tensor_product_matrix.h

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// ---------------------------------------------------------------------
//
// Copyright (C) 2017 - 2020 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE.md at
// the top level directory of deal.II.
//
// ---------------------------------------------------------------------

#ifndef dealii_tensor_product_matrix_h
#define dealii_tensor_product_matrix_h


#include <deal.II/base/config.h>

#include <deal.II/base/array_view.h>
#include <deal.II/base/thread_management.h>

#include <deal.II/lac/lapack_full_matrix.h>

#include <deal.II/matrix_free/tensor_product_kernels.h>

DEAL_II_NAMESPACE_OPEN

// Forward declarations
#ifndef DOXYGEN
template <typename>
class Vector;
template <typename>
class FullMatrix;
#endif

template <int dim, typename Number, int n_rows_1d = -1>
class TensorProductMatrixSymmetricSumBase
{
public:
  using value_type = Number;

  static constexpr int n_rows_1d_static = n_rows_1d;

  unsigned int
  m() const;

  unsigned int
  n() const;

  void
  vmult(const ArrayView<Number> &dst, const ArrayView<const Number> &src) const;

  void
  apply_inverse(const ArrayView<Number> &      dst,
                const ArrayView<const Number> &src) const;

protected:
  TensorProductMatrixSymmetricSumBase() = default;

  std::array<Table<2, Number>, dim> mass_matrix;

  std::array<Table<2, Number>, dim> derivative_matrix;

  std::array<AlignedVector<Number>, dim> eigenvalues;

  std::array<Table<2, Number>, dim> eigenvectors;

private:
  mutable AlignedVector<Number> tmp_array;

  mutable Threads::Mutex mutex;
};



template <int dim, typename Number, int n_rows_1d = -1>
class TensorProductMatrixSymmetricSum
  : public TensorProductMatrixSymmetricSumBase<dim, Number, n_rows_1d>
{
public:
  TensorProductMatrixSymmetricSum() = default;

  TensorProductMatrixSymmetricSum(
    const std::array<Table<2, Number>, dim> &mass_matrix,
    const std::array<Table<2, Number>, dim> &derivative_matrix);

  TensorProductMatrixSymmetricSum(
    const std::array<FullMatrix<Number>, dim> &mass_matrix,
    const std::array<FullMatrix<Number>, dim> &derivative_matrix);

  TensorProductMatrixSymmetricSum(const Table<2, Number> &mass_matrix,
                                  const Table<2, Number> &derivative_matrix);

  void
  reinit(const std::array<Table<2, Number>, dim> &mass_matrix,
         const std::array<Table<2, Number>, dim> &derivative_matrix);

  void
  reinit(const std::array<FullMatrix<Number>, dim> &mass_matrix,
         const std::array<FullMatrix<Number>, dim> &derivative_matrix);

  void
  reinit(const Table<2, Number> &mass_matrix,
         const Table<2, Number> &derivative_matrix);

private:
  template <typename MatrixArray>
  void
  reinit_impl(MatrixArray &&mass_matrix, MatrixArray &&derivative_matrix);
};



template <int dim, typename Number, int n_rows_1d>
class TensorProductMatrixSymmetricSum<dim, VectorizedArray<Number>, n_rows_1d>
  : public TensorProductMatrixSymmetricSumBase<dim,
                                               VectorizedArray<Number>,
                                               n_rows_1d>
{
public:
  TensorProductMatrixSymmetricSum() = default;

  TensorProductMatrixSymmetricSum(
    const std::array<Table<2, VectorizedArray<Number>>, dim> &mass_matrix,
    const std::array<Table<2, VectorizedArray<Number>>, dim>
      &derivative_matrix);

  TensorProductMatrixSymmetricSum(
    const Table<2, VectorizedArray<Number>> &mass_matrix,
    const Table<2, VectorizedArray<Number>> &derivative_matrix);

  void
  reinit(const std::array<Table<2, VectorizedArray<Number>>, dim> &mass_matrix,
         const std::array<Table<2, VectorizedArray<Number>>, dim>
           &derivative_matrix);

  void
  reinit(const Table<2, VectorizedArray<Number>> &mass_matrix,
         const Table<2, VectorizedArray<Number>> &derivative_matrix);

private:
  template <typename MatrixArray>
  void
  reinit_impl(MatrixArray &&mass_matrix, MatrixArray &&derivative_matrix);
};


/*----------------------- Inline functions ----------------------------------*/

#ifndef DOXYGEN

namespace internal
{
  namespace TensorProductMatrix
  {
    template <typename Number>
    void
    spectral_assembly(const Number *     mass_matrix,
                      const Number *     derivative_matrix,
                      const unsigned int n_rows,
                      const unsigned int n_cols,
                      Number *           eigenvalues,
                      Number *           eigenvectors)
    {
      Assert(n_rows == n_cols, ExcNotImplemented());

      auto &&transpose_fill_nm = [](Number *           out,
                                    const Number *     in,
                                    const unsigned int n,
                                    const unsigned int m) {
        for (unsigned int mm = 0; mm < m; ++mm)
          for (unsigned int nn = 0; nn < n; ++nn)
            out[mm + nn * m] = *(in++);
      };

      std::vector<::Vector<Number>> eigenvecs(n_rows);
      LAPACKFullMatrix<Number>            mass_copy(n_rows, n_cols);
      LAPACKFullMatrix<Number>            deriv_copy(n_rows, n_cols);

      transpose_fill_nm(&(mass_copy(0, 0)), mass_matrix, n_rows, n_cols);
      transpose_fill_nm(&(deriv_copy(0, 0)), derivative_matrix, n_rows, n_cols);

      deriv_copy.compute_generalized_eigenvalues_symmetric(mass_copy,
                                                           eigenvecs);
      AssertDimension(eigenvecs.size(), n_rows);
      for (unsigned int i = 0; i < n_rows; ++i)
        for (unsigned int j = 0; j < n_cols; ++j, ++eigenvectors)
          *eigenvectors = eigenvecs[j][i];

      for (unsigned int i = 0; i < n_rows; ++i, ++eigenvalues)
        *eigenvalues = deriv_copy.eigenvalue(i).real();
    }
  } // namespace TensorProductMatrix
} // namespace internal


template <int dim, typename Number, int n_rows_1d>
inline unsigned int
TensorProductMatrixSymmetricSumBase<dim, Number, n_rows_1d>::m() const
{
  unsigned int m = mass_matrix[0].n_rows();
  for (unsigned int d = 1; d < dim; ++d)
    m *= mass_matrix[d].n_rows();
  return m;
}



template <int dim, typename Number, int n_rows_1d>
inline unsigned int
TensorProductMatrixSymmetricSumBase<dim, Number, n_rows_1d>::n() const
{
  unsigned int n = mass_matrix[0].n_cols();
  for (unsigned int d = 1; d < dim; ++d)
    n *= mass_matrix[d].n_cols();
  return n;
}



template <int dim, typename Number, int n_rows_1d>
inline void
TensorProductMatrixSymmetricSumBase<dim, Number, n_rows_1d>::vmult(
  const ArrayView<Number> &      dst_view,
  const ArrayView<const Number> &src_view) const
{
  AssertDimension(dst_view.size(), this->m());
  AssertDimension(src_view.size(), this->n());
  std::lock_guard<std::mutex> lock(this->mutex);
  const unsigned int          n = Utilities::fixed_power<dim>(
    n_rows_1d > 0 ? n_rows_1d : eigenvalues[0].size());
  tmp_array.resize_fast(n * 2);
  constexpr int kernel_size = n_rows_1d > 0 ? n_rows_1d : 0;
  internal::EvaluatorTensorProduct<internal::evaluate_general,
                                   dim,
                                   kernel_size,
                                   kernel_size,
                                   Number>
                eval(AlignedVector<Number>{},
         AlignedVector<Number>{},
         AlignedVector<Number>{},
         mass_matrix[0].n_rows(),
         mass_matrix[0].n_rows());
  Number *      t   = tmp_array.begin();
  const Number *src = src_view.begin();
  Number *      dst = dst_view.data();

  if (dim == 1)
    {
      const Number *A = &derivative_matrix[0](0, 0);
      eval.template apply<0, false, false>(A, src, dst);
    }

  else if (dim == 2)
    {
      const Number *A0 = &derivative_matrix[0](0, 0);
      const Number *M0 = &mass_matrix[0](0, 0);
      const Number *A1 = &derivative_matrix[1](0, 0);
      const Number *M1 = &mass_matrix[1](0, 0);
      eval.template apply<0, false, false>(M0, src, t);
      eval.template apply<1, false, false>(A1, t, dst);
      eval.template apply<0, false, false>(A0, src, t);
      eval.template apply<1, false, true>(M1, t, dst);
    }

  else if (dim == 3)
    {
      const Number *A0 = &derivative_matrix[0](0, 0);
      const Number *M0 = &mass_matrix[0](0, 0);
      const Number *A1 = &derivative_matrix[1](0, 0);
      const Number *M1 = &mass_matrix[1](0, 0);
      const Number *A2 = &derivative_matrix[2](0, 0);
      const Number *M2 = &mass_matrix[2](0, 0);
      eval.template apply<0, false, false>(M0, src, t + n);
      eval.template apply<1, false, false>(M1, t + n, t);
      eval.template apply<2, false, false>(A2, t, dst);
      eval.template apply<1, false, false>(A1, t + n, t);
      eval.template apply<0, false, false>(A0, src, t + n);
      eval.template apply<1, false, true>(M1, t + n, t);
      eval.template apply<2, false, true>(M2, t, dst);
    }

  else
    AssertThrow(false, ExcNotImplemented());
}



template <int dim, typename Number, int n_rows_1d>
inline void
TensorProductMatrixSymmetricSumBase<dim, Number, n_rows_1d>::apply_inverse(
  const ArrayView<Number> &      dst_view,
  const ArrayView<const Number> &src_view) const
{
  AssertDimension(dst_view.size(), this->n());
  AssertDimension(src_view.size(), this->m());
  std::lock_guard<std::mutex> lock(this->mutex);
  const unsigned int n = n_rows_1d > 0 ? n_rows_1d : eigenvalues[0].size();
  tmp_array.resize_fast(Utilities::fixed_power<dim>(n));
  constexpr int kernel_size = n_rows_1d > 0 ? n_rows_1d : 0;
  internal::EvaluatorTensorProduct<internal::evaluate_general,
                                   dim,
                                   kernel_size,
                                   kernel_size,
                                   Number>
                eval(AlignedVector<Number>(),
         AlignedVector<Number>(),
         AlignedVector<Number>(),
         mass_matrix[0].n_rows(),
         mass_matrix[0].n_rows());
  Number *      t   = tmp_array.begin();
  const Number *src = src_view.data();
  Number *      dst = dst_view.data();

  // NOTE: dof_to_quad has to be interpreted as 'dof to eigenvalue index'
  //       --> apply<.,true,.> (S,src,dst) calculates dst = S^T * src,
  //       --> apply<.,false,.> (S,src,dst) calculates dst = S * src,
  //       while the eigenvectors are stored column-wise in S, i.e.
  //       rows correspond to dofs whereas columns to eigenvalue indices!
  if (dim == 1)
    {
      const Number *S = &eigenvectors[0](0, 0);
      eval.template apply<0, true, false>(S, src, t);
      for (unsigned int i = 0; i < n; ++i)
        t[i] /= eigenvalues[0][i];
      eval.template apply<0, false, false>(S, t, dst);
    }

  else if (dim == 2)
    {
      const Number *S0 = &(eigenvectors[0](0, 0));
      const Number *S1 = &(eigenvectors[1](0, 0));
      eval.template apply<0, true, false>(S0, src, t);
      eval.template apply<1, true, false>(S1, t, dst);
      for (unsigned int i1 = 0, c = 0; i1 < n; ++i1)
        for (unsigned int i0 = 0; i0 < n; ++i0, ++c)
          dst[c] /= (eigenvalues[1][i1] + eigenvalues[0][i0]);
      eval.template apply<0, false, false>(S0, dst, t);
      eval.template apply<1, false, false>(S1, t, dst);
    }

  else if (dim == 3)
    {
      const Number *S0 = &eigenvectors[0](0, 0);
      const Number *S1 = &eigenvectors[1](0, 0);
      const Number *S2 = &eigenvectors[2](0, 0);
      eval.template apply<0, true, false>(S0, src, t);
      eval.template apply<1, true, false>(S1, t, dst);
      eval.template apply<2, true, false>(S2, dst, t);
      for (unsigned int i2 = 0, c = 0; i2 < n; ++i2)
        for (unsigned int i1 = 0; i1 < n; ++i1)
          for (unsigned int i0 = 0; i0 < n; ++i0, ++c)
            t[c] /=
              (eigenvalues[2][i2] + eigenvalues[1][i1] + eigenvalues[0][i0]);
      eval.template apply<0, false, false>(S0, t, dst);
      eval.template apply<1, false, false>(S1, dst, t);
      eval.template apply<2, false, false>(S2, t, dst);
    }

  else
    Assert(false, ExcNotImplemented());
}


//---------------------- TensorProductMatrixSymmetricSum ----------------------

template <int dim, typename Number, int n_rows_1d>
inline TensorProductMatrixSymmetricSum<dim, Number, n_rows_1d>::
  TensorProductMatrixSymmetricSum(
    const std::array<Table<2, Number>, dim> &mass_matrix,
    const std::array<Table<2, Number>, dim> &derivative_matrix)
{
  reinit(mass_matrix, derivative_matrix);
}



template <int dim, typename Number, int n_rows_1d>
inline TensorProductMatrixSymmetricSum<dim, Number, n_rows_1d>::
  TensorProductMatrixSymmetricSum(
    const std::array<FullMatrix<Number>, dim> &mass_matrix,
    const std::array<FullMatrix<Number>, dim> &derivative_matrix)
{
  reinit(mass_matrix, derivative_matrix);
}



template <int dim, typename Number, int n_rows_1d>
inline TensorProductMatrixSymmetricSum<dim, Number, n_rows_1d>::
  TensorProductMatrixSymmetricSum(const Table<2, Number> &mass_matrix,
                                  const Table<2, Number> &derivative_matrix)
{
  reinit(mass_matrix, derivative_matrix);
}



template <int dim, typename Number, int n_rows_1d>
template <typename MatrixArray>
inline void
TensorProductMatrixSymmetricSum<dim, Number, n_rows_1d>::reinit_impl(
  MatrixArray &&mass_matrices_,
  MatrixArray &&derivative_matrices_)
{
  auto &&mass_matrices       = std::forward<MatrixArray>(mass_matrices_);
  auto &&derivative_matrices = std::forward<MatrixArray>(derivative_matrices_);
  this->mass_matrix          = mass_matrices;
  this->derivative_matrix    = derivative_matrices;

  for (int dir = 0; dir < dim; ++dir)
    {
      Assert(n_rows_1d == -1 ||
               (n_rows_1d > 0 && static_cast<unsigned int>(n_rows_1d) ==
                                   mass_matrices[dir].n_rows()),
             ExcDimensionMismatch(n_rows_1d, mass_matrices[dir].n_rows()));
      AssertDimension(mass_matrices[dir].n_rows(), mass_matrices[dir].n_cols());
      AssertDimension(mass_matrices[dir].n_rows(),
                      derivative_matrices[dir].n_rows());
      AssertDimension(mass_matrices[dir].n_rows(),
                      derivative_matrices[dir].n_cols());

      this->eigenvectors[dir].reinit(mass_matrices[dir].n_cols(),
                                     mass_matrices[dir].n_rows());
      this->eigenvalues[dir].resize(mass_matrices[dir].n_cols());
      internal::TensorProductMatrix::spectral_assembly<Number>(
        &(mass_matrices[dir](0, 0)),
        &(derivative_matrices[dir](0, 0)),
        mass_matrices[dir].n_rows(),
        mass_matrices[dir].n_cols(),
        this->eigenvalues[dir].begin(),
        &(this->eigenvectors[dir](0, 0)));
    }
}



template <int dim, typename Number, int n_rows_1d>
inline void
TensorProductMatrixSymmetricSum<dim, Number, n_rows_1d>::reinit(
  const std::array<Table<2, Number>, dim> &mass_matrix,
  const std::array<Table<2, Number>, dim> &derivative_matrix)
{
  reinit_impl(mass_matrix, derivative_matrix);
}



template <int dim, typename Number, int n_rows_1d>
inline void
TensorProductMatrixSymmetricSum<dim, Number, n_rows_1d>::reinit(
  const std::array<FullMatrix<Number>, dim> &mass_matrix,
  const std::array<FullMatrix<Number>, dim> &derivative_matrix)
{
  std::array<Table<2, Number>, dim> mass_copy;
  std::array<Table<2, Number>, dim> deriv_copy;

  std::transform(mass_matrix.cbegin(),
                 mass_matrix.cend(),
                 mass_copy.begin(),
                 [](const FullMatrix<Number> &m) -> Table<2, Number> {
                   return m;
                 });
  std::transform(derivative_matrix.cbegin(),
                 derivative_matrix.cend(),
                 deriv_copy.begin(),
                 [](const FullMatrix<Number> &m) -> Table<2, Number> {
                   return m;
                 });

  reinit_impl(std::move(mass_copy), std::move(deriv_copy));
}



template <int dim, typename Number, int n_rows_1d>
inline void
TensorProductMatrixSymmetricSum<dim, Number, n_rows_1d>::reinit(
  const Table<2, Number> &mass_matrix,
  const Table<2, Number> &derivative_matrix)
{
  std::array<Table<2, Number>, dim> mass_matrices;
  std::array<Table<2, Number>, dim> derivative_matrices;

  std::fill(mass_matrices.begin(), mass_matrices.end(), mass_matrix);
  std::fill(derivative_matrices.begin(),
            derivative_matrices.end(),
            derivative_matrix);

  reinit_impl(std::move(mass_matrices), std::move(derivative_matrices));
}



//------------- vectorized spec.: TensorProductMatrixSymmetricSum -------------

template <int dim, typename Number, int n_rows_1d>
inline TensorProductMatrixSymmetricSum<dim,
                                       VectorizedArray<Number>,
                                       n_rows_1d>::
  TensorProductMatrixSymmetricSum(
    const std::array<Table<2, VectorizedArray<Number>>, dim> &mass_matrix,
    const std::array<Table<2, VectorizedArray<Number>>, dim> &derivative_matrix)
{
  reinit(mass_matrix, derivative_matrix);
}



template <int dim, typename Number, int n_rows_1d>
inline TensorProductMatrixSymmetricSum<dim,
                                       VectorizedArray<Number>,
                                       n_rows_1d>::
  TensorProductMatrixSymmetricSum(
    const Table<2, VectorizedArray<Number>> &mass_matrix,
    const Table<2, VectorizedArray<Number>> &derivative_matrix)
{
  reinit(mass_matrix, derivative_matrix);
}



template <int dim, typename Number, int n_rows_1d>
template <typename MatrixArray>
inline void
TensorProductMatrixSymmetricSum<dim, VectorizedArray<Number>, n_rows_1d>::
  reinit_impl(MatrixArray &&mass_matrices_, MatrixArray &&derivative_matrices_)
{
  auto &&mass_matrix       = std::forward<MatrixArray>(mass_matrices_);
  auto &&derivative_matrix = std::forward<MatrixArray>(derivative_matrices_);
  this->mass_matrix        = mass_matrix;
  this->derivative_matrix  = derivative_matrix;

  constexpr unsigned int macro_size = VectorizedArray<Number>::size();
  std::size_t            n_rows_max = (n_rows_1d > 0) ? n_rows_1d : 0;
  if (n_rows_1d == -1)
    for (unsigned int d = 0; d < dim; ++d)
      n_rows_max = std::max(n_rows_max, mass_matrix[d].n_rows());
  const std::size_t nm_flat_size_max = n_rows_max * n_rows_max * macro_size;
  const std::size_t n_flat_size_max  = n_rows_max * macro_size;

  std::vector<Number> mass_matrix_flat;
  std::vector<Number> deriv_matrix_flat;
  std::vector<Number> eigenvalues_flat;
  std::vector<Number> eigenvectors_flat;
  mass_matrix_flat.resize(nm_flat_size_max);
  deriv_matrix_flat.resize(nm_flat_size_max);
  eigenvalues_flat.resize(n_flat_size_max);
  eigenvectors_flat.resize(nm_flat_size_max);
  std::array<unsigned int, macro_size> offsets_nm;
  std::array<unsigned int, macro_size> offsets_n;
  for (int dir = 0; dir < dim; ++dir)
    {
      Assert(n_rows_1d == -1 ||
               (n_rows_1d > 0 && static_cast<unsigned int>(n_rows_1d) ==
                                   mass_matrix[dir].n_rows()),
             ExcDimensionMismatch(n_rows_1d, mass_matrix[dir].n_rows()));
      AssertDimension(mass_matrix[dir].n_rows(), mass_matrix[dir].n_cols());
      AssertDimension(mass_matrix[dir].n_rows(),
                      derivative_matrix[dir].n_rows());
      AssertDimension(mass_matrix[dir].n_rows(),
                      derivative_matrix[dir].n_cols());

      const unsigned int n_rows = mass_matrix[dir].n_rows();
      const unsigned int n_cols = mass_matrix[dir].n_cols();
      const unsigned int nm     = n_rows * n_cols;
      for (unsigned int vv = 0; vv < macro_size; ++vv)
        offsets_nm[vv] = nm * vv;

      vectorized_transpose_and_store(false,
                                     nm,
                                     &(mass_matrix[dir](0, 0)),
                                     offsets_nm.cbegin(),
                                     mass_matrix_flat.data());
      vectorized_transpose_and_store(false,
                                     nm,
                                     &(derivative_matrix[dir](0, 0)),
                                     offsets_nm.cbegin(),
                                     deriv_matrix_flat.data());

      const Number *mass_cbegin    = mass_matrix_flat.data();
      const Number *deriv_cbegin   = deriv_matrix_flat.data();
      Number *      eigenvec_begin = eigenvectors_flat.data();
      Number *      eigenval_begin = eigenvalues_flat.data();
      for (unsigned int lane = 0; lane < macro_size; ++lane)
        internal::TensorProductMatrix::spectral_assembly<Number>(
          mass_cbegin + nm * lane,
          deriv_cbegin + nm * lane,
          n_rows,
          n_cols,
          eigenval_begin + n_rows * lane,
          eigenvec_begin + nm * lane);

      this->eigenvalues[dir].resize(n_rows);
      this->eigenvectors[dir].reinit(n_rows, n_cols);
      for (unsigned int vv = 0; vv < macro_size; ++vv)
        offsets_n[vv] = n_rows * vv;
      vectorized_load_and_transpose(n_rows,
                                    eigenvalues_flat.data(),
                                    offsets_n.cbegin(),
                                    this->eigenvalues[dir].begin());
      vectorized_load_and_transpose(nm,
                                    eigenvectors_flat.data(),
                                    offsets_nm.cbegin(),
                                    &(this->eigenvectors[dir](0, 0)));
    }
}



template <int dim, typename Number, int n_rows_1d>
inline void
TensorProductMatrixSymmetricSum<dim, VectorizedArray<Number>, n_rows_1d>::
  reinit(
    const std::array<Table<2, VectorizedArray<Number>>, dim> &mass_matrix,
    const std::array<Table<2, VectorizedArray<Number>>, dim> &derivative_matrix)
{
  reinit_impl(mass_matrix, derivative_matrix);
}



template <int dim, typename Number, int n_rows_1d>
inline void
TensorProductMatrixSymmetricSum<dim, VectorizedArray<Number>, n_rows_1d>::
  reinit(const Table<2, VectorizedArray<Number>> &mass_matrix,
         const Table<2, VectorizedArray<Number>> &derivative_matrix)
{
  std::array<Table<2, VectorizedArray<Number>>, dim> mass_matrices;
  std::array<Table<2, VectorizedArray<Number>>, dim> derivative_matrices;

  std::fill(mass_matrices.begin(), mass_matrices.end(), mass_matrix);
  std::fill(derivative_matrices.begin(),
            derivative_matrices.end(),
            derivative_matrix);

  reinit_impl(std::move(mass_matrices), std::move(derivative_matrices));
}



#endif

DEAL_II_NAMESPACE_CLOSE

#endif