ILU0Preconditioner.h

// -*- tab-width: 2; indent-tabs-mode: nil; coding: utf-8-with-signature -*-
//-----------------------------------------------------------------------------
// Copyright 2000-2026 CEA (www.cea.fr) IFPEN (www.ifpenergiesnouvelles.com)
// See the top-level COPYRIGHT file for details.
// SPDX-License-Identifier: Apache-2.0
//-----------------------------------------------------------------------------
 
#pragma once
#if defined (ALIEN_USE_EIGEN3)
#if defined(ALIEN_USE_SYCL)
  #ifdef __ACPP__
    //#define EIGEN3_DISABLED
    //#warning "EIGEN USE SYCL"
    #ifdef SYCL_DEVICE_ONLY
      #undef SYCL_DEVICE_ONLY
      #include <Eigen/Dense>
      #define SYCL_DEVICE_ONLY
    #else
      #include <Eigen/Dense>
    #endif
  #else
    #include <Eigen/Dense>
  #endif
#else
  #include <Eigen/Dense>
#endif
#endif
#include <alien/handlers/scalar/CSRModifierViewT.h>
 
namespace Alien
{
 
  template <typename ValueT>
  struct Block1D
  {
    using ValueType = ValueT ;
 
    ValueType& operator()(std::size_t i) {
      return m_values[i];
    }
 
    ValueType operator()(std::size_t i) const {
            return m_values[i];
    }
    ValueType* m_values = nullptr;
  };
 
  template <typename ValueT>
  struct Block2D
  {
    using ValueType = ValueT ;
    Block2D(int N)
    : m_N(N)
    , m_buffer(N*N)
    {
      m_values = m_buffer.data();
    }
 
    Block2D(ValueType* values, int N)
    : m_values(values)
    , m_N(N)
    {}
 
    Block2D(Block2D const& rhs)
    : m_values(rhs.m_values)
    , m_N(rhs.m_N)
    {}
 
    Block2D& operator = (Block2D const& rhs) {
      for(int i=0;i<m_N*m_N;++i)
        m_values[i] = rhs.m_values[i] ;
      return *this ;
    }
 
    Block2D& operator += (Block2D const& rhs) {
      for(int i=0;i<m_N*m_N;++i)
        m_values[i] += rhs.m_values[i] ;
      return *this ;
    }
 
    Block2D& operator -= (Block2D const& rhs) {
      for(int i=0;i<m_N*m_N;++i)
        m_values[i] -= rhs.m_values[i] ;
      return *this ;
    }
 
    ValueType& operator()(std::size_t i, std::size_t j) {
      return m_values[i*m_N+j];
    }
 
    ValueType operator()(std::size_t i, std::size_t j) const {
      return m_values[i*m_N+j];
    }
    ValueType* m_values = nullptr ;
    int m_N = 0;
    std::vector<ValueType> m_buffer ;
  };
 
  template <typename ValueT>
  struct LU
  {
   public:
    using ValueType     = ValueT ;
    using Block2DType   = Block2D<ValueType> ;
 
 
    LU(int N)
    : A(N)
    {
      setZero() ;
    }
 
    LU inv(Block2DType const& r)
    {
      std::copy(r.m_values,r.m_values+r.m_N*r.m_N,A.m_values) ;
      for (int k = 0; k < A.m_N; ++k)
      {
        assert(A(k,k) != 0);
        A(k,k) = 1 / A(k,k);
        for (int i = k + 1; i < A.m_N; ++i) {
          A(i,k) *= A(k,k);
        }
        for (int i = k + 1; i < A.m_N; ++i) {
          for (int j = k + 1; j < A.m_N; ++j) {
            A(i,j) -= A(i,k) * A(k,j);
          }
        }
      }
      return *this ;
    }

    void setIdentity()
    {
      for (int i = 0; i < A.m_N; ++i)
      {
        for (int j = 0; j < i; ++j) {
          A(i,j) = 0;
        }
        A(i,i) = 1;
        for (int j = i + 1; j < A.m_N; ++j) {
          A(i,j) = 0;
        }
      }
    }

    void setZero()
    {
      Real* ptr = A.m_values;
      for (int i = 0; i < A.m_N * A.m_N; ++i) {
        ptr[i] = 0;
      }
    }
 
    void LSolve(Block2DType& X) const
    {
      for (int i = 1; i < A.m_N; ++i)
      {
        for (int j = 0; j < i; ++j)
        {
          for(int k=0;k<A.m_N;++k)
            X(i,k) -= A(i,j) * X(j,k);
        }
      }
    }
 
    void USolve(Block2DType& X) const
    {
      for (int i = A.m_N - 1; i >= 0; --i) {
        for (int j = A.m_N - 1; j > i; --j) {
          for(int k=0;k<A.m_N;++k)
            X(i,k) -= A(i,j) * X(j,k);
        }
        for(int k=0;k<A.m_N;++k)
          X(i,k) *= A(i,i);
      }
    }
 
    Block2DType solve(Block2DType const& r) const
    {
      Block2DType results(A.m_N) ;
      results = r ;
      LSolve(results);
      USolve(results);
      return results ;
    }
 
   private:
    Block2DType A;
  };
 
  template<typename ValueT>
  Block2D<ValueT> operator*(Block2D<ValueT> const& l, Block2D<ValueT> const& r)
  {
    auto N = l.m_N;
    Block2D<ValueT> value{l.m_N} ;
    for(int i=0;i<N;++i)
      for(int j=0;j<N;++j)
        for(int k=0;k<N;++k)
          value(i,k) += l(i,j)*r(j,k) ;
    return value ;
  }
 
 
  template<typename ValueT>
  Block2D<ValueT> operator*(Block2D<ValueT> const& l, LU<ValueT> const& r)
  {
    return r.solve(l) ;
  }
 
template <typename MatrixT, typename VectorT>
class LUFactorisationAlgo
{
 public:
  // clang-format off
  typedef MatrixT                          MatrixType;
  typedef VectorT                          VectorType;
  typedef typename MatrixType::ProfileType ProfileType ;
  typedef typename MatrixType::ValueType   ValueType;
  typedef typename MatrixType::TagType     TagType ;
  // clang-format on
 
  LUFactorisationAlgo()
  {}
 
  virtual ~LUFactorisationAlgo()
  {}
 
  template <typename AlgebraT>
  void baseInit(AlgebraT& algebra, MatrixT const& matrix)
  {
    m_is_parallel = matrix.isParallel();
    m_alloc_size = matrix.getAllocSize();
    if constexpr (requires{matrix.blockSize();})
      m_block_size = matrix.blockSize();
    else
      m_block_size = 1 ;
    m_distribution = matrix.distribution();
    m_lu_matrix.reset(matrix.cloneTo(nullptr));
    m_profile = &m_lu_matrix->getProfile();
    m_work.resize(m_alloc_size);
    m_work.assign(m_work.size(), -1);
    algebra.allocate(AlgebraT::resource(matrix), m_x);
  }
 
  template <typename AlgebraT>
  void init(AlgebraT& algebra, MatrixT const& matrix)
  {
    baseInit(algebra, matrix);
    if(m_block_size==1)
      factorize(*m_lu_matrix);
    else
      blockFactorize(*m_lu_matrix);
 
    m_work.clear();
  }
 
  void factorize(MatrixT& matrix, bool bjacobi = true)
  {
    /*
       *
         For i = 1, . . . ,N Do:
            For k = 1, . . . , i - 1 and if (i, k) 2 NZ(A) Do:
                Compute aik := aik/akk
                For j = k + 1, . . . and if (i, j) 2 NZ(A), Do:
                   compute aij := aij - aik.ak,j.
                EndFor
            EndFor
         EndFor
       *
       */
    m_bjacobi = bjacobi;
    CSRModifierViewT<MatrixT> modifier(matrix);
 
    // clang-format off
    auto nrows  = modifier.nrows() ;
    auto kcol   = modifier.kcol() ;
    auto dcol   = modifier.dcol() ;
    auto cols   = modifier.cols() ;
    auto values = modifier.data() ;
    // clang-format on
    if (m_is_parallel) {
      auto& local_row_size = matrix.getDistStructInfo().m_local_row_size;
      if (m_bjacobi) {
        for (std::size_t irow = 1; irow < nrows; ++irow) // i=1->nrow
        {
          for (int k = kcol[irow]; k < dcol[irow]; ++k) // k=1 ->i-1
          {
            int krow = cols[k];
            ValueType aik = values[k] / values[dcol[krow]]; // aik = aik/akk
            values[k] = aik;
            for (int l = kcol[krow]; l < kcol[krow] + local_row_size[krow]; ++l)
              m_work[cols[l]] = l;
            for (int j = k + 1; j < kcol[krow] + local_row_size[irow]; ++j) // j=k+1->n
            {
              int jcol = cols[j];
              int kj = m_work[jcol];
              if (kj != -1) {
                values[j] -= aik * values[kj]; // aij = aij - aik*akj
              }
            }
            for (int l = kcol[krow]; l < kcol[krow] + local_row_size[krow]; ++l)
              m_work[cols[l]] = -1;
          }
        }
      }
      else {
        typename LUSendRecvTraits<TagType>::matrix_op_type op(matrix, m_distribution, m_work);
        op.recvLowerNeighbLUData(values);
        int first_upper_ghost_index = matrix.getDistStructInfo().m_first_upper_ghost_index;
        for (std::size_t irow = 1; irow < nrows; ++irow) // i=1->nrow
        {
          for (int k = kcol[irow]; k < dcol[irow]; ++k) // k=1 ->i-1
          {
            int krow = cols[k];
            ValueType aik = values[k] / values[dcol[krow]]; // aik = aik/akk
            values[k] = aik;
            for (int l = kcol[krow]; l < kcol[krow + 1]; ++l)
              m_work[cols[l]] = l;
            for (int j = k + 1; j < kcol[irow] + local_row_size[irow]; ++j) // j=k+1->n
            {
              int jcol = cols[j];
              int kj = m_work[jcol];
              if (kj != -1) {
                values[j] -= aik * values[kj]; // aij = aij - aik*akj
              }
            }
            for (int j = kcol[irow] + local_row_size[irow]; j < kcol[irow + 1]; ++j) // j=k+1->n
            {
              int jcol = cols[j];
              int kj = m_work[jcol];
              if ((kj != -1) && (jcol >= first_upper_ghost_index)) {
                values[j] -= aik * values[kj]; // aij = aij - aik*akj
              }
            }
            for (int l = kcol[krow]; l < kcol[krow + 1]; ++l)
              m_work[cols[l]] = -1;
          }
        }
        op.sendUpperNeighbLUData(values);
      }
    }
    else {
      for (std::size_t irow = 1; irow < nrows; ++irow) // i=1->nrow
      {
        for (int k = kcol[irow]; k < dcol[irow]; ++k) // k=1 ->i-1
        {
          int krow = cols[k];
          ValueType aik = values[k] / values[dcol[krow]]; // aik = aik/akk
          values[k] = aik;
          for (int l = kcol[krow]; l < kcol[krow + 1]; ++l)
            m_work[cols[l]] = l;
          for (int j = k + 1; j < kcol[irow + 1]; ++j) // j=k+1->n
          {
            int jcol = cols[j];
            int kj = m_work[jcol];
            if (kj != -1) {
              values[j] -= aik * values[kj]; // aij = aij - aik*akj
            }
          }
          for (int l = kcol[krow]; l < kcol[krow + 1]; ++l)
            m_work[cols[l]] = -1;
        }
      }
    }
  }
 
  void solveL(ValueType const* y, ValueType* x) const
  {
    CSRConstViewT<MatrixT> view(*m_lu_matrix);
    // clang-format off
    auto nrows  = view.nrows() ;
    auto kcol   = view.kcol() ;
    auto dcol   = view.dcol() ;
    auto cols   = view.cols() ;
    auto values = view.data() ;
    // clang-format on
 
    for (std::size_t irow = 0; irow < nrows; ++irow) {
      ValueType val = y[irow];
      for (int k = kcol[irow]; k < dcol[irow]; ++k)
        val -= values[k] * x[cols[k]];
      x[irow] = val;
    }
  }
 
  void solveU(ValueType const* y, ValueType* x) const
  {
    CSRConstViewT<MatrixT> view(*m_lu_matrix);
    // clang-format off
    auto nrows  = view.nrows() ;
    auto kcol   = view.kcol() ;
    auto dcol   = view.dcol() ;
    auto cols   = view.cols() ;
    auto values = view.data() ;
    // clang-format on
    if (m_is_parallel) {
      auto& local_row_size = m_lu_matrix->getDistStructInfo().m_local_row_size;
      for (int irow = (int)nrows - 1; irow > -1; --irow) {
        int dk = dcol[irow];
        ValueType val = y[irow];
        for (int k = dk + 1; k < kcol[irow] + local_row_size[irow]; ++k) {
          val -= values[k] * x[cols[k]];
        }
        x[irow] = val / values[dk];
      }
    }
    else {
      for (int irow = (int)nrows - 1; irow > -1; --irow) {
        int dk = dcol[irow];
        ValueType val = y[irow];
        for (int k = dk + 1; k < kcol[irow + 1]; ++k) {
          val -= values[k] * x[cols[k]];
        }
        x[irow] = val / values[dk];
      }
    }
  }
 
#if defined (ALIEN_USE_EIGEN3) && !defined(EIGEN3_DISABLED)
  inline auto inv(Eigen::Map<Eigen::Matrix<ValueType,Eigen::Dynamic,Eigen::Dynamic,Eigen::RowMajor>>const & block) const
  {
    assert(block.determinant()!=0) ;
    return block.inverse() ;
  }
#endif
 
  void blockFactorize(MatrixT& matrix, bool bjacobi = true)
  {
    /*
       *
         For i = 1, . . . ,N Do:
            For k = 1, . . . , i - 1 and if (i, k) 2 NZ(A) Do:
                Compute aik := aik/akk
                For j = k + 1, . . . and if (i, j) 2 NZ(A), Do:
                   compute aij := aij - aik.ak,j.
                EndFor
            EndFor
         EndFor
       *
       */
    //if constexpr (MatrixType::on_host_only)
    {
#if defined (ALIEN_USE_EIGEN3)  && !defined(EIGEN3_DISABLED)
      using namespace Eigen;
      using Block2D     = Eigen::Matrix<ValueType,Eigen::Dynamic,Eigen::Dynamic,Eigen::RowMajor> ;
      using Block2DView = Eigen::Map<Block2D> ;
 
      int N = m_block_size;
      int N2 = N*N;
 
      m_bjacobi = bjacobi;
      CSRModifierViewT<MatrixT> modifier(matrix);
 
      // clang-format off
      auto nrows  = modifier.nrows() ;
      auto kcol   = modifier.kcol() ;
      auto dcol   = modifier.dcol() ;
      auto cols   = modifier.cols() ;
      auto values = modifier.data() ;
 
      Block2D aik(N,N) ;
      // clang-format on
      /*
      alien_info([&] {
        cout()<<"BLOCK FACTORIZATION : "<<m_is_parallel<<" "<<m_bjacobi;
      });*/
      if (m_is_parallel) {
        auto& local_row_size = matrix.getDistStructInfo().m_local_row_size;
 
        if (m_bjacobi) {
          for (std::size_t irow = 1; irow < nrows; ++irow) // i=1->nrow
          {
            /*alien_info([&] {
              cout()<<"ROW : "<<irow<<" "<<kcol[irow]<<" "<<dcol[irow]<<" "<<kcol[irow+1]-kcol[irow]<<" "<<local_row_size[irow];
            });*/
            for (int k = kcol[irow]; k < dcol[irow]; ++k) // k=1 ->i-1
            {
              int krow = cols[k];
              /*alien_info([&] {
                cout()<<"\tCOL : "<<k<<" "<<cols[k]<<" "<<kcol[krow]<<" "<<local_row_size[krow];
              });*/
 
              //ValueType aik = values[k] / values[dcol[krow]]; // aik = aik/akk
              //values[k] = aik;
              aik = Block2DView(values+k*N2,N,N) * inv(Block2DView(values+dcol[krow]*N2,N,N)) ;
              Block2DView(values+k*N2,N,N) = aik ;
              /*alien_info([&] {
                cout()<<"\t\tVALUES["<<k<<"]=";
                for(int i=0;i<N;++i)
                  for(int j=0;j<N;++j)
                    cout()<<"\t\t\t MAT("<<i<<","<<j<<")="<<values[k*N2+i*N+j];
              });*/
              for (int l = kcol[krow]; l < kcol[krow] + local_row_size[krow]; ++l)
                m_work[cols[l]] = l;
              for (int j = k + 1; j < kcol[irow] + local_row_size[irow]; ++j) // j=k+1->n
              {
                int jcol = cols[j];
                int kj = m_work[jcol];
                if (kj != -1) {
                  //values[j] -= aik * values[kj]; // aij = aij - aik*akj
                  Block2DView(values+j*N2,N,N) -= aik * Block2DView(values+kj*N2,N,N) ;
                  /*alien_info([&] {
                    cout()<<"\t\tVALUES["<<j<<"]=";
                    for(int ii=0;ii<N;++ii)
                      for(int jj=0;jj<N;++jj)
                        cout()<<"\t\t\t MAT("<<ii<<","<<jj<<")="<<values[j*N2+ii*N+jj];
                  });*/
                }
              }
              for (int l = kcol[krow]; l < kcol[krow] + local_row_size[krow]; ++l)
                m_work[cols[l]] = -1;
            }
          }
        }
        else {
          typename LUSendRecvTraits<TagType>::matrix_op_type op(matrix, m_distribution, m_work);
          op.recvLowerNeighbLUData(values);
          int first_upper_ghost_index = matrix.getDistStructInfo().m_first_upper_ghost_index;
          for (std::size_t irow = 1; irow < nrows; ++irow) // i=1->nrow
          {
            for (int k = kcol[irow]; k < dcol[irow]; ++k) // k=1 ->i-1
            {
              int krow = cols[k];
              //ValueType aik = values[k] / values[dcol[krow]]; // aik = aik/akk
              //values[k] = aik;
              aik = Block2DView(values+k*N2,N,N) * inv(Block2DView(values+dcol[krow]*N2,N,N)) ;
              for (int l = kcol[krow]; l < kcol[krow + 1]; ++l)
                m_work[cols[l]] = l;
              for (int j = k + 1; j < kcol[irow] + local_row_size[irow]; ++j) // j=k+1->n
              {
                int jcol = cols[j];
                int kj = m_work[jcol];
                if (kj != -1) {
                  //values[j] -= aik * values[kj]; // aij = aij - aik*akj
                  Block2DView(values+j*N2,N,N) -= aik * Block2DView(values+kj*N2,N,N) ;
                }
              }
              for (int j = kcol[irow] + local_row_size[irow]; j < kcol[irow + 1]; ++j) // j=k+1->n
              {
                int jcol = cols[j];
                int kj = m_work[jcol];
                if ((kj != -1) && (jcol >= first_upper_ghost_index)) {
                  //values[j] -= aik * values[kj]; // aij = aij - aik*akj
                  Block2DView(values+j*N2,N,N) -= aik * Block2DView(values+kj*N2,N,N) ;
                }
              }
              for (int l = kcol[krow]; l < kcol[krow + 1]; ++l)
                m_work[cols[l]] = -1;
            }
          }
          op.sendUpperNeighbLUData(values);
        }
      }
      else {
        for (std::size_t irow = 1; irow < nrows; ++irow) // i=1->nrow
        {
          /*alien_info([&] {
            cout()<<"ROW : "<<irow<<" "<<kcol[irow]<<" "<<dcol[irow]<<" "<<kcol[irow+1]-kcol[irow];
          });*/
          for (int k = kcol[irow]; k < dcol[irow]; ++k) // k=1 ->i-1
          {
            int krow = cols[k];
            /*alien_info([&] {
              cout()<<"\tCOL : "<<k<<" "<<cols[k];
            });*/
 
            //ValueType aik = values[k] / values[dcol[krow]]; // aik = aik/akk
            //values[k] = aik;
            aik = Block2DView(values+k*N2,N,N) * inv(Block2DView(values+dcol[krow]*N2,N,N)) ;
            Block2DView(values+k*N2,N,N) = aik ;
            /*alien_info([&] {
              cout()<<"\t\tVALUES["<<k<<"]=";
              for(int i=0;i<N;++i)
                for(int j=0;j<N;++j)
                  cout()<<"\t\t\t MAT("<<i<<","<<j<<")="<<values[k*N2+i*N+j];
            });*/
 
            for (int l = kcol[krow]; l < kcol[krow + 1]; ++l)
              m_work[cols[l]] = l;
            for (int j = k + 1; j < kcol[irow + 1]; ++j) // j=k+1->n
            {
              int jcol = cols[j];
              int kj = m_work[jcol];
              if (kj != -1) {
                //values[j] -= aik * values[kj]; // aij = aij - aik*akj
                Block2DView(values+j*N2,N,N) -= aik * Block2DView(values+kj*N2,N,N) ;
                /*alien_info([&] {
                  cout()<<"\t\tVALUES["<<j<<"]=";
                  for(int ii=0;ii<N;++ii)
                    for(int jj=0;jj<N;++jj)
                      cout()<<"\t\t\t MAT("<<ii<<","<<jj<<")="<<values[j*N2+ii*N+jj];
                });*/
              }
            }
            for (int l = kcol[krow]; l < kcol[krow + 1]; ++l)
              m_work[cols[l]] = -1;
          }
        }
      }
#else
      throw Arccore::FatalErrorException(
                A_FUNCINFO, "Eigen is required for BlockILU factorization");
#endif
    }
#if defined(EIGEN3_DISABLED)
    else
    {
      using Block2DType     = Block2D<ValueType> ;
 
      int N   = m_block_size;
      int NxN = N*N;
 
      m_bjacobi = bjacobi;
      CSRModifierViewT<MatrixT> modifier(matrix);
 
      // clang-format off
      auto nrows  = modifier.nrows() ;
      auto kcol   = modifier.kcol() ;
      auto dcol   = modifier.dcol() ;
      auto cols   = modifier.cols() ;
      auto values = modifier.data() ;
 
      Block2DType aik{N};
      LU<ValueType> lu{N};
      // clang-format on
      if (m_is_parallel)
      {
        auto& local_row_size = matrix.getDistStructInfo().m_local_row_size;
        if (m_bjacobi) {
          for (std::size_t irow = 1; irow < nrows; ++irow) // i=1->nrow
          {
            for (int k = kcol[irow]; k < dcol[irow]; ++k) // k=1 ->i-1
            {
              int krow = cols[k];
              //ValueType aik = values[k] / values[dcol[krow]]; // aik = aik/akk
              //values[k] = aik;
              aik =  Block2DType{values+k*NxN,N} * lu.inv(Block2DType{values+dcol[krow]*NxN,N});
              Block2DType{values+k*NxN,N} = aik ;
              for (int l = kcol[krow]; l < kcol[krow] + local_row_size[krow]; ++l)
                m_work[cols[l]] = l;
              for (int j = k + 1; j < kcol[krow] + local_row_size[irow]; ++j) // j=k+1->n
              {
                int jcol = cols[j];
                int kj = m_work[jcol];
                if (kj != -1) {
                  //values[j] -= aik * values[kj]; // aij = aij - aik*akj
                  Block2DType{values+j*NxN,N} -= (aik * Block2DType{values+kj*NxN,N}) ;
                }
              }
              for (int l = kcol[krow]; l < kcol[krow] + local_row_size[krow]; ++l)
                m_work[cols[l]] = -1;
            }
          }
        }
        else {
          typename LUSendRecvTraits<TagType>::matrix_op_type op(matrix, m_distribution, m_work);
          op.recvLowerNeighbLUData(values);
          int first_upper_ghost_index = matrix.getDistStructInfo().m_first_upper_ghost_index;
          for (std::size_t irow = 1; irow < nrows; ++irow) // i=1->nrow
          {
            for (int k = kcol[irow]; k < dcol[irow]; ++k) // k=1 ->i-1
            {
              int krow = cols[k];
              //ValueType aik = values[k] / values[dcol[krow]]; // aik = aik/akk
              //values[k] = aik;
              aik = Block2DType{values+k*NxN,N} * lu.inv(Block2DType{values+dcol[krow]*NxN,N}) ;
              for (int l = kcol[krow]; l < kcol[krow + 1]; ++l)
                m_work[cols[l]] = l;
              for (int j = k + 1; j < kcol[irow] + local_row_size[irow]; ++j) // j=k+1->n
              {
                int jcol = cols[j];
                int kj = m_work[jcol];
                if (kj != -1) {
                  //values[j] -= aik * values[kj]; // aij = aij - aik*akj
                  Block2DType{values+j*NxN,N} -= aik * Block2DType{values+kj*NxN,N} ;
                }
              }
              for (int j = kcol[irow] + local_row_size[irow]; j < kcol[irow + 1]; ++j) // j=k+1->n
              {
                int jcol = cols[j];
                int kj = m_work[jcol];
                if ((kj != -1) && (jcol >= first_upper_ghost_index)) {
                  //values[j] -= aik * values[kj]; // aij = aij - aik*akj
                  Block2DType{values+j*NxN,N} -= aik * Block2DType{values+kj*NxN,N} ;
                }
              }
              for (int l = kcol[krow]; l < kcol[krow + 1]; ++l)
                m_work[cols[l]] = -1;
            }
          }
          op.sendUpperNeighbLUData(values);
        }
      }
      else
      {
        for (std::size_t irow = 1; irow < nrows; ++irow) // i=1->nrow
        {
          for (int k = kcol[irow]; k < dcol[irow]; ++k) // k=1 ->i-1
          {
            int krow = cols[k];
            //ValueType aik = values[k] / values[dcol[krow]]; // aik = aik/akk
            //values[k] = aik;
            aik = Block2DType{values+k*NxN,N} * lu.inv(Block2DType{values+dcol[krow]*NxN,N}) ;
            Block2DType{values+k*NxN,N} = aik ;
            for (int l = kcol[krow]; l < kcol[krow + 1]; ++l)
              m_work[cols[l]] = l;
            for (int j = k + 1; j < kcol[irow + 1]; ++j) // j=k+1->n
            {
              int jcol = cols[j];
              int kj = m_work[jcol];
              if (kj != -1) {
                //values[j] -= aik * values[kj]; // aij = aij - aik*akj
                Block2DType{values+j*NxN,N} -= aik * Block2DType{values+kj*NxN,N} ;
              }
            }
            for (int l = kcol[krow]; l < kcol[krow + 1]; ++l)
              m_work[cols[l]] = -1;
          }
        }
      }
    }
#endif
  }
 
  void blockSolveL(ValueType const* y, ValueType* x) const
  {
    if constexpr (MatrixType::on_host_only)
    {
#if defined (ALIEN_USE_EIGEN3)  && !defined(EIGEN3_DISABLED)
      using namespace Eigen;
      using Block2D     = Eigen::Matrix<ValueType,Eigen::Dynamic,Eigen::Dynamic,Eigen::RowMajor> ;
      using Block2DView = Eigen::Map<Block2D> ;
      using Block1D     = Eigen::Matrix<ValueType,Dynamic,1> ;
      using Block1DView = Eigen::Map<Block1D> ;
 
      int N = m_block_size ;
      int N2 = N*N;
 
 
      Block1D val(N);
 
      CSRConstViewT<MatrixT> view(*m_lu_matrix);
      // clang-format off
      auto nrows  = view.nrows() ;
      auto kcol   = view.kcol() ;
      auto dcol   = view.dcol() ;
      auto cols   = view.cols() ;
      auto values = view.data() ;
      // clang-format on
 
      for (std::size_t irow = 0; irow < nrows; ++irow) {
        //ValueType val = y[irow];
        val = Block1DView(const_cast<ValueType*>(y+irow*N),N) ;
        for (int k = kcol[irow]; k < dcol[irow]; ++k)
        {
          //val -= values[k] * x[cols[k]];
          val -= Block2DView(const_cast<ValueType*>(values+k*N2),N,N) * Block1DView(x+cols[k]*N,N) ;
        }
        //x[irow] = val;
        Block1DView(x+irow*N,N) = val ;
      }
#endif
    }
  }
 
  void blockSolveU(ValueType const* y, ValueType* x) const
  {
    if constexpr (MatrixType::on_host_only)
    {
#if defined (ALIEN_USE_EIGEN3)  && !defined(EIGEN3_DISABLED)
      using namespace Eigen;
      using Block2D     = Eigen::Matrix<ValueType,Eigen::Dynamic,Eigen::Dynamic,Eigen::RowMajor> ;
      using Block2DView = Eigen::Map<Block2D> ;
      using Block1D     = Eigen::Matrix<ValueType,Dynamic,1> ;
      using Block1DView = Eigen::Map<Block1D> ;
 
      int N = m_block_size;
      int N2 = N*N;
 
      Block1D val(N);
 
      CSRConstViewT<MatrixT> view(*m_lu_matrix);
      // clang-format off
      auto nrows  = view.nrows() ;
      auto kcol   = view.kcol() ;
      auto dcol   = view.dcol() ;
      auto cols   = view.cols() ;
      auto values = view.data() ;
      // clang-format on
      if (m_is_parallel) {
        auto& local_row_size = m_lu_matrix->getDistStructInfo().m_local_row_size;
        for (int irow = (int)nrows - 1; irow > -1; --irow) {
          int dk = dcol[irow];
          //ValueType val = y[irow];
          val = Block1DView(const_cast<ValueType*>(y+irow*N),N) ;
          for (int k = dk + 1; k < kcol[irow] + local_row_size[irow]; ++k) {
            //val -= values[k] * x[cols[k]];
            val -= Block2DView(const_cast<ValueType*>(values+k*N2),N,N) * Block1DView(x+cols[k]*N,N) ;
          }
          //x[irow] = val / values[dk];
          Block1DView(x+irow*N,N) = inv(Block2DView(const_cast<ValueType*>(values+dk*N2),N,N)) * val;
        }
      }
      else {
        for (int irow = (int)nrows - 1; irow > -1; --irow) {
          int dk = dcol[irow];
          //ValueType val = y[irow];
          val = Block1DView(const_cast<ValueType*>(y+irow*N),N) ;
          for (int k = dk + 1; k < kcol[irow + 1]; ++k) {
            //val -= values[k] * x[cols[k]];
            val -= Block2DView(const_cast<ValueType*>(values+k*N2),N,N) * Block1DView(x+cols[k]*N,N) ;
          }
          //x[irow] = val / values[dk];
          Block1DView(x+irow*N,N) = inv(Block2DView(const_cast<ValueType*>(values+dk*N2),N,N)) * val;
        }
      }
#endif
    }
  }
 
 
  template <typename AlgebraT>
  void solve([[maybe_unused]] AlgebraT& algebra, VectorType const& y, VectorType& x) const
  {
    if(m_block_size==1)
    {
      //
      //     L.X1 = Y
      //
      solveL(y.data(), m_x.data());

      //
      //     U.X = X1
      //
      solveU(m_x.data(), x.data());
    }
    else
    {
      //
      //     L.X1 = Y
      //
      blockSolveL(y.data(), m_x.data());

      //
      //     U.X = X1
      //
      blockSolveU(m_x.data(), x.data());
    }
  }
 
  const MatrixType& getLUMatrix() const
  {
    return *m_lu_matrix;
  }
 
 protected:
  // clang-format off
  std::unique_ptr<MatrixType>   m_lu_matrix ;
  int                           m_block_size                  = 1;
  ProfileType const*            m_profile                     = nullptr;
  mutable VectorType            m_x ;
 
  MatrixDistribution            m_distribution ;
  std::vector<int>              m_work ;
  std::size_t                   m_alloc_size                  = 0 ;
  bool                          m_is_parallel                 = false ;
  bool                          m_bjacobi                     = false ;
 
  std::vector<int>                    m_send_lu_ibuffer ;
  std::vector<std::vector<ValueType>> m_send_lu_buffer ;
  // clang-format on
};
 
template <typename AlgebraT>
class ILU0Preconditioner
{
 public:
  // clang-format off
  typedef AlgebraT                         AlgebraType ;
  typedef typename AlgebraType::Matrix     MatrixType;
  typedef typename AlgebraType::Vector     VectorType;
  typedef typename MatrixType::ProfileType ProfileType ;
  typedef typename MatrixType::ValueType   ValueType;
  // clang-format on
 
  typedef LUFactorisationAlgo<MatrixType, VectorType> AlgoType;
 
  ILU0Preconditioner(AlgebraType& algebra, MatrixType const& matrix, ITraceMng* trace_mng = nullptr)
  : m_algebra(algebra)
  , m_matrix(matrix)
  , m_trace_mng(trace_mng)
  {
  }
 
  virtual ~ILU0Preconditioner(){};
 
  void init()
  {
    m_algo.init(m_algebra, m_matrix);
  }
 
  void solve(VectorType const& y, VectorType& x) const
  {
    m_algo.solve(m_algebra, y, x);
  }
 
  void solve(AlgebraType& algebra, VectorType const& y, VectorType& x) const
  {
    m_algo.solve(algebra, y, x);
  }
 
  void update()
  {
    // update value from m_matrix
  }
 
 private:
  // clang-format off
  AlgebraType&                  m_algebra ;
  MatrixType const&             m_matrix;
  AlgoType                      m_algo ;
 
  ITraceMng*                    m_trace_mng = nullptr ;
  // clang-format on
};
 
} // namespace Alien