MatrixCSR.h

// Copyright (C) 2021-2022 Garth N. Wells and Chris N. Richardson
//
// This file is part of DOLFINx (https://www.fenicsproject.org)
//
// SPDX-License-Identifier:    LGPL-3.0-or-later
 
#pragma once
 
#include "SparsityPattern.h"
#include "Vector.h"
#include "matrix_csr_impl.h"
#include <algorithm>
#include <dolfinx/common/IndexMap.h>
#include <dolfinx/common/MPI.h>
#include <dolfinx/graph/AdjacencyList.h>
#include <mpi.h>
#include <numeric>
#include <span>
#include <utility>
#include <vector>
 
// Define requirements on sparsity pattern required for MatrixCSR constructor
// allowing alternative implementations that can provide these essentials.
template <typename T>
concept SparsityImplementation = requires(T sp, int i) {
  { sp.graph() };
  requires std::forward_iterator<typename decltype(sp.graph().first)::iterator>;
  requires std::convertible_to<std::int32_t,
                               typename decltype(sp.graph().first)::value_type>;
  requires std::forward_iterator<
      typename decltype(sp.graph().second)::iterator>;
  requires std::convertible_to<
      std::int64_t, typename decltype(sp.graph().second)::value_type>;
 
  { sp.block_size(i) } -> std::same_as<int>;
  {
    sp.index_map(i)
  } -> std::same_as<std::shared_ptr<const dolfinx::common::IndexMap>>;
};
 
namespace dolfinx::la
{
enum class BlockMode : int
{
  compact = 0, 
  expanded = 1 
};

template <typename Scalar, typename Container = std::vector<Scalar>,
          typename ColContainer = std::vector<std::int32_t>,
          typename RowPtrContainer = std::vector<std::int64_t>>
class MatrixCSR
{
  static_assert(std::is_same_v<typename Container::value_type, Scalar>);
  static_assert(std::is_integral_v<typename ColContainer::value_type>);
  static_assert(std::is_integral_v<typename RowPtrContainer::value_type>);
 
  template <typename, typename, typename, typename>
  friend class MatrixCSR;
 
public:
  using value_type = Scalar;

  using container_type = Container;

  using column_container_type = ColContainer;

  using rowptr_container_type = RowPtrContainer;

  template <int BS0 = 1, int BS1 = 1>
  auto mat_set_values()
  {
    if ((BS0 != _bs[0] and BS0 > 1 and _bs[0] > 1)
        or (BS1 != _bs[1] and BS1 > 1 and _bs[1] > 1))
    {
      throw std::runtime_error(
          "Cannot insert blocks of different size than matrix block size");
    }
 
    return [this](std::span<const std::int32_t> rows,
                  std::span<const std::int32_t> cols,
                  std::span<const value_type> data) -> int
    {
      this->set<BS0, BS1>(data, rows, cols);
      return 0;
    };
  }

  template <int BS0 = 1, int BS1 = 1>
  auto mat_add_values()
  {
    if ((BS0 != _bs[0] and BS0 > 1 and _bs[0] > 1)
        or (BS1 != _bs[1] and BS1 > 1 and _bs[1] > 1))
    {
      throw std::runtime_error(
          "Cannot insert blocks of different size than matrix block size");
    }
 
    return [this](std::span<const std::int32_t> rows,
                  std::span<const std::int32_t> cols,
                  std::span<const value_type> data) -> int
    {
      this->add<BS0, BS1>(data, rows, cols);
      return 0;
    };
  }

  template <SparsityImplementation T>
  MatrixCSR(const T& p, BlockMode mode = BlockMode::compact);

  MatrixCSR(MatrixCSR&& A) = default;

  MatrixCSR(const MatrixCSR& A) = default;

  template <typename Scalar0, typename Container0, typename ColContainer0,
            typename RowPtrContainer0>
  explicit MatrixCSR(
      const MatrixCSR<Scalar0, Container0, ColContainer0, RowPtrContainer0>& A)
      : _index_maps(A._index_maps), _block_mode(A.block_mode()),
        _bs(A.block_size()), _data(A._data.begin(), A._data.end()),
        _cols(A.cols().begin(), A.cols().end()),
        _row_ptr(A.row_ptr().begin(), A.row_ptr().end()),
        _off_diagonal_offset(A.off_diag_offset().begin(),
                             A.off_diag_offset().end()),
        _comm(A.comm()), _request(MPI_REQUEST_NULL), _unpack_pos(A._unpack_pos),
        _val_send_disp(A._val_send_disp), _val_recv_disp(A._val_recv_disp),
        _ghost_row_to_rank(A._ghost_row_to_rank), _finalized(A._finalized)
  {
  }

  [[deprecated("Use std::ranges::fill(A.values(), v) instead.")]]
  void set(value_type x)
  {
    check_not_finalized();
    std::ranges::fill(_data, x);
  }

  template <int BS0, int BS1>
  void set(std::span<const value_type> x, std::span<const std::int32_t> rows,
           std::span<const std::int32_t> cols)
  {
    check_not_finalized();
    auto set_fn = [](value_type& y, const value_type& x) { y = x; };
 
    std::int32_t num_rows
        = _index_maps[0]->size_local() + _index_maps[0]->num_ghosts();
    assert(x.size() == rows.size() * cols.size() * BS0 * BS1);
    if (_bs[0] == BS0 and _bs[1] == BS1)
    {
      impl::insert_csr<BS0, BS1>(_data, _cols, _row_ptr, x, rows, cols, set_fn,
                                 num_rows);
    }
    else if (_bs[0] == 1 and _bs[1] == 1)
    {
      // Set blocked data in a regular CSR matrix (_bs[0]=1, _bs[1]=1)
      // with correct sparsity
      impl::insert_blocked_csr<BS0, BS1>(_data, _cols, _row_ptr, x, rows, cols,
                                         set_fn, num_rows);
    }
    else
    {
      assert(BS0 == 1 and BS1 == 1);
      // Set non-blocked data in a blocked CSR matrix (BS0=1, BS1=1)
      impl::insert_nonblocked_csr(_data, _cols, _row_ptr, x, rows, cols, set_fn,
                                  num_rows, _bs[0], _bs[1]);
    }
  }

  template <int BS0 = 1, int BS1 = 1>
  void add(std::span<const value_type> x, std::span<const std::int32_t> rows,
           std::span<const std::int32_t> cols)
  {
    check_not_finalized();
    auto add_fn = [](value_type& y, const value_type& x) { y += x; };
 
    assert(x.size() == rows.size() * cols.size() * BS0 * BS1);
    if (_bs[0] == BS0 and _bs[1] == BS1)
    {
      impl::insert_csr<BS0, BS1>(_data, _cols, _row_ptr, x, rows, cols, add_fn,
                                 _row_ptr.size());
    }
    else if (_bs[0] == 1 and _bs[1] == 1)
    {
      // Add blocked data to a regular CSR matrix (_bs[0]=1, _bs[1]=1)
      impl::insert_blocked_csr<BS0, BS1>(_data, _cols, _row_ptr, x, rows, cols,
                                         add_fn, _row_ptr.size());
    }
    else
    {
      assert(BS0 == 1 and BS1 == 1);
      // Add non-blocked data to a blocked CSR matrix (BS0=1, BS1=1)
      impl::insert_nonblocked_csr(_data, _cols, _row_ptr, x, rows, cols, add_fn,
                                  _row_ptr.size(), _bs[0], _bs[1]);
    }
  }

  std::int32_t num_owned_rows() const { return _index_maps[0]->size_local(); }

  std::int32_t num_all_rows() const { return _row_ptr.size() - 1; }

  std::vector<value_type> to_dense() const
  {
    const std::size_t nrows = num_all_rows();
    const std::size_t ncols = _index_maps[1]->size_global();
    std::vector<value_type> A(nrows * ncols * _bs[0] * _bs[1], value_type(0));
    for (std::size_t r = 0; r < nrows; ++r)
    {
      for (std::int32_t j = _row_ptr[r]; j < _row_ptr[r + 1]; ++j)
      {
        for (int i0 = 0; i0 < _bs[0]; ++i0)
        {
          for (int i1 = 0; i1 < _bs[1]; ++i1)
          {
            std::array<std::int32_t, 1> local_col{_cols[j]};
            std::array<std::int64_t, 1> global_col{0};
            _index_maps[1]->local_to_global(local_col, global_col);
            A[(r * _bs[0] + i0) * ncols * _bs[1] + global_col[0] * _bs[1] + i1]
                = _data[j * _bs[0] * _bs[1] + i0 * _bs[1] + i1];
          }
        }
      }
    }
 
    return A;
  }

  void scatter_rev()
  {
    scatter_rev_begin();
    scatter_rev_end();
  }

  void scatter_rev_begin()
  {
    check_not_finalized();
    const std::int32_t local_size0 = _index_maps[0]->size_local();
    const std::int32_t num_ghosts0 = _index_maps[0]->num_ghosts();
    const int bs2 = _bs[0] * _bs[1];
 
    // For each ghost row, pack and send values to send to neighborhood
    std::vector<int> insert_pos = _val_send_disp;
    _ghost_value_data.resize(_val_send_disp.back());
    for (int i = 0; i < num_ghosts0; ++i)
    {
      int rank = _ghost_row_to_rank[i];
 
      // Get position in send buffer to place data to send to this
      // neighbour
      std::int32_t val_pos = insert_pos[rank];
      std::copy(std::next(_data.data(), _row_ptr[local_size0 + i] * bs2),
                std::next(_data.data(), _row_ptr[local_size0 + i + 1] * bs2),
                std::next(_ghost_value_data.begin(), val_pos));
      insert_pos[rank]
          += bs2 * (_row_ptr[local_size0 + i + 1] - _row_ptr[local_size0 + i]);
    }
 
    _ghost_value_data_in.resize(_val_recv_disp.back());
 
    // Compute data sizes for send and receive from displacements
    std::vector<int> val_send_count(_val_send_disp.size() - 1);
    std::adjacent_difference(std::next(_val_send_disp.begin()),
                             _val_send_disp.end(), val_send_count.begin());
 
    std::vector<int> val_recv_count(_val_recv_disp.size() - 1);
    std::adjacent_difference(std::next(_val_recv_disp.begin()),
                             _val_recv_disp.end(), val_recv_count.begin());
 
    int status = MPI_Ineighbor_alltoallv(
        _ghost_value_data.data(), val_send_count.data(), _val_send_disp.data(),
        dolfinx::MPI::mpi_t<value_type>, _ghost_value_data_in.data(),
        val_recv_count.data(), _val_recv_disp.data(),
        dolfinx::MPI::mpi_t<value_type>, _comm.comm(), &_request);
    dolfinx::MPI::check_error(_comm.comm(), status);
  }

  void scatter_rev_end()
  {
    check_not_finalized();
    int status = MPI_Wait(&_request, MPI_STATUS_IGNORE);
    dolfinx::MPI::check_error(_comm.comm(), status);
 
    _ghost_value_data.clear();
    _ghost_value_data.shrink_to_fit();
 
    // Add to local rows
    int bs2 = _bs[0] * _bs[1];
    assert(_ghost_value_data_in.size() == _unpack_pos.size() * bs2);
    for (std::size_t i = 0; i < _unpack_pos.size(); ++i)
      for (int j = 0; j < bs2; ++j)
        _data[_unpack_pos[i] * bs2 + j] += _ghost_value_data_in[i * bs2 + j];
 
    _ghost_value_data_in.clear();
    _ghost_value_data_in.shrink_to_fit();
 
    // Set ghost row data to zero
    std::int32_t local_size0 = _index_maps[0]->size_local();
    std::fill(std::next(_data.begin(), _row_ptr[local_size0] * bs2),
              _data.end(), 0);
  }

  double squared_norm() const
  {
    const std::size_t num_owned_rows = _index_maps[0]->size_local();
    const int bs2 = _bs[0] * _bs[1];
    assert(num_owned_rows < _row_ptr.size());
    double norm_sq_local = std::accumulate(
        _data.cbegin(),
        std::next(_data.cbegin(), _row_ptr[num_owned_rows] * bs2), double(0),
        [](auto norm, value_type y) { return norm + std::norm(y); });
    double norm_sq;
    MPI_Allreduce(&norm_sq_local, &norm_sq, 1, MPI_DOUBLE, MPI_SUM,
                  _comm.comm());
    return norm_sq;
  }

  void mult(Vector<value_type>& x, Vector<value_type>& y) const;

  void multT(Vector<value_type>& x, Vector<value_type>& y) const;

  MPI_Comm comm() const { return _comm.comm(); }

  std::shared_ptr<const common::IndexMap> index_map(int dim) const
  {
    return _index_maps.at(dim);
  }

  container_type& values() { return _data; }

  const container_type& values() const { return _data; }

  const rowptr_container_type& row_ptr() const { return _row_ptr; }

  const column_container_type& cols() const { return _cols; }

  const rowptr_container_type& off_diag_offset() const
  {
    return _off_diagonal_offset;
  }

  std::array<int, 2> block_size() const { return _bs; }

  BlockMode block_mode() const { return _block_mode; }

  void eliminate_zeros(value_type tol = 0)
  {
    // Remove any zero entries (blocks, where all entries in the block
    // are within tolerance of zero) in data, and update the column
    // indices and row pointers accordingly.
    const std::size_t bs2 = _bs[0] * _bs[1];
 
    // True if every entry of the block starting at block index j is
    // within tolerance of zero, i.e. the whole block can be dropped.
    auto is_zero_block = [this, bs2, tol](std::int64_t j)
    {
      return std::all_of(std::next(_data.begin(), j * bs2),
                         std::next(_data.begin(), (j + 1) * bs2),
                         [tol](value_type x)
                         { return std::abs(x) <= std::abs(tol); });
    };
 
    std::int64_t ptr_out = 0;
    std::vector<std::int64_t> new_row_ptr = {0};
    std::vector<std::int64_t> new_off_diagonal_offset;
    new_row_ptr.reserve(_row_ptr.size());
    new_off_diagonal_offset.reserve(_off_diagonal_offset.size());
    for (std::size_t i = 0; i < _row_ptr.size() - 1; ++i)
    {
      for (std::int64_t j = _row_ptr[i]; j < _off_diagonal_offset[i]; ++j)
      {
        if (!is_zero_block(j))
        {
          _cols[ptr_out] = _cols[j];
          std::copy_n(std::next(_data.begin(), j * bs2), bs2,
                      std::next(_data.begin(), ptr_out * bs2));
          ++ptr_out;
        }
      }
      new_off_diagonal_offset.push_back(ptr_out);
      for (std::int64_t j = _off_diagonal_offset[i]; j < _row_ptr[i + 1]; ++j)
      {
        if (!is_zero_block(j))
        {
          _cols[ptr_out] = _cols[j];
          std::copy_n(std::next(_data.begin(), j * bs2), bs2,
                      std::next(_data.begin(), ptr_out * bs2));
          ++ptr_out;
        }
      }
      new_row_ptr.push_back(ptr_out);
    }
    _data.resize(ptr_out * bs2);
    _cols.resize(ptr_out);
    _row_ptr = new_row_ptr;
    _off_diagonal_offset = new_off_diagonal_offset;
    _finalized = true;
  }
 
private:
  // Parallel distribution of the rows and columns
  std::array<std::shared_ptr<const common::IndexMap>, 2> _index_maps;
 
  // Block mode (compact or expanded)
  BlockMode _block_mode;
 
  // Block sizes
  std::array<int, 2> _bs;
 
  // Matrix data
  container_type _data;
  column_container_type _cols;
  rowptr_container_type _row_ptr;
 
  // Start of off-diagonal (unowned columns) on each row
  rowptr_container_type _off_diagonal_offset;
 
  // Communicator with neighborhood (ghost->owner communicator for rows)
  dolfinx::MPI::Comm _comm;
 
  // -- Precomputed data for scatter_rev/update
 
  // Request in non-blocking communication
  MPI_Request _request;
 
  // Position in _data to add received data
  std::vector<std::size_t> _unpack_pos;
 
  // Displacements for alltoall for each neighbor when sending and
  // receiving
  std::vector<int> _val_send_disp, _val_recv_disp;
 
  // Ownership of each row, by neighbor (for the neighbourhood defined
  // on _comm)
  std::vector<int> _ghost_row_to_rank;
 
  // Temporary stores for data during non-blocking communication
  container_type _ghost_value_data;
  container_type _ghost_value_data_in;
 
  // Set by eliminate_zeros(). Once true, the sparsity may have been
  // reduced and the precomputed scatter_rev communication pattern
  // (_unpack_pos, _val_send_disp, _val_recv_disp) is no longer valid,
  // so further modification of the matrix is disallowed.
  bool _finalized = false;
 
  // Throw if the matrix has been finalized by eliminate_zeros().
  void check_not_finalized() const
  {
    if (_finalized)
    {
      throw std::runtime_error(
          "MatrixCSR has been finalized by eliminate_zeros() and can no "
          "longer be modified or scattered.");
    }
  }
};
//-----------------------------------------------------------------------------

template <typename U, typename V, typename W, typename X>
template <SparsityImplementation SparsityType>
MatrixCSR<U, V, W, X>::MatrixCSR(const SparsityType& p, BlockMode mode)
    : _index_maps({p.index_map(0), p.index_map(1)}), _block_mode(mode),
      _bs({p.block_size(0), p.block_size(1)}),
      _data(p.graph().first.size() * _bs[0] * _bs[1], 0),
      _cols(p.graph().first.begin(), p.graph().first.end()),
      _row_ptr(p.graph().second.begin(), p.graph().second.end()),
      _comm(MPI_COMM_NULL)
{
  if (_block_mode == BlockMode::expanded)
  {
    // Rebuild IndexMaps
    for (int i = 0; i < 2; ++i)
    {
      auto im = _index_maps[i];
      std::int32_t size_local = im->size_local() * _bs[i];
      std::span ghost_i = im->ghosts();
      std::vector<std::int64_t> ghosts;
      const std::vector<int> ghost_owner_i(im->owners().begin(),
                                           im->owners().end());
      std::vector<int> src_rank;
      for (std::size_t j = 0; j < ghost_i.size(); ++j)
      {
        for (int k = 0; k < _bs[i]; ++k)
        {
          ghosts.push_back(ghost_i[j] * _bs[i] + k);
          src_rank.push_back(ghost_owner_i[j]);
        }
      }
 
      std::array<std::vector<int>, 2> src_dest0
          = {std::vector(_index_maps[i]->src().begin(),
                         _index_maps[i]->src().end()),
             std::vector(_index_maps[i]->dest().begin(),
                         _index_maps[i]->dest().end())};
      _index_maps[i] = std::make_shared<common::IndexMap>(
          _index_maps[i]->comm(), size_local, src_dest0, ghosts, src_rank);
    }
 
    // Convert sparsity pattern and set _bs to 1
 
    column_container_type new_cols;
    new_cols.reserve(_data.size());
    rowptr_container_type new_row_ptr{0};
    new_row_ptr.reserve(_row_ptr.size() * _bs[0]);
    std::span<const std::int32_t> num_diag_nnz = p.off_diagonal_offsets();
    for (std::size_t i = 0; i < _row_ptr.size() - 1; ++i)
    {
      // Repeat row _bs[0] times
      for (int q0 = 0; q0 < _bs[0]; ++q0)
      {
        _off_diagonal_offset.push_back(new_row_ptr.back()
                                       + num_diag_nnz[i] * _bs[1]);
        for (auto j = _row_ptr[i]; j < _row_ptr[i + 1]; ++j)
        {
          for (int q1 = 0; q1 < _bs[1]; ++q1)
            new_cols.push_back(_cols[j] * _bs[1] + q1);
        }
        new_row_ptr.push_back(new_cols.size());
      }
    }
    _cols = new_cols;
    _row_ptr = new_row_ptr;
    _bs[0] = 1;
    _bs[1] = 1;
  }
  else
  {
    // Compute off-diagonal offset for each row (compact)
    std::span<const std::int32_t> num_diag_nnz = p.off_diagonal_offsets();
    _off_diagonal_offset.reserve(num_diag_nnz.size());
    std::ranges::transform(num_diag_nnz, _row_ptr,
                           std::back_inserter(_off_diagonal_offset),
                           std::plus{});
  }
 
  // Some short-hand
  std::array local_size
      = {_index_maps[0]->size_local(), _index_maps[1]->size_local()};
  std::array local_range
      = {_index_maps[0]->local_range(), _index_maps[1]->local_range()};
  std::span ghosts1 = _index_maps[1]->ghosts();
 
  std::span ghosts0 = _index_maps[0]->ghosts();
  std::span src_ranks = _index_maps[0]->src();
  std::span dest_ranks = _index_maps[0]->dest();
 
  // Create neighbourhood communicator (owner <- ghost)
  MPI_Comm comm;
  MPI_Dist_graph_create_adjacent(_index_maps[0]->comm(), dest_ranks.size(),
                                 dest_ranks.data(), MPI_UNWEIGHTED,
                                 src_ranks.size(), src_ranks.data(),
                                 MPI_UNWEIGHTED, MPI_INFO_NULL, false, &comm);
  _comm = dolfinx::MPI::Comm(comm, false);
 
  // Build map from ghost row index position to owning (neighborhood)
  // rank
  _ghost_row_to_rank.reserve(_index_maps[0]->owners().size());
  for (int r : _index_maps[0]->owners())
  {
    auto it = std::ranges::lower_bound(src_ranks, r);
    assert(it != src_ranks.end() and *it == r);
    std::size_t pos = std::ranges::distance(src_ranks.begin(), it);
    _ghost_row_to_rank.push_back(pos);
  }
 
  // Compute size of data to send to each neighbor
  std::vector<std::int32_t> data_per_proc(src_ranks.size(), 0);
  for (std::size_t i = 0; i < _ghost_row_to_rank.size(); ++i)
  {
    assert(_ghost_row_to_rank[i] < (int)data_per_proc.size());
    std::size_t pos = local_size[0] + i;
    data_per_proc[_ghost_row_to_rank[i]] += _row_ptr[pos + 1] - _row_ptr[pos];
  }
 
  // Compute send displacements
  _val_send_disp.resize(src_ranks.size() + 1, 0);
  std::partial_sum(data_per_proc.begin(), data_per_proc.end(),
                   std::next(_val_send_disp.begin()));
 
  // For each ghost row, pack and send indices to neighborhood
  std::vector<std::int64_t> ghost_index_data(2 * _val_send_disp.back());
  {
    std::vector<int> insert_pos = _val_send_disp;
    for (std::size_t i = 0; i < _ghost_row_to_rank.size(); ++i)
    {
      int rank = _ghost_row_to_rank[i];
      std::int32_t row_id = local_size[0] + i;
      for (int j = _row_ptr[row_id]; j < _row_ptr[row_id + 1]; ++j)
      {
        // Get position in send buffer
        std::int32_t idx_pos = 2 * insert_pos[rank];
 
        // Pack send data (row, col) as global indices
        ghost_index_data[idx_pos] = ghosts0[i];
        if (std::int32_t col_local = _cols[j]; col_local < local_size[1])
          ghost_index_data[idx_pos + 1] = col_local + local_range[1][0];
        else
          ghost_index_data[idx_pos + 1] = ghosts1[col_local - local_size[1]];
 
        insert_pos[rank] += 1;
      }
    }
  }
 
  // Communicate data with neighborhood
  std::vector<std::int64_t> ghost_index_array;
  std::vector<int> recv_disp;
  {
    std::vector<int> send_sizes;
    std::ranges::transform(data_per_proc, std::back_inserter(send_sizes),
                           [](auto x) { return 2 * x; });
 
    std::vector<int> recv_sizes(dest_ranks.size());
    send_sizes.reserve(1);
    recv_sizes.reserve(1);
    MPI_Neighbor_alltoall(send_sizes.data(), 1, MPI_INT, recv_sizes.data(), 1,
                          MPI_INT, _comm.comm());
 
    // Build send/recv displacement
    std::vector<int> send_disp{0};
    std::partial_sum(send_sizes.begin(), send_sizes.end(),
                     std::back_inserter(send_disp));
    recv_disp = {0};
    std::partial_sum(recv_sizes.begin(), recv_sizes.end(),
                     std::back_inserter(recv_disp));
 
    ghost_index_array.resize(recv_disp.back());
    MPI_Neighbor_alltoallv(ghost_index_data.data(), send_sizes.data(),
                           send_disp.data(), MPI_INT64_T,
                           ghost_index_array.data(), recv_sizes.data(),
                           recv_disp.data(), MPI_INT64_T, _comm.comm());
  }
 
  // Store receive displacements for future use, when transferring
  // data values
  _val_recv_disp.resize(recv_disp.size());
  int bs2 = _bs[0] * _bs[1];
  std::ranges::transform(recv_disp, _val_recv_disp.begin(),
                         [&bs2](auto d) { return bs2 * d / 2; });
  std::ranges::transform(_val_send_disp, _val_send_disp.begin(),
                         [&bs2](auto d) { return d * bs2; });
 
  // Global-to-local map for ghost columns
  std::vector<std::pair<std::int64_t, std::int32_t>> global_to_local;
  global_to_local.reserve(ghosts1.size());
  for (std::int64_t idx : ghosts1)
    global_to_local.push_back({idx, global_to_local.size() + local_size[1]});
  std::ranges::sort(global_to_local);
 
  // Compute location in which data for each index should be stored
  // when received
  for (std::size_t i = 0; i < ghost_index_array.size(); i += 2)
  {
    // Row must be on this process
    std::int32_t local_row = ghost_index_array[i] - local_range[0][0];
    assert(local_row >= 0 and local_row < local_size[0]);
 
    // Column may be owned or unowned
    std::int32_t local_col = ghost_index_array[i + 1] - local_range[1][0];
    if (local_col < 0 or local_col >= local_size[1])
    {
      auto it = std::ranges::lower_bound(
          global_to_local, std::pair(ghost_index_array[i + 1], -1),
          [](auto a, auto b) { return a.first < b.first; });
      assert(it != global_to_local.end()
             and it->first == ghost_index_array[i + 1]);
      local_col = it->second;
    }
    auto cit0 = std::next(_cols.begin(), _row_ptr[local_row]);
    auto cit1 = std::next(_cols.begin(), _row_ptr[local_row + 1]);
 
    // Find position of column index and insert data
    auto cit = std::lower_bound(cit0, cit1, local_col);
    assert(cit != cit1);
    assert(*cit == local_col);
    std::size_t d = std::ranges::distance(_cols.begin(), cit);
    _unpack_pos.push_back(d);
  }
 
  _unpack_pos.shrink_to_fit();
}
//-----------------------------------------------------------------------------
 
// The matrix A is distributed across P  processes by blocks of rows:
//  A = |   A_0  |
//      |   A_1  |
//      |   ...  |
//      |  A_P-1 |
//
// Each submatrix A_i is owned by a single process "i" and can be further
// decomposed into diagonal (Ai[0]) and off diagonal (Ai[1]) blocks:
//  Ai = |Ai[0] Ai[1]|
//
// If A is square, the diagonal block Ai[0] is also square and contains
// only owned columns and rows. The block Ai[1] contains ghost columns
// (unowned dofs).
 
// Likewise, a local vector x can be decomposed into owned and ghost blocks:
// xi = |   x[0]  |
//      |   x[1]  |
//
// So the product y = Ax can be computed into two separate steps:
//  y[0] = |Ai[0] Ai[1]| |   x[0]  | = Ai[0] x[0] + Ai[1] x[1]
//                       |   x[1]  |
//
template <typename Scalar, typename V, typename W, typename X>
void MatrixCSR<Scalar, V, W, X>::mult(la::Vector<Scalar>& x,
                                      la::Vector<Scalar>& y) const
{
  // start communication (update ghosts)
  x.scatter_fwd_begin();
 
  std::int32_t nrowslocal = num_owned_rows();
  std::span<const std::int64_t> Arow_ptr(row_ptr().data(), nrowslocal + 1);
  std::span<const std::int32_t> Acols(cols().data(), Arow_ptr[nrowslocal]);
  std::span<const std::int64_t> Aoff_diag_offset(off_diag_offset().data(),
                                                 nrowslocal);
  std::span<const Scalar> Avalues(values().data(),
                                  Arow_ptr[nrowslocal] * _bs[0] * _bs[1]);
 
  std::span<const Scalar> _x = x.array();
  std::span<Scalar> _y = y.array();
 
  std::span<const std::int64_t> Arow_begin(Arow_ptr.data(), nrowslocal);
  std::span<const std::int64_t> Arow_end(Arow_ptr.data() + 1, nrowslocal);
 
  // First stage:  spmv - diagonal
  // yi[0] += Ai[0] * xi[0]
  if (_bs[1] == 1)
  {
    impl::spmv<Scalar>(Avalues, Arow_begin, Aoff_diag_offset, Acols, _x, _y,
                       _bs[0], std::integral_constant<int, 1>{});
  }
  else if (_bs[1] == 2)
  {
    impl::spmv<Scalar>(Avalues, Arow_begin, Aoff_diag_offset, Acols, _x, _y,
                       _bs[0], std::integral_constant<int, 2>{});
  }
  else if (_bs[1] == 3)
  {
    impl::spmv<Scalar>(Avalues, Arow_begin, Aoff_diag_offset, Acols, _x, _y,
                       _bs[0], std::integral_constant<int, 3>{});
  }
  else
  {
    impl::spmv<Scalar>(Avalues, Arow_begin, Aoff_diag_offset, Acols, _x, _y,
                       _bs[0], _bs[1]);
  }
 
  // finalize ghost update
  x.scatter_fwd_end();
 
  // Second stage:  spmv - off-diagonal
  // yi[0] += Ai[1] * xi[1]
  if (_bs[1] == 1)
  {
    impl::spmv<Scalar>(Avalues, Aoff_diag_offset, Arow_end, Acols, _x, _y,
                       _bs[0], std::integral_constant<int, 1>{});
  }
  else if (_bs[1] == 2)
  {
    impl::spmv<Scalar>(Avalues, Aoff_diag_offset, Arow_end, Acols, _x, _y,
                       _bs[0], std::integral_constant<int, 2>{});
  }
  else if (_bs[1] == 3)
  {
    impl::spmv<Scalar>(Avalues, Aoff_diag_offset, Arow_end, Acols, _x, _y,
                       _bs[0], std::integral_constant<int, 3>{});
  }
  else
  {
    impl::spmv<Scalar>(Avalues, Aoff_diag_offset, Arow_end, Acols, _x, _y,
                       _bs[0], _bs[1]);
  }
}

template <typename Scalar, typename V, typename W, typename X>
void MatrixCSR<Scalar, V, W, X>::multT(la::Vector<Scalar>& x,
                                       la::Vector<Scalar>& y) const
{
  std::int32_t nrowslocal = num_owned_rows();
  std::span<const std::int64_t> Arow_ptr(row_ptr().data(), nrowslocal + 1);
  std::span<const std::int32_t> Acols(cols().data(), Arow_ptr[nrowslocal]);
  std::span<const std::int64_t> Aoff_diag_offset(off_diag_offset().data(),
                                                 nrowslocal);
  std::span<const Scalar> Avalues(values().data(),
                                  Arow_ptr[nrowslocal] * _bs[0] * _bs[1]);
 
  std::span<const Scalar> _x = x.array();
  std::span<Scalar> _y = y.array();
 
  std::span<const std::int64_t> Arow_begin(Arow_ptr.data(), nrowslocal);
  std::span<const std::int64_t> Arow_end(Arow_ptr.data() + 1, nrowslocal);
 
  // Compute ghost region contribution and scatter back. Zero only the
  // ghost portion of y so the caller's owned values are preserved (multT
  // accumulates).
  std::int32_t ncolslocal = index_map(1)->size_local();
  std::fill(std::next(_y.begin(), ncolslocal * _bs[1]), _y.end(), Scalar(0));
  if (_bs[1] == 1)
  {
    impl::spmvT<Scalar>(Avalues, Aoff_diag_offset, Arow_end, Acols, _x, _y,
                        _bs[0], std::integral_constant<int, 1>{});
  }
  else if (_bs[1] == 2)
  {
    impl::spmvT<Scalar>(Avalues, Aoff_diag_offset, Arow_end, Acols, _x, _y,
                        _bs[0], std::integral_constant<int, 2>{});
  }
  else if (_bs[1] == 3)
  {
    impl::spmvT<Scalar>(Avalues, Aoff_diag_offset, Arow_end, Acols, _x, _y,
                        _bs[0], std::integral_constant<int, 3>{});
  }
  else
  {
    impl::spmvT<Scalar>(Avalues, Aoff_diag_offset, Arow_end, Acols, _x, _y,
                        _bs[0], _bs[1]);
  }
 
  y.scatter_rev(std::plus<Scalar>{});
 
  if (_bs[1] == 1)
  {
    impl::spmvT<Scalar>(Avalues, Arow_begin, Aoff_diag_offset, Acols, _x, _y,
                        _bs[0], std::integral_constant<int, 1>{});
  }
  else if (_bs[1] == 2)
  {
    impl::spmvT<Scalar>(Avalues, Arow_begin, Aoff_diag_offset, Acols, _x, _y,
                        _bs[0], std::integral_constant<int, 2>{});
  }
  else if (_bs[1] == 3)
  {
    impl::spmvT<Scalar>(Avalues, Arow_begin, Aoff_diag_offset, Acols, _x, _y,
                        _bs[0], std::integral_constant<int, 3>{});
  }
  else
  {
    impl::spmvT<Scalar>(Avalues, Arow_begin, Aoff_diag_offset, Acols, _x, _y,
                        _bs[0], _bs[1]);
  }
}
} // namespace dolfinx::la