd << endl;
          gmm::clear(xi);
          gmm::copy(u, x);
          NS.compute_sub_F(fi, x, isd); gmm::scale(fi, value_type(-1));
          mtype r = gmm::vect_norm2(fi), r_t(r);
          if (r > value_type(0)) {
            iternc.set_rhsnorm(std::max(r, mtype(1)));
            while(!iternc.finished(r)) {
              NS.compute_sub_tangent_matrix(Mloc, x, isd);
 
              PP.build_with(Mloc);
              iter3.init();
              AS_local_solve(local_solver(), Mloc, di, fi, PP, iter3);
              
              internal_ls.init_search(r, iternc.get_iteration());
              do {
                alpha = internal_ls.next_try();
                gmm::add(xi, gmm::scaled(di, -alpha), xii);
                gmm::mult(Bi, gmm::scaled(xii, -1.0), u, x);
                NS.compute_sub_F(fi, x, isd); gmm::scale(fi, value_type(-1));
                r_t = gmm::vect_norm2(fi);
              } while (!internal_ls.is_converged(r_t));
              
              if (alpha != internal_ls.converged_value()) {
                alpha = internal_ls.converged_value();
                gmm::add(xi, gmm::scaled(di, -alpha), xii);
                gmm::mult(Bi, gmm::scaled(xii, -1.0), u, x);
                NS.compute_sub_F(fi, x, isd); gmm::scale(fi, value_type(-1));
                r_t = gmm::vect_norm2(fi);
              }
              gmm::copy(x, vx[isd]); // for exact gradient
 
              if (iternc.get_noisy()) cout << "(step=" << alpha << ")\t";
              ++iternc; r = r_t; gmm::copy(xii, xi); 
            }
            if (SOR_step) gmm::mult(Bi, gmm::scaled(xii, -1.0), u, u);
            gmm::mult(Bi, gmm::scaled(xii, -1.0), rhs, rhs);
          }
        }
        precond_res = gmm::vect_norm2(rhs);
        if (SOR_step) cout << "SOR step residual = " << precond_res << endl;
        if (precond_res < residual) break;
        cout << "Precond residual = " << precond_res << endl;
      }
 
      iter2.init();
      // solving linear system for the global Newton method
      if (0) {
        NS.compute_tangent_matrix(M, u);
        add_schwarz_mat<Matrixt, MatrixBi, Precond, local_solver>
          ASM(M, NS.get_vB(), iter4, P, iter.get_resmax());
        AS_global_solve(global_solver(), ASM, d, rhs, iter2);
      }
      else {  // for exact gradient
        AS_exact_gradient<Matrixt, MatrixBi> eg(NS.get_vB());
        for (size_type i = 0; i < NS.get_vB().size(); ++i) {
          NS.compute_tangent_matrix(eg.vM[i], vx[i]);
        }
        eg.init();
        gmres(eg, d, rhs, gmm::identity_matrix(), 50, iter2);
      }
 
      //      gmm::add(gmm::scaled(rhs, 0.1), u); ++iter;
      external_ls.init_search(act_res, iter.get_iteration());
      do {
        alpha = external_ls.next_try();
        gmm::add(gmm::scaled(d, alpha), u, x);
        NS.compute_F(rhs, x);
        act_res_new = gmm::vect_norm2(rhs);
      } while (!external_ls.is_converged(act_res_new));
      
      if (alpha != external_ls.converged_value()) {
        alpha = external_ls.converged_value();
        gmm::add(gmm::scaled(d, alpha), u, x);
        NS.compute_F(rhs, x);
        act_res_new = gmm::vect_norm2(rhs);
      }
 
      if (iter.get_noisy() > 1) cout << endl;
      act_res = act_res_new; 
      if (iter.get_noisy()) cout << "(step=" << alpha << ")\t unprecond res = " << act_res << " ";
      
      
      ++iter; gmm::copy(x, u);
    }
  }
 
}
 
 
#endif //  GMM_SOLVERS_SCHWARZ_ADDITIVE_H__