interpolate.h

// Copyright (C) 2020-2024 Garth N. Wells, Igor A. Baratta, Massimiliano Leoni
// and Jørgen S.Dokken
//
// This file is part of DOLFINx (https://www.fenicsproject.org)
//
// SPDX-License-Identifier:    LGPL-3.0-or-later
 
#pragma once
 
#include "CoordinateElement.h"
#include "DofMap.h"
#include "FiniteElement.h"
#include "FunctionSpace.h"
#include <algorithm>
#include <basix/mdspan.hpp>
#include <concepts>
#include <dolfinx/common/IndexMap.h>
#include <dolfinx/common/types.h>
#include <dolfinx/geometry/utils.h>
#include <dolfinx/mesh/Mesh.h>
#include <functional>
#include <numeric>
#include <ranges>
#include <span>
#include <vector>
 
namespace dolfinx::fem
{
template <dolfinx::scalar T, std::floating_point U>
class Function;
 
template <typename T>
concept MDSpan = requires(T x, std::size_t idx) {
  x(idx, idx);
  { x.extent(0) } -> std::integral;
  { x.extent(1) } -> std::integral;
};

template <std::floating_point T>
std::vector<T> interpolation_coords(const fem::FiniteElement<T>& element,
                                    const mesh::Geometry<T>& geometry,
                                    std::span<const std::int32_t> cells)
{
  // Find CoordinateElement appropriate to element
  const std::vector<CoordinateElement<T>>& cmaps = geometry.cmaps();
  mesh::CellType cell_type = element.cell_type();
  auto it
      = std::find_if(cmaps.begin(), cmaps.end(), [&cell_type](const auto& cm)
                     { return cell_type == cm.cell_shape(); });
  if (it == cmaps.end())
    throw std::runtime_error("Cannot find CoordinateElement for FiniteElement");
  int index = std::distance(cmaps.begin(), it);
 
  // Get geometry data and the element coordinate map
  const std::size_t gdim = geometry.dim();
  auto x_dofmap = geometry.dofmap(index);
  std::span<const T> x_g = geometry.x();
 
  const CoordinateElement<T>& cmap = cmaps.at(index);
  const std::size_t num_dofs_g = cmap.dim();
 
  // Get the interpolation points on the reference cells
  const auto [X, Xshape] = element.interpolation_points();
 
  // Evaluate coordinate element basis at reference points
  std::array<std::size_t, 4> phi_shape = cmap.tabulate_shape(0, Xshape[0]);
  std::vector<T> phi_b(
      std::reduce(phi_shape.begin(), phi_shape.end(), 1, std::multiplies{}));
  md::mdspan<const T, md::extents<std::size_t, 1, md::dynamic_extent,
                                  md::dynamic_extent, 1>>
      phi_full(phi_b.data(), phi_shape);
  cmap.tabulate(0, X, Xshape, phi_b);
  auto phi = md::submdspan(phi_full, 0, md::full_extent, md::full_extent, 0);
 
  // Push reference coordinates (X) forward to the physical coordinates
  // (x) for each cell
  std::vector<T> coordinate_dofs(num_dofs_g * gdim, 0);
  std::vector<T> x(3 * (cells.size() * Xshape[0]), 0);
  for (std::size_t c = 0; c < cells.size(); ++c)
  {
    // Get geometry data for current cell
    auto x_dofs = md::submdspan(x_dofmap, cells[c], md::full_extent);
    for (std::size_t i = 0; i < x_dofs.size(); ++i)
    {
      std::copy_n(std::next(x_g.begin(), 3 * x_dofs[i]), gdim,
                  std::next(coordinate_dofs.begin(), i * gdim));
    }
 
    // Push forward coordinates (X -> x)
    for (std::size_t p = 0; p < Xshape[0]; ++p)
    {
      for (std::size_t j = 0; j < gdim; ++j)
      {
        T acc = 0;
        for (std::size_t k = 0; k < num_dofs_g; ++k)
          acc += phi(p, k) * coordinate_dofs[k * gdim + j];
        x[j * (cells.size() * Xshape[0]) + c * Xshape[0] + p] = acc;
      }
    }
  }
 
  return x;
}

template <dolfinx::scalar T, std::floating_point U>
void interpolate(Function<T, U>& u, std::span<const T> f,
                 std::array<std::size_t, 2> fshape,
                 std::span<const std::int32_t> cells);
 
namespace impl
{
template <typename T, std::size_t D>
using mdspan_t = md::mdspan<T, md::dextents<std::size_t, D>>;

template <dolfinx::scalar T>
void scatter_values(MPI_Comm comm, std::span<const std::int32_t> src_ranks,
                    std::span<const std::int32_t> dest_ranks,
                    mdspan_t<const T, 2> send_values, std::span<T> recv_values)
{
  const std::size_t block_size = send_values.extent(1);
  assert(src_ranks.size() * block_size == send_values.size());
  assert(recv_values.size() == dest_ranks.size() * block_size);
 
  // Build unique set of the sorted src_ranks
  std::vector<std::int32_t> out_ranks(src_ranks.size());
  out_ranks.assign(src_ranks.begin(), src_ranks.end());
  auto [unique_end, range_end] = std::ranges::unique(out_ranks);
  out_ranks.erase(unique_end, range_end);
  out_ranks.reserve(out_ranks.size() + 1);
 
  // Remove negative entries from dest_ranks
  std::vector<std::int32_t> in_ranks;
  in_ranks.reserve(dest_ranks.size());
  std::copy_if(dest_ranks.begin(), dest_ranks.end(),
               std::back_inserter(in_ranks),
               [](auto rank) { return rank >= 0; });
 
  // Create unique set of sorted in-ranks
  {
    std::ranges::sort(in_ranks);
    auto [unique_end, range_end] = std::ranges::unique(in_ranks);
    in_ranks.erase(unique_end, range_end);
  }
  in_ranks.reserve(in_ranks.size() + 1);
 
  // Create neighborhood communicator
  MPI_Comm reverse_comm;
  MPI_Dist_graph_create_adjacent(
      comm, in_ranks.size(), in_ranks.data(), MPI_UNWEIGHTED, out_ranks.size(),
      out_ranks.data(), MPI_UNWEIGHTED, MPI_INFO_NULL, false, &reverse_comm);
 
  std::vector<std::int32_t> comm_to_output;
  std::vector<std::int32_t> recv_sizes(in_ranks.size());
  recv_sizes.reserve(1);
  std::vector<std::int32_t> recv_offsets(in_ranks.size() + 1, 0);
  {
    // Build map from parent to neighborhood communicator ranks
    std::vector<std::pair<std::int32_t, std::int32_t>> rank_to_neighbor;
    rank_to_neighbor.reserve(in_ranks.size());
    for (std::size_t i = 0; i < in_ranks.size(); i++)
      rank_to_neighbor.push_back({in_ranks[i], i});
    std::ranges::sort(rank_to_neighbor);
 
    // Compute receive sizes
    std::ranges::for_each(
        dest_ranks,
        [&rank_to_neighbor, &recv_sizes, block_size](auto rank)
        {
          if (rank >= 0)
          {
            auto it = std::ranges::lower_bound(rank_to_neighbor, rank,
                                               std::ranges::less(),
                                               [](auto e) { return e.first; });
            assert(it != rank_to_neighbor.end() and it->first == rank);
            recv_sizes[it->second] += block_size;
          }
        });
 
    // Compute receiving offsets
    std::partial_sum(recv_sizes.begin(), recv_sizes.end(),
                     std::next(recv_offsets.begin(), 1));
 
    // Compute map from receiving values to position in recv_values
    comm_to_output.resize(recv_offsets.back() / block_size);
    std::vector<std::int32_t> recv_counter(recv_sizes.size(), 0);
    for (std::size_t i = 0; i < dest_ranks.size(); ++i)
    {
      if (const std::int32_t rank = dest_ranks[i]; rank >= 0)
      {
        auto it = std::ranges::lower_bound(rank_to_neighbor, rank,
                                           std::ranges::less(),
                                           [](auto e) { return e.first; });
        assert(it != rank_to_neighbor.end() and it->first == rank);
        int insert_pos = recv_offsets[it->second] + recv_counter[it->second];
        comm_to_output[insert_pos / block_size] = i * block_size;
        recv_counter[it->second] += block_size;
      }
    }
  }
 
  std::vector<std::int32_t> send_sizes(out_ranks.size());
  send_sizes.reserve(1);
  {
    // Compute map from parent MPI rank to neighbor rank for outgoing
    // data. `out_ranks` is sorted, so rank_to_neighbor will be sorted
    // too.
    std::vector<std::pair<std::int32_t, std::int32_t>> rank_to_neighbor;
    rank_to_neighbor.reserve(out_ranks.size());
    for (std::size_t i = 0; i < out_ranks.size(); i++)
      rank_to_neighbor.push_back({out_ranks[i], i});
 
    // Compute send sizes. As `src_ranks` is sorted, we can move 'start'
    // in search forward.
    auto start = rank_to_neighbor.begin();
    std::ranges::for_each(
        src_ranks,
        [&rank_to_neighbor, &send_sizes, block_size, &start](auto rank)
        {
          auto it = std::ranges::lower_bound(start, rank_to_neighbor.end(),
                                             rank, std::ranges::less(),
                                             [](auto e) { return e.first; });
          assert(it != rank_to_neighbor.end() and it->first == rank);
          send_sizes[it->second] += block_size;
          start = it;
        });
  }
 
  // Compute sending offsets
  std::vector<std::int32_t> send_offsets(send_sizes.size() + 1, 0);
  std::partial_sum(send_sizes.begin(), send_sizes.end(),
                   std::next(send_offsets.begin(), 1));
 
  // Send values to dest ranks
  std::vector<T> values(recv_offsets.back());
  values.reserve(1);
  MPI_Neighbor_alltoallv(send_values.data_handle(), send_sizes.data(),
                         send_offsets.data(), dolfinx::MPI::mpi_t<T>,
                         values.data(), recv_sizes.data(), recv_offsets.data(),
                         dolfinx::MPI::mpi_t<T>, reverse_comm);
  MPI_Comm_free(&reverse_comm);
 
  // Insert values received from neighborhood communicator in output
  // span
  std::ranges::fill(recv_values, T(0));
  for (std::size_t i = 0; i < comm_to_output.size(); i++)
  {
    auto vals = std::next(recv_values.begin(), comm_to_output[i]);
    auto vals_from = std::next(values.begin(), i * block_size);
    std::copy_n(vals_from, block_size, vals);
  }
};

template <MDSpan U, MDSpan V, dolfinx::scalar T>
void interpolation_apply(U&& Pi, V&& data, std::span<T> coeffs, int bs)
{
  using X = typename dolfinx::scalar_value_t<T>;
 
  // Compute coefficients = Pi * x (matrix-vector multiply)
  if (bs == 1)
  {
    assert(data.extent(0) * data.extent(1) == Pi.extent(1));
    for (std::size_t i = 0; i < Pi.extent(0); ++i)
    {
      coeffs[i] = 0.0;
      for (std::size_t k = 0; k < data.extent(1); ++k)
        for (std::size_t j = 0; j < data.extent(0); ++j)
          coeffs[i]
              += static_cast<X>(Pi(i, k * data.extent(0) + j)) * data(j, k);
    }
  }
  else
  {
    const std::size_t cols = Pi.extent(1);
    assert(data.extent(0) == Pi.extent(1));
    assert(data.extent(1) == bs);
    for (int k = 0; k < bs; ++k)
    {
      for (std::size_t i = 0; i < Pi.extent(0); ++i)
      {
        T acc = 0;
        for (std::size_t j = 0; j < cols; ++j)
          acc += static_cast<X>(Pi(i, j)) * data(j, k);
        coeffs[bs * i + k] = acc;
      }
    }
  }
}

template <dolfinx::scalar T, std::floating_point U>
void interpolate_same_map(Function<T, U>& u1, const Function<T, U>& u0,
                          std::span<const std::int32_t> cells1,
                          std::span<const std::int32_t> cells0)
{
  auto V0 = u0.function_space();
  assert(V0);
  auto V1 = u1.function_space();
  assert(V1);
  auto mesh0 = V0->mesh();
  assert(mesh0);
 
  auto mesh1 = V1->mesh();
  assert(mesh1);
 
  auto element0 = V0->element();
  assert(element0);
  auto element1 = V1->element();
  assert(element1);
 
  assert(mesh0->topology()->dim());
  const int tdim = mesh0->topology()->dim();
  auto map = mesh0->topology()->index_map(tdim);
  assert(map);
  std::span<T> u1_array = u1.x()->mutable_array();
  std::span<const T> u0_array = u0.x()->array();
 
  std::span<const std::uint32_t> cell_info0;
  std::span<const std::uint32_t> cell_info1;
  if (element1->needs_dof_transformations()
      or element0->needs_dof_transformations())
  {
    mesh0->topology_mutable()->create_entity_permutations();
    cell_info0 = std::span(mesh0->topology()->get_cell_permutation_info());
    mesh1->topology_mutable()->create_entity_permutations();
    cell_info1 = std::span(mesh1->topology()->get_cell_permutation_info());
  }
 
  // Get dofmaps
  auto dofmap1 = V1->dofmap();
  auto dofmap0 = V0->dofmap();
 
  // Get block sizes and dof transformation operators
  const int bs1 = dofmap1->bs();
  const int bs0 = dofmap0->bs();
  auto apply_dof_transformation = element0->template dof_transformation_fn<T>(
      doftransform::transpose, false);
  auto apply_inverse_dof_transform
      = element1->template dof_transformation_fn<T>(
          doftransform::inverse_transpose, false);
 
  // Create working array
  std::vector<T> local0(element0->space_dimension());
  std::vector<T> local1(element1->space_dimension());
 
  // Create interpolation operator
  const auto [i_m, im_shape]
      = element1->create_interpolation_operator(*element0);
 
  // Iterate over mesh and interpolate on each cell
  using X = typename dolfinx::scalar_value_t<T>;
  for (std::size_t c = 0; c < cells0.size(); c++)
  {
    // Pack and transform cell dofs to reference ordering
    std::span<const std::int32_t> dofs0 = dofmap0->cell_dofs(cells0[c]);
    for (std::size_t i = 0; i < dofs0.size(); ++i)
      for (int k = 0; k < bs0; ++k)
        local0[bs0 * i + k] = u0_array[bs0 * dofs0[i] + k];
 
    apply_dof_transformation(local0, cell_info0, cells0[c], 1);
 
    // FIXME: Get compile-time ranges from Basix
    // Apply interpolation operator
    std::ranges::fill(local1, 0);
    for (std::size_t i = 0; i < im_shape[0]; ++i)
      for (std::size_t j = 0; j < im_shape[1]; ++j)
        local1[i] += static_cast<X>(i_m[im_shape[1] * i + j]) * local0[j];
 
    apply_inverse_dof_transform(local1, cell_info1, cells1[c], 1);
    std::span<const std::int32_t> dofs1 = dofmap1->cell_dofs(cells1[c]);
    for (std::size_t i = 0; i < dofs1.size(); ++i)
      for (int k = 0; k < bs1; ++k)
        u1_array[bs1 * dofs1[i] + k] = local1[bs1 * i + k];
  }
}

template <dolfinx::scalar T, std::floating_point U>
void interpolate_nonmatching_maps(Function<T, U>& u1,
                                  std::span<const std::int32_t> cells1,
                                  const Function<T, U>& u0,
                                  std::span<const std::int32_t> cells0)
{
  // Get mesh
  auto V0 = u0.function_space();
  assert(V0);
  auto mesh0 = V0->mesh();
  assert(mesh0);
 
  // Mesh dims
  const int tdim = mesh0->topology()->dim();
  const int gdim = mesh0->geometry().dim();
 
  // Get elements
  auto V1 = u1.function_space();
  assert(V1);
  auto mesh1 = V1->mesh();
  assert(mesh1);
  auto element0 = V0->element();
  assert(element0);
  auto element1 = V1->element();
  assert(element1);
 
  std::span<const std::uint32_t> cell_info0;
  std::span<const std::uint32_t> cell_info1;
  if (element1->needs_dof_transformations()
      or element0->needs_dof_transformations())
  {
    mesh0->topology_mutable()->create_entity_permutations();
    cell_info0 = std::span(mesh0->topology()->get_cell_permutation_info());
    mesh1->topology_mutable()->create_entity_permutations();
    cell_info1 = std::span(mesh1->topology()->get_cell_permutation_info());
  }
 
  // Get dofmaps
  auto dofmap0 = V0->dofmap();
  auto dofmap1 = V1->dofmap();
 
  const auto [X, Xshape] = element1->interpolation_points();
 
  // Get block sizes and dof transformation operators
  const int bs0 = element0->block_size();
  const int bs1 = element1->block_size();
  auto apply_dof_transformation0 = element0->template dof_transformation_fn<U>(
      doftransform::standard, false);
  auto apply_inverse_dof_transform1
      = element1->template dof_transformation_fn<T>(
          doftransform::inverse_transpose, false);
 
  // Get sizes of elements
  const std::size_t dim0 = element0->space_dimension() / bs0;
  const std::size_t value_size_ref0 = element0->reference_value_size();
  const std::size_t value_size0 = V0->element()->reference_value_size();
 
  const CoordinateElement<U>& cmap = mesh0->geometry().cmap();
  auto x_dofmap = mesh0->geometry().dofmap();
  const std::size_t num_dofs_g = cmap.dim();
  std::span<const U> x_g = mesh0->geometry().x();
 
  // (0) is derivative index, (1) is the point index, (2) is the basis
  // function index and (3) is the basis function component.
 
  // Evaluate coordinate map basis at reference interpolation points
  const std::array<std::size_t, 4> phi_shape
      = cmap.tabulate_shape(1, Xshape[0]);
  std::vector<U> phi_b(
      std::reduce(phi_shape.begin(), phi_shape.end(), 1, std::multiplies{}));
  md::mdspan<const U, md::extents<std::size_t, md::dynamic_extent,
                                  md::dynamic_extent, md::dynamic_extent, 1>>
      phi(phi_b.data(), phi_shape);
  cmap.tabulate(1, X, Xshape, phi_b);
 
  // Evaluate v basis functions at reference interpolation points
  const auto [_basis_derivatives_reference0, b0shape]
      = element0->tabulate(X, Xshape, 0);
  md::mdspan<const U, std::extents<std::size_t, 1, md::dynamic_extent,
                                   md::dynamic_extent, md::dynamic_extent>>
      basis_derivatives_reference0(_basis_derivatives_reference0.data(),
                                   b0shape);
 
  // Create working arrays
  std::vector<T> local1(element1->space_dimension());
  std::vector<T> coeffs0(element0->space_dimension());
 
  std::vector<U> basis0_b(Xshape[0] * dim0 * value_size0);
  md::mdspan<U, std::dextents<std::size_t, 3>> basis0(
      basis0_b.data(), Xshape[0], dim0, value_size0);
 
  std::vector<U> basis_reference0_b(Xshape[0] * dim0 * value_size_ref0);
  md::mdspan<U, std::dextents<std::size_t, 3>> basis_reference0(
      basis_reference0_b.data(), Xshape[0], dim0, value_size_ref0);
 
  std::vector<T> values0_b(Xshape[0] * 1 * V1->element()->value_size());
  md::mdspan<
      T, md::extents<std::size_t, md::dynamic_extent, 1, md::dynamic_extent>>
      values0(values0_b.data(), Xshape[0], 1, V1->element()->value_size());
 
  std::vector<T> mapped_values_b(Xshape[0] * 1 * V1->element()->value_size());
  md::mdspan<
      T, md::extents<std::size_t, md::dynamic_extent, 1, md::dynamic_extent>>
      mapped_values0(mapped_values_b.data(), Xshape[0], 1,
                     V1->element()->value_size());
 
  std::vector<U> coord_dofs_b(num_dofs_g * gdim);
  md::mdspan<U, std::dextents<std::size_t, 2>> coord_dofs(coord_dofs_b.data(),
                                                          num_dofs_g, gdim);
 
  std::vector<U> J_b(Xshape[0] * gdim * tdim);
  md::mdspan<U, std::dextents<std::size_t, 3>> J(J_b.data(), Xshape[0], gdim,
                                                 tdim);
  std::vector<U> K_b(Xshape[0] * tdim * gdim);
  md::mdspan<U, std::dextents<std::size_t, 3>> K(K_b.data(), Xshape[0], tdim,
                                                 gdim);
  std::vector<U> detJ(Xshape[0]);
  std::vector<U> det_scratch(2 * gdim * tdim);
 
  // Get interpolation operator
  const auto [_Pi_1, pi_shape] = element1->interpolation_operator();
  impl::mdspan_t<const U, 2> Pi_1(_Pi_1.data(), pi_shape);
 
  using u_t = md::mdspan<U, std::dextents<std::size_t, 2>>;
  using U_t = md::mdspan<const U, std::dextents<std::size_t, 2>>;
  using J_t = md::mdspan<const U, std::dextents<std::size_t, 2>>;
  using K_t = md::mdspan<const U, std::dextents<std::size_t, 2>>;
  auto push_forward_fn0
      = element0->basix_element().template map_fn<u_t, U_t, J_t, K_t>();
 
  using v_t = md::mdspan<const T, std::dextents<std::size_t, 2>>;
  using V_t = decltype(md::submdspan(mapped_values0, 0, md::full_extent,
                                     md::full_extent));
  auto pull_back_fn1
      = element1->basix_element().template map_fn<V_t, v_t, K_t, J_t>();
 
  // Iterate over mesh and interpolate on each cell
  std::span<const T> array0 = u0.x()->array();
  std::span<T> array1 = u1.x()->mutable_array();
  for (std::size_t c = 0; c < cells0.size(); c++)
  {
    // Get cell geometry (coordinate dofs)
    auto x_dofs = md::submdspan(x_dofmap, cells0[c], md::full_extent);
    for (std::size_t i = 0; i < num_dofs_g; ++i)
    {
      const int pos = 3 * x_dofs[i];
      for (int j = 0; j < gdim; ++j)
        coord_dofs(i, j) = x_g[pos + j];
    }
 
    // Compute Jacobians and reference points for current cell
    std::ranges::fill(J_b, 0);
    for (std::size_t p = 0; p < Xshape[0]; ++p)
    {
      auto dphi
          = md::submdspan(phi, std::pair(1, tdim + 1), p, md::full_extent, 0);
      auto _J = md::submdspan(J, p, md::full_extent, md::full_extent);
      cmap.compute_jacobian(dphi, coord_dofs, _J);
      auto _K = md::submdspan(K, p, md::full_extent, md::full_extent);
      cmap.compute_jacobian_inverse(_J, _K);
      detJ[p] = cmap.compute_jacobian_determinant(_J, det_scratch);
    }
 
    // Copy evaluated basis on reference, apply DOF transformations, and
    // push forward to physical element
    for (std::size_t k0 = 0; k0 < basis_reference0.extent(0); ++k0)
      for (std::size_t k1 = 0; k1 < basis_reference0.extent(1); ++k1)
        for (std::size_t k2 = 0; k2 < basis_reference0.extent(2); ++k2)
          basis_reference0(k0, k1, k2)
              = basis_derivatives_reference0(0, k0, k1, k2);
 
    for (std::size_t p = 0; p < Xshape[0]; ++p)
    {
      apply_dof_transformation0(
          std::span(basis_reference0_b.data() + p * dim0 * value_size_ref0,
                    dim0 * value_size_ref0),
          cell_info0, cells0[c], value_size_ref0);
    }
 
    for (std::size_t i = 0; i < basis0.extent(0); ++i)
    {
      auto _u = md::submdspan(basis0, i, md::full_extent, md::full_extent);
      auto _U = md::submdspan(basis_reference0, i, md::full_extent,
                              md::full_extent);
      auto _K = md::submdspan(K, i, md::full_extent, md::full_extent);
      auto _J = md::submdspan(J, i, md::full_extent, md::full_extent);
      push_forward_fn0(_u, _U, _J, detJ[i], _K);
    }
 
    // Copy expansion coefficients for v into local array
    const int dof_bs0 = dofmap0->bs();
    std::span<const std::int32_t> dofs0 = dofmap0->cell_dofs(cells0[c]);
    for (std::size_t i = 0; i < dofs0.size(); ++i)
      for (int k = 0; k < dof_bs0; ++k)
        coeffs0[dof_bs0 * i + k] = array0[dof_bs0 * dofs0[i] + k];
 
    // Evaluate v at the interpolation points (physical space values)
    using X = typename dolfinx::scalar_value_t<T>;
    for (std::size_t p = 0; p < Xshape[0]; ++p)
    {
      for (int k = 0; k < bs0; ++k)
      {
        for (std::size_t j = 0; j < value_size0; ++j)
        {
          T acc = 0;
          for (std::size_t i = 0; i < dim0; ++i)
            acc += coeffs0[bs0 * i + k] * static_cast<X>(basis0(p, i, j));
          values0(p, 0, j * bs0 + k) = acc;
        }
      }
    }
 
    // Pull back the physical values to the u reference
    for (std::size_t i = 0; i < values0.extent(0); ++i)
    {
      auto _u = md::submdspan(values0, i, md::full_extent, md::full_extent);
      auto _U
          = md::submdspan(mapped_values0, i, md::full_extent, md::full_extent);
      auto _K = md::submdspan(K, i, md::full_extent, md::full_extent);
      auto _J = md::submdspan(J, i, md::full_extent, md::full_extent);
      pull_back_fn1(_U, _u, _K, 1.0 / detJ[i], _J);
    }
 
    auto values
        = md::submdspan(mapped_values0, md::full_extent, 0, md::full_extent);
    interpolation_apply(Pi_1, values, std::span(local1), bs1);
    apply_inverse_dof_transform1(local1, cell_info1, cells1[c], 1);
 
    // Copy local coefficients to the correct position in u dof array
    const int dof_bs1 = dofmap1->bs();
    std::span<const std::int32_t> dofs1 = dofmap1->cell_dofs(c);
    for (std::size_t i = 0; i < dofs1.size(); ++i)
      for (int k = 0; k < dof_bs1; ++k)
        array1[dof_bs1 * dofs1[i] + k] = local1[dof_bs1 * i + k];
  }
}

template <dolfinx::scalar T, std::floating_point U>
void point_evaluation(const FiniteElement<U>& element, bool symmetric,
                      const DofMap& dofmap, std::span<const std::int32_t> cells,
                      std::span<const std::uint32_t> cell_info,
                      std::span<const T> f, std::array<std::size_t, 2> fshape,
                      std::span<T> coeffs)
{
  // Point evaluation element *and* the geometric map is the identity,
  // e.g. not Piola mapped
 
  const int element_bs = element.block_size();
  const int num_scalar_dofs = element.space_dimension() / element_bs;
  const int dofmap_bs = dofmap.bs();
 
  std::vector<T> _coeffs(num_scalar_dofs);
 
  auto apply_inv_transpose_dof_transformation
      = element.template dof_transformation_fn<T>(
          doftransform::inverse_transpose, true);
 
  if (symmetric)
  {
    std::size_t matrix_size = 0;
    while (matrix_size * matrix_size < fshape[0])
      ++matrix_size;
 
    spdlog::info("Loop over cells");
    // Loop over cells
    for (std::size_t c = 0; c < cells.size(); ++c)
    {
      // The entries of a symmetric matrix are numbered (for an
      // example 4x4 element):
      //  0 * * *
      //  1 2 * *
      //  3 4 5 *
      //  6 7 8 9
      // The loop extracts these elements. In this loop, row is the
      // row of this matrix, and (k - rowstart) is the column
      std::size_t row = 0;
      std::size_t rowstart = 0;
      const std::int32_t cell = cells[c];
      std::span<const std::int32_t> dofs = dofmap.cell_dofs(cell);
      for (int k = 0; k < element_bs; ++k)
      {
        if (k - rowstart > row)
        {
          ++row;
          rowstart = k;
        }
 
        // num_scalar_dofs is the number of interpolation points per
        // cell in this case (interpolation matrix is identity)
        std::copy_n(
            std::next(f.begin(), (row * matrix_size + k - rowstart) * fshape[1]
                                     + c * num_scalar_dofs),
            num_scalar_dofs, _coeffs.begin());
        apply_inv_transpose_dof_transformation(_coeffs, cell_info, cell, 1);
        for (int i = 0; i < num_scalar_dofs; ++i)
        {
          const int dof = i * element_bs + k;
          std::div_t pos = std::div(dof, dofmap_bs);
          coeffs[dofmap_bs * dofs[pos.quot] + pos.rem] = _coeffs[i];
        }
      }
    }
  }
  else
  {
    // Loop over cells
    for (std::size_t c = 0; c < cells.size(); ++c)
    {
      const std::int32_t cell = cells[c];
      std::span<const std::int32_t> dofs = dofmap.cell_dofs(cell);
      for (int k = 0; k < element_bs; ++k)
      {
        // num_scalar_dofs is the number of interpolation points per
        // cell in this case (interpolation matrix is identity)
        std::copy_n(std::next(f.begin(), k * fshape[1] + c * num_scalar_dofs),
                    num_scalar_dofs, _coeffs.begin());
        apply_inv_transpose_dof_transformation(_coeffs, cell_info, cell, 1);
        for (int i = 0; i < num_scalar_dofs; ++i)
        {
          const int dof = i * element_bs + k;
          std::div_t pos = std::div(dof, dofmap_bs);
          coeffs[dofmap_bs * dofs[pos.quot] + pos.rem] = _coeffs[i];
        }
      }
    }
  }
}

template <dolfinx::scalar T, std::floating_point U>
void identity_mapped_evaluation(const FiniteElement<U>& element, bool symmetric,
                                const DofMap& dofmap,
                                std::span<const std::int32_t> cells,
                                std::span<const std::uint32_t> cell_info,
                                std::span<const T> f,
                                std::array<std::size_t, 2> fshape,
                                std::span<T> coeffs)
{
  // Not a point evaluation, but the geometric map is the identity,
  // e.g. not Piola mapped
 
  const int element_bs = element.block_size();
  const int num_scalar_dofs = element.space_dimension() / element_bs;
  const int dofmap_bs = dofmap.bs();
 
  std::vector<T> _coeffs(num_scalar_dofs);
 
  if (symmetric)
  {
    throw std::runtime_error("Interpolation into this element not supported.");
  }
 
  const int element_vs = element.reference_value_size();
 
  if (element_vs > 1 and element_bs > 1)
  {
    throw std::runtime_error("Interpolation into this element not supported.");
  }
 
  // Get interpolation operator
  const auto [_Pi, pi_shape] = element.interpolation_operator();
  md::mdspan<const U, std::dextents<std::size_t, 2>> Pi(_Pi.data(), pi_shape);
  const std::size_t num_interp_points = Pi.extent(1);
  assert(Pi.extent(0) == num_scalar_dofs);
 
  auto apply_inv_transpose_dof_transformation
      = element.template dof_transformation_fn<T>(
          doftransform::inverse_transpose, true);
 
  // Loop over cells
  std::vector<T> ref_data_b(num_interp_points);
  md::mdspan<T, md::extents<std::size_t, md::dynamic_extent, 1>> ref_data(
      ref_data_b.data(), num_interp_points, 1);
  for (std::size_t c = 0; c < cells.size(); ++c)
  {
    const std::int32_t cell = cells[c];
    std::span<const std::int32_t> dofs = dofmap.cell_dofs(cell);
    for (int k = 0; k < element_bs; ++k)
    {
      for (int i = 0; i < element_vs; ++i)
      {
        std::copy_n(
            std::next(f.begin(),
                      (i + k) * fshape[1] + c * num_interp_points / element_vs),
            num_interp_points / element_vs,
            std::next(ref_data_b.begin(), i * num_interp_points / element_vs));
      }
      impl::interpolation_apply(Pi, ref_data, std::span(_coeffs), 1);
      apply_inv_transpose_dof_transformation(_coeffs, cell_info, cell, 1);
      for (int i = 0; i < num_scalar_dofs; ++i)
      {
        const int dof = i * element_bs + k;
        std::div_t pos = std::div(dof, dofmap_bs);
        coeffs[dofmap_bs * dofs[pos.quot] + pos.rem] = _coeffs[i];
      }
    }
  }
}

template <dolfinx::scalar T, std::floating_point U>
void piola_mapped_evaluation(const FiniteElement<U>& element, bool symmetric,
                             const DofMap& dofmap,
                             std::span<const std::int32_t> cells,
                             std::span<const std::uint32_t> cell_info,
                             std::span<const T> f,
                             std::array<std::size_t, 2> fshape,
                             const mesh::Mesh<U>& mesh, std::span<T> coeffs)
{
 
  const int gdim = mesh.geometry().dim();
  const int tdim = mesh.topology()->dim();
 
  const int element_bs = element.block_size();
  const int num_scalar_dofs = element.space_dimension() / element_bs;
  const int value_size = element.reference_value_size();
  const int dofmap_bs = dofmap.bs();
 
  std::vector<T> _coeffs(num_scalar_dofs);
  md::mdspan<const T, md::dextents<std::size_t, 2>> _f(f.data(), fshape);
 
  if (symmetric)
  {
    throw std::runtime_error("Interpolation into this element not supported.");
  }
  // Get the interpolation points on the reference cells
  const auto [X, Xshape] = element.interpolation_points();
  if (X.empty())
  {
    throw std::runtime_error(
        "Interpolation into this space is not yet supported.");
  }
 
  if (_f.extent(1) != cells.size() * Xshape[0])
    throw std::runtime_error("Interpolation data has the wrong shape.");
 
  // Get coordinate map
  const CoordinateElement<U>& cmap = mesh.geometry().cmap();
 
  // Get geometry data
  auto x_dofmap = mesh.geometry().dofmap();
  const int num_dofs_g = cmap.dim();
  std::span<const U> x_g = mesh.geometry().x();
 
  // Create data structures for Jacobian info
  std::vector<U> J_b(Xshape[0] * gdim * tdim);
  md::mdspan<U, std::dextents<std::size_t, 3>> J(J_b.data(), Xshape[0], gdim,
                                                 tdim);
  std::vector<U> K_b(Xshape[0] * tdim * gdim);
  md::mdspan<U, std::dextents<std::size_t, 3>> K(K_b.data(), Xshape[0], tdim,
                                                 gdim);
  std::vector<U> detJ(Xshape[0]);
  std::vector<U> det_scratch(2 * gdim * tdim);
 
  std::vector<U> coord_dofs_b(num_dofs_g * gdim);
  md::mdspan<U, std::dextents<std::size_t, 2>> coord_dofs(coord_dofs_b.data(),
                                                          num_dofs_g, gdim);
  const std::size_t value_size_ref = element.reference_value_size();
  std::vector<T> ref_data_b(Xshape[0] * 1 * value_size_ref);
  md::mdspan<
      T, md::extents<std::size_t, md::dynamic_extent, 1, md::dynamic_extent>>
      ref_data(ref_data_b.data(), Xshape[0], 1, value_size_ref);
 
  std::vector<T> _vals_b(Xshape[0] * 1 * value_size);
  md::mdspan<
      T, md::extents<std::size_t, md::dynamic_extent, 1, md::dynamic_extent>>
      _vals(_vals_b.data(), Xshape[0], 1, value_size);
 
  // Tabulate 1st derivative of shape functions at interpolation
  // coords
  std::array<std::size_t, 4> phi_shape = cmap.tabulate_shape(1, Xshape[0]);
  std::vector<U> phi_b(
      std::reduce(phi_shape.begin(), phi_shape.end(), 1, std::multiplies{}));
  md::mdspan<const U, md::extents<std::size_t, md::dynamic_extent,
                                  md::dynamic_extent, md::dynamic_extent, 1>>
      phi(phi_b.data(), phi_shape);
  cmap.tabulate(1, X, Xshape, phi_b);
  auto dphi = md::submdspan(phi, std::pair(1, tdim + 1), md::full_extent,
                            md::full_extent, 0);
 
  const std::function<void(std::span<T>, std::span<const std::uint32_t>,
                           std::int32_t, int)>
      apply_inverse_transpose_dof_transformation
      = element.template dof_transformation_fn<T>(
          doftransform::inverse_transpose);
 
  // Get interpolation operator
  const auto [_Pi, pi_shape] = element.interpolation_operator();
  md::mdspan<const U, std::dextents<std::size_t, 2>> Pi(_Pi.data(), pi_shape);
 
  using u_t = md::mdspan<const T, md::dextents<std::size_t, 2>>;
  using U_t
      = decltype(md::submdspan(ref_data, 0, md::full_extent, md::full_extent));
  using J_t = md::mdspan<const U, md::dextents<std::size_t, 2>>;
  using K_t = md::mdspan<const U, md::dextents<std::size_t, 2>>;
  auto pull_back_fn
      = element.basix_element().template map_fn<U_t, u_t, J_t, K_t>();
 
  for (std::size_t c = 0; c < cells.size(); ++c)
  {
    const std::int32_t cell = cells[c];
    auto x_dofs = md::submdspan(x_dofmap, cell, md::full_extent);
    for (int i = 0; i < num_dofs_g; ++i)
    {
      const int pos = 3 * x_dofs[i];
      for (int j = 0; j < gdim; ++j)
        coord_dofs(i, j) = x_g[pos + j];
    }
 
    // Compute J, detJ and K
    std::ranges::fill(J_b, 0);
    for (std::size_t p = 0; p < Xshape[0]; ++p)
    {
      auto _dphi = md::submdspan(dphi, md::full_extent, p, md::full_extent);
      auto _J = md::submdspan(J, p, md::full_extent, md::full_extent);
      cmap.compute_jacobian(_dphi, coord_dofs, _J);
      auto _K = md::submdspan(K, p, md::full_extent, md::full_extent);
      cmap.compute_jacobian_inverse(_J, _K);
      detJ[p] = cmap.compute_jacobian_determinant(_J, det_scratch);
    }
 
    std::span<const std::int32_t> dofs = dofmap.cell_dofs(cell);
    for (int k = 0; k < element_bs; ++k)
    {
      // Extract computed expression values for element block k
      for (int m = 0; m < value_size; ++m)
      {
        for (std::size_t k0 = 0; k0 < Xshape[0]; ++k0)
        {
          _vals(k0, 0, m)
              = f[fshape[1] * (k * value_size + m) + c * Xshape[0] + k0];
        }
      }
 
      // Get element degrees of freedom for block
      for (std::size_t i = 0; i < Xshape[0]; ++i)
      {
        auto _u = md::submdspan(_vals, i, md::full_extent, md::full_extent);
        auto _U = md::submdspan(ref_data, i, md::full_extent, md::full_extent);
        auto _K = md::submdspan(K, i, md::full_extent, md::full_extent);
        auto _J = md::submdspan(J, i, md::full_extent, md::full_extent);
        pull_back_fn(_U, _u, _K, 1.0 / detJ[i], _J);
      }
 
      auto ref = md::submdspan(ref_data, md::full_extent, 0, md::full_extent);
      impl::interpolation_apply(Pi, ref, std::span(_coeffs), element_bs);
      apply_inverse_transpose_dof_transformation(_coeffs, cell_info, cell, 1);
 
      // Copy interpolation dofs into coefficient vector
      assert(_coeffs.size() == num_scalar_dofs);
      for (int i = 0; i < num_scalar_dofs; ++i)
      {
        const int dof = i * element_bs + k;
        std::div_t pos = std::div(dof, dofmap_bs);
        coeffs[dofmap_bs * dofs[pos.quot] + pos.rem] = _coeffs[i];
      }
    }
  }
}
 
//----------------------------------------------------------------------------
} // namespace impl
 
template <dolfinx::scalar T, std::floating_point U>
void interpolate(Function<T, U>& u, std::span<const T> f,
                 std::array<std::size_t, 2> fshape,
                 std::span<const std::int32_t> cells)
{
  // TODO: Index for mixed-topology, zero for now
  const int index = 0;
  auto element = u.function_space()->elements(index);
  assert(element);
  const int element_bs = element->block_size();
  if (int num_sub = element->num_sub_elements();
      num_sub > 0 and num_sub != element_bs)
  {
    throw std::runtime_error("Cannot directly interpolate a mixed space. "
                             "Interpolate into subspaces.");
  }
 
  // Get mesh
  assert(u.function_space());
  auto mesh = u.function_space()->mesh();
  assert(mesh);
 
  if (fshape[0]
          != (std::size_t)u.function_space()->elements(index)->value_size()
      or f.size() != fshape[0] * fshape[1])
    throw std::runtime_error("Interpolation data has the wrong shape/size.");
 
  spdlog::debug("Check for dof transformation");
  std::span<const std::uint32_t> cell_info;
  if (element->needs_dof_transformations())
  {
    mesh->topology_mutable()->create_entity_permutations();
    cell_info = std::span(mesh->topology()->get_cell_permutation_info());
  }
 
  spdlog::debug("Interpolate: get dofmap");
  // Get dofmap
  const auto dofmap = u.function_space()->dofmaps(index);
  assert(dofmap);
 
  // Result will be stored to coeffs
  std::span<T> coeffs = u.x()->mutable_array();
 
  const bool symmetric = u.function_space()->symmetric();
  if (element->map_ident() && element->interpolation_ident())
  {
    spdlog::debug("Interpolate: point evaluation");
    // This assumes that any element with an identity interpolation matrix
    // is a point evaluation
    impl::point_evaluation(*element, symmetric, *dofmap, cells, cell_info, f,
                           fshape, coeffs);
  }
  else if (element->map_ident())
  {
    spdlog::debug("Interpolate: identity-mapped evaluation");
    impl::identity_mapped_evaluation(*element, symmetric, *dofmap, cells,
                                     cell_info, f, fshape, coeffs);
  }
  else
  {
    spdlog::debug("Interpolate: Piola-mapped evaluation");
    impl::piola_mapped_evaluation(*element, symmetric, *dofmap, cells,
                                  cell_info, f, fshape, *mesh, coeffs);
  }
}

template <std::floating_point T>
geometry::PointOwnershipData<T> create_interpolation_data(
    const mesh::Geometry<T>& geometry0, const FiniteElement<T>& element0,
    const mesh::Mesh<T>& mesh1, std::span<const std::int32_t> cells, T padding)
{
  // Collect all the points at which values are needed to define the
  // interpolating function
  std::vector<T> coords = interpolation_coords(element0, geometry0, cells);
 
  // Transpose interpolation coords
  std::vector<T> x(coords.size());
  std::size_t num_points = coords.size() / 3;
  for (std::size_t i = 0; i < num_points; ++i)
    for (std::size_t j = 0; j < 3; ++j)
      x[3 * i + j] = coords[i + j * num_points];
 
  // Determine ownership of each point
  return geometry::determine_point_ownership<T>(mesh1, x, padding);
}

template <dolfinx::scalar T, std::floating_point U>
void interpolate(Function<T, U>& u, const Function<T, U>& v,
                 std::span<const std::int32_t> cells,
                 const geometry::PointOwnershipData<U>& interpolation_data)
{
  auto mesh = u.function_space()->mesh();
  assert(mesh);
  MPI_Comm comm = mesh->comm();
  {
    auto mesh_v = v.function_space()->mesh();
    assert(mesh_v);
    int result;
    MPI_Comm_compare(comm, mesh_v->comm(), &result);
    if (result == MPI_UNEQUAL)
    {
      throw std::runtime_error("Interpolation on different meshes is only "
                               "supported on the same communicator.");
    }
  }
 
  assert(mesh->topology());
  auto cell_map = mesh->topology()->index_map(mesh->topology()->dim());
  assert(cell_map);
  auto element_u = u.function_space()->element();
  assert(element_u);
  const std::size_t value_size = u.function_space()->element()->value_size();
 
  const std::vector<int>& dest_ranks = interpolation_data.src_owner;
  const std::vector<int>& src_ranks = interpolation_data.dest_owners;
  const std::vector<U>& recv_points = interpolation_data.dest_points;
  const std::vector<std::int32_t>& evaluation_cells
      = interpolation_data.dest_cells;
 
  // Evaluate the interpolating function where possible
  std::vector<T> send_values(recv_points.size() / 3 * value_size);
  v.eval(recv_points, {recv_points.size() / 3, (std::size_t)3},
         evaluation_cells, send_values, {recv_points.size() / 3, value_size});
 
  // Send values back to owning process
  std::vector<T> values_b(dest_ranks.size() * value_size);
  md::mdspan<const T, md::dextents<std::size_t, 2>> _send_values(
      send_values.data(), src_ranks.size(), value_size);
  impl::scatter_values(comm, src_ranks, dest_ranks, _send_values,
                       std::span(values_b));
 
  // Transpose received data
  md::mdspan<const T, md::dextents<std::size_t, 2>> values(
      values_b.data(), dest_ranks.size(), value_size);
  std::vector<T> valuesT_b(value_size * dest_ranks.size());
  md::mdspan<T, md::dextents<std::size_t, 2>> valuesT(
      valuesT_b.data(), value_size, dest_ranks.size());
  for (std::size_t i = 0; i < values.extent(0); ++i)
    for (std::size_t j = 0; j < values.extent(1); ++j)
      valuesT(j, i) = values(i, j);
 
  // Call local interpolation operator
  fem::interpolate<T>(u, valuesT_b, {valuesT.extent(0), valuesT.extent(1)},
                      cells);
}

template <dolfinx::scalar T, std::floating_point U>
void interpolate(Function<T, U>& u1, std::span<const std::int32_t> cells1,
                 const Function<T, U>& u0, std::span<const std::int32_t> cells0)
{
  if (cells0.size() != cells1.size())
    throw std::runtime_error("Length of cell lists do not match.");
 
  assert(u1.function_space());
  assert(u0.function_space());
  auto mesh = u1.function_space()->mesh();
  assert(mesh);
  assert(cells0.size() == cells1.size());
 
  auto cell_map0 = mesh->topology()->index_map(mesh->topology()->dim());
  assert(cell_map0);
  std::size_t num_cells0 = cell_map0->size_local() + cell_map0->num_ghosts();
  if (u1.function_space() == u0.function_space()
      and cells1.size() == num_cells0)
  {
    // Same function spaces and on whole mesh
    std::span<T> u1_array = u1.x()->mutable_array();
    std::span<const T> u0_array = u0.x()->array();
    std::ranges::copy(u0_array, u1_array.begin());
  }
  else
  {
    // Get elements and check value shape
    auto fs0 = u0.function_space();
    auto element0 = fs0->element();
    assert(element0);
    auto fs1 = u1.function_space();
    auto element1 = fs1->element();
    assert(element1);
    if (!std::ranges::equal(fs0->element()->value_shape(),
                            fs1->element()->value_shape()))
    {
      throw std::runtime_error(
          "Interpolation: elements have different value dimensions");
    }
 
    if (element1 == element0 or *element1 == *element0)
    {
      // Same element, different dofmaps (or just a subset of cells)
      const int tdim = mesh->topology()->dim();
      auto cell_map1 = mesh->topology()->index_map(tdim);
      assert(cell_map1);
      assert(element1->block_size() == element0->block_size());
 
      // Get dofmaps
      std::shared_ptr<const DofMap> dofmap0 = u0.function_space()->dofmap();
      assert(dofmap0);
      std::shared_ptr<const DofMap> dofmap1 = u1.function_space()->dofmap();
      assert(dofmap1);
 
      std::span<T> u1_array = u1.x()->mutable_array();
      std::span<const T> u0_array = u0.x()->array();
 
      // Iterate over mesh and interpolate on each cell
      const int bs0 = dofmap0->bs();
      const int bs1 = dofmap1->bs();
      for (std::size_t c = 0; c < cells1.size(); ++c)
      {
        std::span<const std::int32_t> dofs0 = dofmap0->cell_dofs(cells0[c]);
        std::span<const std::int32_t> dofs1 = dofmap1->cell_dofs(cells1[c]);
        assert(bs0 * dofs0.size() == bs1 * dofs1.size());
        for (std::size_t i = 0; i < dofs0.size(); ++i)
        {
          for (int k = 0; k < bs0; ++k)
          {
            int index = bs0 * i + k;
            std::div_t dv1 = std::div(index, bs1);
            u1_array[bs1 * dofs1[dv1.quot] + dv1.rem]
                = u0_array[bs0 * dofs0[i] + k];
          }
        }
      }
    }
    else if (element1->map_type() == element0->map_type())
    {
      // Different elements, same basis function map type
      impl::interpolate_same_map(u1, u0, cells1, cells0);
    }
    else
    {
      //  Different elements with different maps for basis functions
      impl::interpolate_nonmatching_maps(u1, cells1, u0, cells0);
    }
  }
}
} // namespace dolfinx::fem