rb_evaluation.C

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// rbOOmit: An implementation of the Certified Reduced Basis method.
// Copyright (C) 2009, 2010 David J. Knezevic

// This file is part of rbOOmit.

// rbOOmit is free software; you can 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.

// rbOOmit is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// Lesser General Public License for more details.

// You should have received a copy of the GNU Lesser General Public
// License along with this library; if not, write to the Free Software
// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA

// rbOOmit includes
#include "libmesh/rb_evaluation.h"
#include "libmesh/rb_theta_expansion.h"

// libMesh includes
#include "libmesh/libmesh_common.h"
#include "libmesh/libmesh_version.h"
#include "libmesh/system.h"
#include "libmesh/numeric_vector.h"
#include "libmesh/libmesh_logging.h"
#include "libmesh/xdr_cxx.h"
#include "libmesh/mesh_tools.h"
#include "libmesh/utility.h"

// TIMPI includes
#include "timpi/communicator.h"

// C/C++ includes
#include <sys/types.h>
#include <sys/stat.h>
#include <errno.h>
#include <fstream>
#include <sstream>
#include <iomanip>

namespace libMesh
{

RBEvaluation::RBEvaluation (const Parallel::Communicator & comm_in)
  :
  ParallelObject(comm_in),
  evaluate_RB_error_bound(true),
  compute_RB_inner_product(false),
  rb_theta_expansion(nullptr)
{
}

RBEvaluation::~RBEvaluation() = default;

void RBEvaluation::clear()
{
  LOG_SCOPE("clear()", "RBEvaluation");

  // Clear the basis functions
  basis_functions.clear();
  set_n_basis_functions(0);

  clear_riesz_representors();

  // Clear the Greedy param list
  for (auto & plist : greedy_param_list)
    plist.clear();
  greedy_param_list.clear();
}

void RBEvaluation::set_n_basis_functions(unsigned int n_bfs)
{
  basis_functions.resize(n_bfs);
}

void RBEvaluation::set_rb_theta_expansion(RBThetaExpansion & rb_theta_expansion_in)
{
  rb_theta_expansion = &rb_theta_expansion_in;
}

RBThetaExpansion & RBEvaluation::get_rb_theta_expansion()
{
  libmesh_error_msg_if(!is_rb_theta_expansion_initialized(),
                       "Error: rb_theta_expansion hasn't been initialized yet");

  return *rb_theta_expansion;
}

const RBThetaExpansion & RBEvaluation::get_rb_theta_expansion() const
{
  libmesh_error_msg_if(!is_rb_theta_expansion_initialized(),
                       "Error: rb_theta_expansion hasn't been initialized yet");

  return *rb_theta_expansion;
}

bool RBEvaluation::is_rb_theta_expansion_initialized() const
{
  if (rb_theta_expansion)
    {
      return true;
    }
  else
    {
      return false;
    }
}

void RBEvaluation::resize_data_structures(const unsigned int Nmax,
                                          bool resize_error_bound_data)
{
  LOG_SCOPE("resize_data_structures()", "RBEvaluation");

  libmesh_error_msg_if(Nmax < this->get_n_basis_functions(),
                       "Error: Cannot set Nmax to be less than the current number of basis functions.");

  // Resize/clear inner product matrix
  if (compute_RB_inner_product)
    RB_inner_product_matrix.resize(Nmax,Nmax);

  // Allocate dense matrices for RB solves
  RB_Aq_vector.resize(rb_theta_expansion->get_n_A_terms());

  for (unsigned int q=0; q<rb_theta_expansion->get_n_A_terms(); q++)
    {
      // Initialize the memory for the RB matrices
      RB_Aq_vector[q].resize(Nmax,Nmax);
    }

  RB_Fq_vector.resize(rb_theta_expansion->get_n_F_terms());

  for (unsigned int q=0; q<rb_theta_expansion->get_n_F_terms(); q++)
    {
      // Initialize the memory for the RB vectors
      RB_Fq_vector[q].resize(Nmax);
    }


  // Initialize the RB output vectors
  RB_output_vectors.resize(rb_theta_expansion->get_n_outputs());
  for (unsigned int n=0; n<rb_theta_expansion->get_n_outputs(); n++)
    {
      RB_output_vectors[n].resize(rb_theta_expansion->get_n_output_terms(n));
      for (unsigned int q_l=0; q_l<rb_theta_expansion->get_n_output_terms(n); q_l++)
        {
          RB_output_vectors[n][q_l].resize(Nmax);
        }
    }

  // Initialize vectors storing output data
  RB_outputs.resize(rb_theta_expansion->get_n_outputs(), 0.);


  if (resize_error_bound_data)
    {
      // Initialize vectors for the norms of the Fq representors
      unsigned int Q_f_hat = rb_theta_expansion->get_n_F_terms()*(rb_theta_expansion->get_n_F_terms()+1)/2;
      Fq_representor_innerprods.resize(Q_f_hat);

      // Initialize vectors for the norms of the representors
      Fq_Aq_representor_innerprods.resize(rb_theta_expansion->get_n_F_terms());
      for (unsigned int i=0; i<rb_theta_expansion->get_n_F_terms(); i++)
        {
          Fq_Aq_representor_innerprods[i].resize(rb_theta_expansion->get_n_A_terms());
          for (unsigned int j=0; j<rb_theta_expansion->get_n_A_terms(); j++)
            {
              Fq_Aq_representor_innerprods[i][j].resize(Nmax, 0.);
            }
        }

      unsigned int Q_a_hat = rb_theta_expansion->get_n_A_terms()*(rb_theta_expansion->get_n_A_terms()+1)/2;
      Aq_Aq_representor_innerprods.resize(Q_a_hat);
      for (unsigned int i=0; i<Q_a_hat; i++)
        {
          Aq_Aq_representor_innerprods[i].resize(Nmax);
          for (unsigned int j=0; j<Nmax; j++)
            {
              Aq_Aq_representor_innerprods[i][j].resize(Nmax, 0.);
            }
        }

      RB_output_error_bounds.resize(rb_theta_expansion->get_n_outputs(), 0.);

      // Resize the output dual norm vectors
      output_dual_innerprods.resize(rb_theta_expansion->get_n_outputs());
      for (unsigned int n=0; n<rb_theta_expansion->get_n_outputs(); n++)
        {
          unsigned int Q_l_hat = rb_theta_expansion->get_n_output_terms(n)*(rb_theta_expansion->get_n_output_terms(n)+1)/2;
          output_dual_innerprods[n].resize(Q_l_hat);
        }

      // Clear and resize the vector of Aq_representors
      clear_riesz_representors();

      Aq_representor.resize(rb_theta_expansion->get_n_A_terms());
      for (unsigned int q_a=0; q_a<rb_theta_expansion->get_n_A_terms(); q_a++)
        {
          Aq_representor[q_a].resize(Nmax);
        }
    }
}

NumericVector<Number> & RBEvaluation::get_basis_function(unsigned int i)
{
  libmesh_assert_less (i, basis_functions.size());

  return *(basis_functions[i]);
}

const NumericVector<Number> & RBEvaluation::get_basis_function(unsigned int i) const
{
  libmesh_assert_less (i, basis_functions.size());

  return *(basis_functions[i]);
}

Real RBEvaluation::rb_solve(unsigned int N)
{
  return rb_solve(N, nullptr);
}

Real RBEvaluation::rb_solve(unsigned int N,
                            const std::vector<Number> * evaluated_thetas)
{
  LOG_SCOPE("rb_solve()", "RBEvaluation");

  libmesh_error_msg_if(N > get_n_basis_functions(),
                       "ERROR: N cannot be larger than the number of basis functions in rb_solve");

  // In case the theta functions have been pre-evaluated, first check the size for consistency. The
  // size of the input "evaluated_thetas" vector must match the sum of the "A", "F", and "output" terms
  // in the expansion.
  this->check_evaluated_thetas_size(evaluated_thetas);

  const RBParameters & mu = get_parameters();

  // Resize (and clear) the solution vector
  RB_solution.resize(N);

  // Assemble the RB system
  DenseMatrix<Number> RB_system_matrix(N,N);
  RB_system_matrix.zero();

  DenseMatrix<Number> RB_Aq_a;
  for (unsigned int q_a=0; q_a<rb_theta_expansion->get_n_A_terms(); q_a++)
    {
      RB_Aq_vector[q_a].get_principal_submatrix(N, RB_Aq_a);

      if (evaluated_thetas)
        RB_system_matrix.add((*evaluated_thetas)[q_a], RB_Aq_a);
      else
        RB_system_matrix.add(rb_theta_expansion->eval_A_theta(q_a, mu), RB_Aq_a);
    }

  // Assemble the RB rhs
  DenseVector<Number> RB_rhs(N);
  RB_rhs.zero();

  DenseVector<Number> RB_Fq_f;
  for (unsigned int q_f=0; q_f<rb_theta_expansion->get_n_F_terms(); q_f++)
    {
      RB_Fq_vector[q_f].get_principal_subvector(N, RB_Fq_f);

      if (evaluated_thetas)
        RB_rhs.add((*evaluated_thetas)[q_f+rb_theta_expansion->get_n_A_terms()], RB_Fq_f);
      else
        RB_rhs.add(rb_theta_expansion->eval_F_theta(q_f, mu), RB_Fq_f);
    }

  // Solve the linear system
  if (N > 0)
    {
      RB_system_matrix.lu_solve(RB_rhs, RB_solution);
    }

  // Place to store the output of get_principal_subvector() calls
  DenseVector<Number> RB_output_vector_N;

  // Evaluate RB outputs
  unsigned int output_counter = 0;
  for (unsigned int n=0; n<rb_theta_expansion->get_n_outputs(); n++)
    {
      RB_outputs[n] = 0.;
      for (unsigned int q_l=0; q_l<rb_theta_expansion->get_n_output_terms(n); q_l++)
        {
          RB_output_vectors[n][q_l].get_principal_subvector(N, RB_output_vector_N);

          // Compute dot product with current output vector and RB_solution
          auto dot_prod = RB_output_vector_N.dot(RB_solution);

          // Determine the coefficient depending on whether or not
          // pre-evaluated thetas were provided. Note that if
          // pre-evaluated thetas were provided, they must come after
          // the "A" and "F" thetas and be in "row-major" order. In
          // other words, there is a possibly "ragged" 2D array of
          // output thetas ordered by output index "n" and term index
          // "q_l" which is accessed in row-major order.
          auto coeff = evaluated_thetas ?
            (*evaluated_thetas)[output_counter + rb_theta_expansion->get_n_A_terms() + rb_theta_expansion->get_n_F_terms()] :
            rb_theta_expansion->eval_output_theta(n, q_l, mu);

          // Finally, accumulate the result in RB_outputs[n]
          RB_outputs[n] += coeff * dot_prod;

          // Go to next output
          output_counter++;
        }
    }

  if (evaluate_RB_error_bound) // Calculate the error bounds
    {
      // Evaluate the dual norm of the residual for RB_solution_vector
      Real epsilon_N = compute_residual_dual_norm(N, evaluated_thetas);

      // Get lower bound for coercivity constant
      const Real alpha_LB = get_stability_lower_bound();
      // alpha_LB needs to be positive to get a valid error bound
      libmesh_assert_greater ( alpha_LB, 0. );

      // Evaluate the (absolute) error bound
      Real abs_error_bound = epsilon_N / residual_scaling_denom(alpha_LB);

      // Now compute the output error bounds
      for (unsigned int n=0; n<rb_theta_expansion->get_n_outputs(); n++)
        RB_output_error_bounds[n] = abs_error_bound * this->eval_output_dual_norm(n, evaluated_thetas);

      return abs_error_bound;
    }
  else // Don't calculate the error bounds
    {
      // Just return -1. if we did not compute the error bound
      return -1.;
    }
}

Real RBEvaluation::get_error_bound_normalization()
{
  // Normalize the error based on the error bound in the
  // case of an empty reduced basis. The error bound is based
  // on the residual F - AU, so with an empty basis this gives
  // a value based on the norm of F at the current parameters.

  Real normalization = rb_solve(0);
  return normalization;
}

Real RBEvaluation::compute_residual_dual_norm(const unsigned int N)
{
  return compute_residual_dual_norm(N, nullptr);
}

Real RBEvaluation::compute_residual_dual_norm(const unsigned int N,
                                              const std::vector<Number> * evaluated_thetas)
{
  LOG_SCOPE("compute_residual_dual_norm()", "RBEvaluation");

  // In case the theta functions have been pre-evaluated, first check the size for consistency
  this->check_evaluated_thetas_size(evaluated_thetas);

  // If evaluated_thetas is provided, then mu is not actually used for anything
  const RBParameters & mu = get_parameters();

  const unsigned int n_A_terms = rb_theta_expansion->get_n_A_terms();
  const unsigned int n_F_terms = rb_theta_expansion->get_n_F_terms();

  // Use the stored representor inner product values
  // to evaluate the residual norm
  Number residual_norm_sq = 0.;

  // Lambdas to help with evaluating F_theta and A_theta functions
  // using either the pre-evaluated thetas (if provided) or by calling
  // eval_{F,A}_theta()
  auto eval_F = [&](unsigned int index)
    {
      return (evaluated_thetas) ? (*evaluated_thetas)[index + n_A_terms] : rb_theta_expansion->eval_F_theta(index, mu);
    };
  auto eval_A = [&](unsigned int index)
    {
      return (evaluated_thetas) ? (*evaluated_thetas)[index] : rb_theta_expansion->eval_A_theta(index, mu);
    };

  unsigned int q=0;
  for (unsigned int q_f1=0; q_f1<n_F_terms; q_f1++)
    {
      const Number val_q_f1 = eval_F(q_f1);

      for (unsigned int q_f2=q_f1; q_f2<n_F_terms; q_f2++)
        {
          const Number val_q_f2 = eval_F(q_f2);

          Real delta = (q_f1==q_f2) ? 1. : 2.;
          residual_norm_sq += delta * libmesh_real(val_q_f1 * libmesh_conj(val_q_f2) * Fq_representor_innerprods[q] );

          q++;
        }
    }

  for (unsigned int q_f=0; q_f<n_F_terms; q_f++)
    {
      const Number val_q_f = eval_F(q_f);

      for (unsigned int q_a=0; q_a<n_A_terms; q_a++)
        {
          const Number val_q_a = eval_A(q_a);

          for (unsigned int i=0; i<N; i++)
            {
              Real delta = 2.;
              residual_norm_sq +=
                delta * libmesh_real( val_q_f * libmesh_conj(val_q_a) *
                                      libmesh_conj(RB_solution(i)) * Fq_Aq_representor_innerprods[q_f][q_a][i] );
            }
        }
    }

  q=0;
  for (unsigned int q_a1=0; q_a1<n_A_terms; q_a1++)
    {
      const Number val_q_a1 = eval_A(q_a1);

      for (unsigned int q_a2=q_a1; q_a2<n_A_terms; q_a2++)
        {
          const Number val_q_a2 = eval_A(q_a2);

          Real delta = (q_a1==q_a2) ? 1. : 2.;

          for (unsigned int i=0; i<N; i++)
            {
              for (unsigned int j=0; j<N; j++)
                {
                  residual_norm_sq +=
                    delta * libmesh_real( libmesh_conj(val_q_a1) * val_q_a2 *
                                          libmesh_conj(RB_solution(i)) * RB_solution(j) * Aq_Aq_representor_innerprods[q][i][j] );
                }
            }

          q++;
        }
    }

  if (libmesh_real(residual_norm_sq) < 0.)
    {
      //    libMesh::out << "Warning: Square of residual norm is negative "
      //                 << "in RBSystem::compute_residual_dual_norm()" << std::endl;

      //     Sometimes this is negative due to rounding error,
      //     but when this occurs the error is on the order of 1.e-10,
      //     so shouldn't affect error bound much...
      residual_norm_sq = std::abs(residual_norm_sq);
    }

  return std::sqrt( libmesh_real(residual_norm_sq) );
}

Real RBEvaluation::get_stability_lower_bound()
{
  // Return a default value of 1, this function should
  // be overloaded to specify a problem-dependent stability
  // factor lower bound
  return 1.;
}

Real RBEvaluation::residual_scaling_denom(Real alpha_LB)
{
  // Here we implement the residual scaling for a coercive
  // problem.
  return alpha_LB;
}

#ifdef LIBMESH_ENABLE_DEPRECATED
Real RBEvaluation::eval_output_dual_norm(unsigned int n, const RBParameters & /*mu*/)
{
  libmesh_deprecated();

  // Call non-deprecated version of this function, ignoring input mu
  return this->eval_output_dual_norm(n, nullptr);
}
#endif // LIBMESH_ENABLE_DEPRECATED

Real RBEvaluation::eval_output_dual_norm(unsigned int n,
                                         const std::vector<Number> * evaluated_thetas)
{
  // Return value
  Number output_bound_sq = 0.;

  // mu is only used if evaluated_thetas == nullptr
  const RBParameters & mu = this->get_parameters();

  // Index into output_dual_innerprods
  unsigned int q=0;
  for (unsigned int q_l1=0; q_l1<rb_theta_expansion->get_n_output_terms(n); q_l1++)
    {
      for (unsigned int q_l2=q_l1; q_l2<rb_theta_expansion->get_n_output_terms(n); q_l2++)
        {
          Real delta = (q_l1==q_l2) ? 1. : 2.;

          Number val_l1 =
            evaluated_thetas ?
            (*evaluated_thetas)[rb_theta_expansion->output_index_1D(n, q_l1) + rb_theta_expansion->get_n_A_terms() + rb_theta_expansion->get_n_F_terms()] :
            rb_theta_expansion->eval_output_theta(n, q_l1, mu);

          Number val_l2 =
            evaluated_thetas ?
            (*evaluated_thetas)[rb_theta_expansion->output_index_1D(n, q_l2) + rb_theta_expansion->get_n_A_terms() + rb_theta_expansion->get_n_F_terms()] :
            rb_theta_expansion->eval_output_theta(n, q_l2, mu);

          output_bound_sq += delta * libmesh_real(
            libmesh_conj(val_l1) * val_l2 * output_dual_innerprods[n][q]);

          q++;
        }
    }

  return libmesh_real(std::sqrt( output_bound_sq ));
}

void RBEvaluation::clear_riesz_representors()
{
  Aq_representor.clear();
}

void RBEvaluation::legacy_write_offline_data_to_files(const std::string & directory_name,
                                                      const bool write_binary_data)
{
  LOG_SCOPE("legacy_write_offline_data_to_files()", "RBEvaluation");

  // Get the number of basis functions
  unsigned int n_bfs = get_n_basis_functions();

  // The writing mode: ENCODE for binary, WRITE for ASCII
  XdrMODE mode = write_binary_data ? ENCODE : WRITE;

  // The suffix to use for all the files that are written out
  const std::string suffix = write_binary_data ? ".xdr" : ".dat";

  if (this->processor_id() == 0)
    {

      // Make a directory to store all the data files
      Utility::mkdir(directory_name.c_str());
      //    if (mkdir(directory_name.c_str(), 0777) == -1)
      //    {
      //      libMesh::out << "In RBEvaluation::write_offline_data_to_files, directory "
      //                   << directory_name << " already exists, overwriting contents." << std::endl;
      //    }

      // First, write out how many basis functions we have generated
      std::ostringstream file_name;
      {
        file_name << directory_name << "/n_bfs" << suffix;
        Xdr n_bfs_out(file_name.str(), mode);
        n_bfs_out << n_bfs;
        n_bfs_out.close();
      }

      // Write out the parameter ranges
      file_name.str("");
      file_name << directory_name << "/parameter_ranges" << suffix;
      std::string continuous_param_file_name = file_name.str();

      // Write out the discrete parameter values
      file_name.str("");
      file_name << directory_name << "/discrete_parameter_values" << suffix;
      std::string discrete_param_file_name = file_name.str();

      write_parameter_data_to_files(continuous_param_file_name,
                                    discrete_param_file_name,
                                    write_binary_data);

      // Write out Fq representor norm data
      file_name.str("");
      file_name << directory_name << "/Fq_innerprods" << suffix;
      Xdr RB_Fq_innerprods_out(file_name.str(), mode);
      unsigned int Q_f_hat = rb_theta_expansion->get_n_F_terms()*(rb_theta_expansion->get_n_F_terms()+1)/2;
      for (unsigned int i=0; i<Q_f_hat; i++)
        {
          RB_Fq_innerprods_out << Fq_representor_innerprods[i];
        }
      RB_Fq_innerprods_out.close();

      // Write out output data
      for (unsigned int n=0; n<rb_theta_expansion->get_n_outputs(); n++)
        {
          file_name.str("");
          file_name << directory_name << "/output_";
          file_name << std::setw(3)
                    << std::setprecision(0)
                    << std::setfill('0')
                    << std::right
                    << n;

          file_name << "_dual_innerprods" << suffix;
          Xdr output_dual_innerprods_out(file_name.str(), mode);

          unsigned int Q_l_hat = rb_theta_expansion->get_n_output_terms(n)*(rb_theta_expansion->get_n_output_terms(n)+1)/2;
          for (unsigned int q=0; q<Q_l_hat; q++)
            {
              output_dual_innerprods_out << output_dual_innerprods[n][q];
            }
          output_dual_innerprods_out.close();
        }


      // Write out output data to multiple files
      for (unsigned int n=0; n<rb_theta_expansion->get_n_outputs(); n++)
        {
          for (unsigned int q_l=0; q_l<rb_theta_expansion->get_n_output_terms(n); q_l++)
            {
              file_name.str("");
              file_name << directory_name << "/output_";
              file_name << std::setw(3)
                        << std::setprecision(0)
                        << std::setfill('0')
                        << std::right
                        << n;
              file_name << "_";
              file_name << std::setw(3)
                        << std::setprecision(0)
                        << std::setfill('0')
                        << std::right
                        << q_l;
              file_name << suffix;
              Xdr output_n_out(file_name.str(), mode);

              for (unsigned int j=0; j<n_bfs; j++)
                {
                  output_n_out << RB_output_vectors[n][q_l](j);
                }
              output_n_out.close();
            }
        }

      if (compute_RB_inner_product)
        {
          // Next write out the inner product matrix
          file_name.str("");
          file_name << directory_name << "/RB_inner_product_matrix" << suffix;
          Xdr RB_inner_product_matrix_out(file_name.str(), mode);
          for (unsigned int i=0; i<n_bfs; i++)
            {
              for (unsigned int j=0; j<n_bfs; j++)
                {
                  RB_inner_product_matrix_out << RB_inner_product_matrix(i,j);
                }
            }
          RB_inner_product_matrix_out.close();
        }

      // Next write out the Fq vectors
      for (unsigned int q_f=0; q_f<rb_theta_expansion->get_n_F_terms(); q_f++)
        {
          file_name.str("");
          file_name << directory_name << "/RB_F_";
          file_name << std::setw(3)
                    << std::setprecision(0)
                    << std::setfill('0')
                    << std::right
                    << q_f;
          file_name << suffix;
          Xdr RB_Fq_f_out(file_name.str(), mode);

          for (unsigned int i=0; i<n_bfs; i++)
            {
              RB_Fq_f_out << RB_Fq_vector[q_f](i);
            }
          RB_Fq_f_out.close();
        }

      // Next write out the Aq matrices
      for (unsigned int q_a=0; q_a<rb_theta_expansion->get_n_A_terms(); q_a++)
        {
          file_name.str("");
          file_name << directory_name << "/RB_A_";
          file_name << std::setw(3)
                    << std::setprecision(0)
                    << std::setfill('0')
                    << std::right
                    << q_a;
          file_name << suffix;
          Xdr RB_Aq_a_out(file_name.str(), mode);

          for (unsigned int i=0; i<n_bfs; i++)
            {
              for (unsigned int j=0; j<n_bfs; j++)
                {
                  RB_Aq_a_out << RB_Aq_vector[q_a](i,j);
                }
            }
          RB_Aq_a_out.close();
        }

      // Next write out Fq_Aq representor norm data
      file_name.str("");
      file_name << directory_name << "/Fq_Aq_innerprods" << suffix;
      Xdr RB_Fq_Aq_innerprods_out(file_name.str(), mode);

      for (unsigned int q_f=0; q_f<rb_theta_expansion->get_n_F_terms(); q_f++)
        {
          for (unsigned int q_a=0; q_a<rb_theta_expansion->get_n_A_terms(); q_a++)
            {
              for (unsigned int i=0; i<n_bfs; i++)
                {
                  RB_Fq_Aq_innerprods_out << Fq_Aq_representor_innerprods[q_f][q_a][i];
                }
            }
        }
      RB_Fq_Aq_innerprods_out.close();

      // Next write out Aq_Aq representor norm data
      file_name.str("");
      file_name << directory_name << "/Aq_Aq_innerprods" << suffix;
      Xdr RB_Aq_Aq_innerprods_out(file_name.str(), mode);

      unsigned int Q_a_hat = rb_theta_expansion->get_n_A_terms()*(rb_theta_expansion->get_n_A_terms()+1)/2;
      for (unsigned int i=0; i<Q_a_hat; i++)
        {
          for (unsigned int j=0; j<n_bfs; j++)
            {
              for (unsigned int l=0; l<n_bfs; l++)
                {
                  RB_Aq_Aq_innerprods_out << Aq_Aq_representor_innerprods[i][j][l];
                }
            }
        }
      RB_Aq_Aq_innerprods_out.close();

      // Also, write out the greedily selected parameters
      {
        file_name.str("");
        file_name << directory_name << "/greedy_params" << suffix;
        Xdr greedy_params_out(file_name.str(), mode);

        for (const auto & param : greedy_param_list)
          for (const auto & pr : param)
            for (const auto & value_vector : pr.second)
              {
                // Need to make a copy of the value so that it's not const
                // Xdr is not templated on const's
                libmesh_error_msg_if(value_vector.size() != 1,
                                     "Error: multi-value RB parameters are not yet supported here.");
                Real param_value = value_vector[0];
                greedy_params_out << param_value;
              }
        greedy_params_out.close();
      }

    }
}

void RBEvaluation::legacy_read_offline_data_from_files(const std::string & directory_name,
                                                       bool read_error_bound_data,
                                                       const bool read_binary_data)
{
  LOG_SCOPE("legacy_read_offline_data_from_files()", "RBEvaluation");

  // The reading mode: DECODE for binary, READ for ASCII
  XdrMODE mode = read_binary_data ? DECODE : READ;

  // The suffix to use for all the files that are written out
  const std::string suffix = read_binary_data ? ".xdr" : ".dat";

  // The string stream we'll use to make the file names
  std::ostringstream file_name;

  // First, find out how many basis functions we had when Greedy terminated
  unsigned int n_bfs;
  {
    file_name << directory_name << "/n_bfs" << suffix;
    assert_file_exists(file_name.str());

    Xdr n_bfs_in(file_name.str(), mode);
    n_bfs_in >> n_bfs;
    n_bfs_in.close();
  }
  resize_data_structures(n_bfs, read_error_bound_data);

  // Read in the parameter ranges
  file_name.str("");
  file_name << directory_name << "/parameter_ranges" << suffix;
  std::string continuous_param_file_name = file_name.str();

  // Read in the discrete parameter values
  file_name.str("");
  file_name << directory_name << "/discrete_parameter_values" << suffix;
  std::string discrete_param_file_name = file_name.str();
  read_parameter_data_from_files(continuous_param_file_name,
                                 discrete_param_file_name,
                                 read_binary_data);

  // Read in output data in multiple files
  for (unsigned int n=0; n<rb_theta_expansion->get_n_outputs(); n++)
    {
      for (unsigned int q_l=0; q_l<rb_theta_expansion->get_n_output_terms(n); q_l++)
        {
          file_name.str("");
          file_name << directory_name << "/output_";
          file_name << std::setw(3)
                    << std::setprecision(0)
                    << std::setfill('0')
                    << std::right
                    << n;
          file_name << "_";
          file_name << std::setw(3)
                    << std::setprecision(0)
                    << std::setfill('0')
                    << std::right
                    << q_l;
          file_name << suffix;
          assert_file_exists(file_name.str());

          Xdr output_n_in(file_name.str(), mode);

          for (unsigned int j=0; j<n_bfs; j++)
            {
              Number value;
              output_n_in >> value;
              RB_output_vectors[n][q_l](j) = value;
            }
          output_n_in.close();
        }
    }

  if (compute_RB_inner_product)
    {
      // Next read in the inner product matrix
      file_name.str("");
      file_name << directory_name << "/RB_inner_product_matrix" << suffix;
      assert_file_exists(file_name.str());

      Xdr RB_inner_product_matrix_in(file_name.str(), mode);

      for (unsigned int i=0; i<n_bfs; i++)
        {
          for (unsigned int j=0; j<n_bfs; j++)
            {
              Number value;
              RB_inner_product_matrix_in >> value;
              RB_inner_product_matrix(i,j) = value;
            }
        }
      RB_inner_product_matrix_in.close();
    }

  // Next read in the Fq vectors
  for (unsigned int q_f=0; q_f<rb_theta_expansion->get_n_F_terms(); q_f++)
    {
      file_name.str("");
      file_name << directory_name << "/RB_F_";
      file_name << std::setw(3)
                << std::setprecision(0)
                << std::setfill('0')
                << std::right
                << q_f;
      file_name << suffix;
      assert_file_exists(file_name.str());

      Xdr RB_Fq_f_in(file_name.str(), mode);

      for (unsigned int i=0; i<n_bfs; i++)
        {
          Number value;
          RB_Fq_f_in >> value;
          RB_Fq_vector[q_f](i) = value;
        }
      RB_Fq_f_in.close();
    }

  // Next read in the Aq matrices
  for (unsigned int q_a=0; q_a<rb_theta_expansion->get_n_A_terms(); q_a++)
    {
      file_name.str("");
      file_name << directory_name << "/RB_A_";
      file_name << std::setw(3)
                << std::setprecision(0)
                << std::setfill('0')
                << std::right
                << q_a;
      file_name << suffix;
      assert_file_exists(file_name.str());

      Xdr RB_Aq_a_in(file_name.str(), mode);

      for (unsigned int i=0; i<n_bfs; i++)
        {
          for (unsigned int j=0; j<n_bfs; j++)
            {
              Number  value;
              RB_Aq_a_in >> value;
              RB_Aq_vector[q_a](i,j) = value;
            }
        }
      RB_Aq_a_in.close();
    }


  if (read_error_bound_data)
    {
      // Next read in Fq representor norm data
      file_name.str("");
      file_name << directory_name << "/Fq_innerprods" << suffix;
      assert_file_exists(file_name.str());

      Xdr RB_Fq_innerprods_in(file_name.str(), mode);

      unsigned int Q_f_hat = rb_theta_expansion->get_n_F_terms()*(rb_theta_expansion->get_n_F_terms()+1)/2;
      for (unsigned int i=0; i<Q_f_hat; i++)
        {
          RB_Fq_innerprods_in >> Fq_representor_innerprods[i];
        }
      RB_Fq_innerprods_in.close();

      // Read in output data
      for (unsigned int n=0; n<rb_theta_expansion->get_n_outputs(); n++)
        {
          file_name.str("");
          file_name << directory_name << "/output_";
          file_name << std::setw(3)
                    << std::setprecision(0)
                    << std::setfill('0')
                    << std::right
                    << n;
          file_name << "_dual_innerprods" << suffix;
          assert_file_exists(file_name.str());

          Xdr output_dual_innerprods_in(file_name.str(), mode);

          unsigned int Q_l_hat = rb_theta_expansion->get_n_output_terms(n)*(rb_theta_expansion->get_n_output_terms(n)+1)/2;
          for (unsigned int q=0; q<Q_l_hat; q++)
            {
              output_dual_innerprods_in >> output_dual_innerprods[n][q];
            }
          output_dual_innerprods_in.close();
        }


      // Next read in Fq_Aq representor norm data
      file_name.str("");
      file_name << directory_name << "/Fq_Aq_innerprods" << suffix;
      assert_file_exists(file_name.str());

      Xdr RB_Fq_Aq_innerprods_in(file_name.str(), mode);

      for (unsigned int q_f=0; q_f<rb_theta_expansion->get_n_F_terms(); q_f++)
        {
          for (unsigned int q_a=0; q_a<rb_theta_expansion->get_n_A_terms(); q_a++)
            {
              for (unsigned int i=0; i<n_bfs; i++)
                {
                  RB_Fq_Aq_innerprods_in >> Fq_Aq_representor_innerprods[q_f][q_a][i];
                }
            }
        }
      RB_Fq_Aq_innerprods_in.close();

      // Next read in Aq_Aq representor norm data
      file_name.str("");
      file_name << directory_name << "/Aq_Aq_innerprods" << suffix;
      assert_file_exists(file_name.str());

      Xdr RB_Aq_Aq_innerprods_in(file_name.str(), mode);

      unsigned int Q_a_hat = rb_theta_expansion->get_n_A_terms()*(rb_theta_expansion->get_n_A_terms()+1)/2;
      for (unsigned int i=0; i<Q_a_hat; i++)
        {
          for (unsigned int j=0; j<n_bfs; j++)
            {
              for (unsigned int l=0; l<n_bfs; l++)
                {
                  RB_Aq_Aq_innerprods_in >> Aq_Aq_representor_innerprods[i][j][l];
                }
            }
        }
      RB_Aq_Aq_innerprods_in.close();
    }

  // Resize basis_functions even if we don't read them in so that
  // get_n_bfs() returns the correct value. Initialize the pointers
  // to nullptr.
  basis_functions.clear();
  set_n_basis_functions(n_bfs);
}

void RBEvaluation::assert_file_exists(const std::string & file_name)
{
  libmesh_error_msg_if(!std::ifstream(file_name.c_str()), "File missing: " << file_name);
}

void RBEvaluation::write_out_basis_functions(System & sys,
                                             const std::string & directory_name,
                                             const bool write_binary_basis_functions)
{
  LOG_SCOPE("write_out_basis_functions()", "RBEvaluation");

  std::vector<NumericVector<Number>*> basis_functions_ptrs;
  for (std::size_t i=0; i<basis_functions.size(); i++)
  {
    basis_functions_ptrs.push_back(basis_functions[i].get());
  }

  write_out_vectors(sys,
                    basis_functions_ptrs,
                    directory_name,
                    "bf",
                    write_binary_basis_functions);
}

void RBEvaluation::write_out_vectors(System & sys,
                                     std::vector<NumericVector<Number>*> & vectors,
                                     const std::string & directory_name,
                                     const std::string & data_name,
                                     const bool write_binary_vectors)
{
  LOG_SCOPE("write_out_vectors()", "RBEvaluation");

  if (sys.comm().rank() == 0)
    {
      // Make a directory to store all the data files
      Utility::mkdir(directory_name.c_str());
    }

  // Make sure processors are synced up before we begin
  sys.comm().barrier();

  std::ostringstream file_name;
  const std::string basis_function_suffix = (write_binary_vectors ? ".xdr" : ".dat");

  file_name << directory_name << "/" << data_name << "_data" << basis_function_suffix;
  Xdr bf_data(file_name.str(),
              write_binary_vectors ? ENCODE : WRITE);

  // Following EquationSystems::write(), we should only write the header information
  // if we're proc 0
  if (sys.comm().rank() == 0)
    {
      std::string version("libMesh-" + libMesh::get_io_compatibility_version());
#ifdef LIBMESH_ENABLE_INFINITE_ELEMENTS
      version += " with infinite elements";
#endif
      bf_data.data(version ,"# File Format Identifier");

      sys.write_header(bf_data, /*(unused arg)*/ version, /*write_additional_data=*/false);
    }

  // Following EquationSystemsIO::write, we use a temporary numbering (node major)
  // before writing out the data
  MeshTools::Private::globally_renumber_nodes_and_elements(sys.get_mesh());

  // Write all vectors at once.
  {
    // Note the API wants pointers to constant vectors, hence this...
    std::vector<const NumericVector<Number> *> bf_out;
    for (const auto & vec : vectors)
      bf_out.push_back(vec);

    // for (auto & val : vectors)
    //   bf_out.push_back(val);
    sys.write_serialized_vectors (bf_data, bf_out);
  }


  // set the current version
  bf_data.set_version(LIBMESH_VERSION_ID(LIBMESH_MAJOR_VERSION,
                                         LIBMESH_MINOR_VERSION,
                                         LIBMESH_MICRO_VERSION));


  // Undo the temporary renumbering
  sys.get_mesh().fix_broken_node_and_element_numbering();
}

void RBEvaluation::read_in_basis_functions(System & sys,
                                           const std::string & directory_name,
                                           const bool read_binary_basis_functions)
{
  LOG_SCOPE("read_in_basis_functions()", "RBEvaluation");

  read_in_vectors(sys,
                  basis_functions,
                  directory_name,
                  "bf",
                  read_binary_basis_functions);
}

void RBEvaluation::read_in_vectors(System & sys,
                                   std::vector<std::unique_ptr<NumericVector<Number>>> & vectors,
                                   const std::string & directory_name,
                                   const std::string & data_name,
                                   const bool read_binary_vectors)
{
  std::vector<std::vector<std::unique_ptr<NumericVector<Number>>> *> vectors_vec;
  vectors_vec.push_back(&vectors);

  std::vector<std::string> directory_name_vec;
  directory_name_vec.push_back(directory_name);

  std::vector<std::string> data_name_vec;
  data_name_vec.push_back(data_name);

  read_in_vectors_from_multiple_files(sys,
                                      vectors_vec,
                                      directory_name_vec,
                                      data_name_vec,
                                      read_binary_vectors);
}

void RBEvaluation::read_in_vectors_from_multiple_files(System & sys,
                                                       std::vector<std::vector<std::unique_ptr<NumericVector<Number>>> *> multiple_vectors,
                                                       const std::vector<std::string> & multiple_directory_names,
                                                       const std::vector<std::string> & multiple_data_names,
                                                       const bool read_binary_vectors)
{
  LOG_SCOPE("read_in_vectors_from_multiple_files()", "RBEvaluation");

  std::size_t n_files = multiple_vectors.size();
  std::size_t n_directories = multiple_directory_names.size();
  libmesh_assert((n_files == n_directories) && (n_files == multiple_data_names.size()));

  if (n_files == 0)
    return;

  // Make sure processors are synced up before we begin
  sys.comm().barrier();

  std::ostringstream file_name;
  const std::string basis_function_suffix = (read_binary_vectors ? ".xdr" : ".dat");

  // Following EquationSystemsIO::read, we use a temporary numbering (node major)
  // before writing out the data. For the sake of efficiency, we do this once for
  // all the vectors that we read in.
  MeshTools::Private::globally_renumber_nodes_and_elements(sys.get_mesh());

  for (std::size_t data_index=0; data_index<n_directories; data_index++)
    {
      std::vector<std::unique_ptr<NumericVector<Number>>> & vectors = *multiple_vectors[data_index];

      // Allocate storage for each vector
      for (auto & vec : vectors)
        {
          // vectors should all be nullptr, otherwise we get a memory leak when
          // we create the new vectors in RBEvaluation::read_in_vectors.
          libmesh_error_msg_if(vec, "Non-nullptr vector passed to read_in_vectors_from_multiple_files");

          vec = NumericVector<Number>::build(sys.comm());

          vec->init (sys.n_dofs(),
                     sys.n_local_dofs(),
                     false,
                     PARALLEL);
        }

      file_name.str("");
      file_name << multiple_directory_names[data_index]
                << "/" << multiple_data_names[data_index]
                << "_data" << basis_function_suffix;

      // On processor zero check to be sure the file exists
      if (sys.comm().rank() == 0)
        {
          struct stat stat_info;
          int stat_result = stat(file_name.str().c_str(), &stat_info);

          libmesh_error_msg_if(stat_result != 0, "File does not exist: " << file_name.str());
        }

      assert_file_exists(file_name.str());
      Xdr vector_data(file_name.str(),
                      read_binary_vectors ? DECODE : READ);

      // Read the header data. This block of code is based on EquationSystems::_read_impl.
      {
        std::string version;
        vector_data.data(version);

        const std::string libMesh_label = "libMesh-";
        std::string::size_type lm_pos = version.find(libMesh_label);
        libmesh_error_msg_if(lm_pos == std::string::npos, "version info missing in Xdr header");

        std::istringstream iss(version.substr(lm_pos + libMesh_label.size()));
        int ver_major = 0, ver_minor = 0, ver_patch = 0;
        char dot;
        iss >> ver_major >> dot >> ver_minor >> dot >> ver_patch;
        vector_data.set_version(LIBMESH_VERSION_ID(ver_major, ver_minor, ver_patch));

        // Actually read the header data. When we do this, set read_header=false
        // so that we do not reinit sys, since we assume that it has already been
        // set up properly (e.g. the appropriate variables have already been added).
        sys.read_header(vector_data, version, /*read_header=*/false, /*read_additional_data=*/false);
      }

      // Comply with the System::read_serialized_vectors() interface which uses dumb pointers.
      std::vector<NumericVector<Number> *> vec_in;
      for (auto & vec : vectors)
        vec_in.push_back(vec.get());

      sys.read_serialized_vectors (vector_data, vec_in);
    }

  // Undo the temporary renumbering
  sys.get_mesh().fix_broken_node_and_element_numbering();
}

void RBEvaluation::check_evaluated_thetas_size(const std::vector<Number> * evaluated_thetas) const
{
  if (rb_theta_expansion && evaluated_thetas)
    {
      auto actual_size = evaluated_thetas->size();
      auto expected_size =
        rb_theta_expansion->get_n_A_terms() +
        rb_theta_expansion->get_n_F_terms() +
        rb_theta_expansion->get_total_n_output_terms();

      libmesh_error_msg_if(actual_size != expected_size,
                           "ERROR: Evaluated thetas vector has size " <<
                           actual_size << ", but we expected " <<
                           rb_theta_expansion->get_n_A_terms() << " A term(s), " <<
                           rb_theta_expansion->get_n_F_terms() << " F term(s), and " <<
                           rb_theta_expansion->get_total_n_output_terms() << " output term(s), " <<
                           "for a total of " << expected_size << " terms.");
    }
}

} // namespace libMesh