balance.cc

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/*!
 *
 * Copyright (C) 2015 Technical University of Liberec.  All rights reserved.
 * 
 * This program is free software; you can redistribute it and/or modify it under
 * the terms of the GNU General Public License version 3 as published by the
 * Free Software Foundation. (http://www.gnu.org/licenses/gpl-3.0.en.html)
 * 
 * This program 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 General Public License for more details.
 *
 * 
 * @file    balance.cc
 * @ingroup transport
 * @brief   Mass balance
 */
 
#include <iostream>
#include <iomanip>
#include <unordered_map>
 
#include "system/system.hh"
#include "system/sys_profiler.hh"
 
#include <petscmat.h>
#include "mesh/side_impl.hh"
#include "mesh/mesh.h"
#include "mesh/long_idx.hh"
#include "mesh/accessors.hh"
#include "io/output_time_set.hh"
#include "coupling/balance.hh"
#include "tools/unit_si.hh"
#include "tools/time_governor.hh"
#include "la/distribution.hh"
 
using namespace Input::Type;
 
bool Balance::do_yaml_output_ = true;
 
const Selection & Balance::get_format_selection_input_type() {
    return Selection("Balance_output_format", "Format of output file for balance.")
        .add_value(Balance::legacy, "legacy", "Legacy format used by previous program versions.")
        .add_value(Balance::txt, "txt", "Excel format with tab delimiter.")
        .add_value(Balance::gnuplot, "gnuplot", "Format compatible with GnuPlot datafile with fixed column width.")
        .close();
}
 
const Record & Balance::get_input_type() {
    return Record("Balance", "Balance of a conservative quantity, boundary fluxes and sources.")
        .declare_key("times", OutputTimeSet::get_input_type(), Default("[]"), "" )
        .declare_key("add_output_times", Bool(), Default("true"), "Add all output times of the balanced equation to the balance output times set. "
                "Note that this is not the time set of the output stream.")
        //.declare_key("balance_on", Bool(), Default("true"), "Balance is computed if the value is true.")
        .declare_key("format", Balance::get_format_selection_input_type(), Default("\"txt\""), "Format of output file.")
        .declare_key("cumulative", Bool(), Default("false"), "Compute cumulative balance over time. "
                "If true, then balance is calculated at each computational time step, which can slow down the program.")
        .declare_key("file", FileName::output(), Default::read_time("File name generated from the balanced quantity: <quantity_name>_balance.*"), "File name for output of balance.")
        .close();
}
 
void Balance::set_yaml_output() {
    Balance::do_yaml_output_ = true;
}
 
/*
std::shared_ptr<Balance> Balance::make_balance(
        const std::string &file_prefix,
        const Mesh *mesh,
        const Input::Record &in_rec,
        TimeGovernor &tg)
{
    auto ptr = make_shared<Balance>(file_prefix, mesh, in_rec, tg);
    auto &marks = TimeGovernor::marks();
    auto balance_output_type = tg.equation_mark_type() | TimeGovernor::marks().type_balance_output();
    if (marks.begin(balance_output_type) == marks.end(balance_output_type) ) {
        // no balance output time => force balance off
        ptr.reset();
    }
    return ptr;
}*/
 
 
 
Balance::Balance(const std::string &file_prefix, const Mesh *mesh)
    :     file_prefix_(file_prefix),
          mesh_(mesh),
          last_time_(),
          initial_(true),
          allocation_done_(false),
          balance_on_(true),
          output_line_counter_(0),
          output_yaml_header_(false)
 
{
    MPI_Comm_rank(PETSC_COMM_WORLD, &rank_);
    ASSERT_PTR(mesh_);
}
 
 
 
Balance::~Balance()
{
    if (rank_ == 0) {
        output_.close();
        if (do_yaml_output_) output_yaml_.close();
    }
    if (! allocation_done_) return;
 
    for (unsigned int c=0; c<quantities_.size(); ++c)
    {
        chkerr(MatDestroy(&(region_mass_matrix_[c])));
        chkerr(MatDestroy(&(be_flux_matrix_[c])));
        chkerr(MatDestroy(&(region_source_matrix_[c])));
        chkerr(MatDestroy(&(region_source_rhs_[c])));
        chkerr(VecDestroy(&(be_flux_vec_[c])));
        chkerr(VecDestroy(&(region_source_vec_[c])));
        chkerr(VecDestroy(&(region_mass_vec_[c])));
    }
    delete[] region_mass_matrix_;
    delete[] be_flux_matrix_;
    delete[] be_flux_vec_;
    delete[] region_source_matrix_;
    delete[] region_source_rhs_;
    delete[] region_source_vec_;
    delete[] region_mass_vec_;
 
    chkerr(MatDestroy(&region_be_matrix_));
    chkerr(VecDestroy(&ones_));
    chkerr(VecDestroy(&ones_be_));
}
 
 
void Balance::init_from_input(
        const Input::Record &in_rec,
        TimeGovernor &tg)
{
    ASSERT(! allocation_done_);
 
    time_ = &tg;
    
    auto &marks = TimeGovernor::marks();
 
    output_mark_type_ = tg.equation_mark_type() | marks.type_output(),
    balance_output_type_ = tg.equation_fixed_mark_type() | marks.type_balance_output();
 
    cumulative_ = in_rec.val<bool>("cumulative");
    output_format_ = in_rec.val<OutputFormat>("format");
 
    OutputTimeSet time_set;
    time_set.read_from_input( in_rec.val<Input::Array>("times"), tg, balance_output_type_);
    add_output_times_ = in_rec.val<bool>("add_output_times");
 
    input_record_ = in_rec;
 
}
 
 
void Balance::units(const UnitSI &unit)
{
    ASSERT(! allocation_done_);
 
    units_ = unit;
    units_.undef(false);
}
 
unsigned int Balance::add_quantity(const string &name)
{
    ASSERT(! allocation_done_);
 
    Quantity q(quantities_.size(), name);
    quantities_.push_back(q);
 
    return q.index_;
}
 
 
std::vector<unsigned int> Balance::add_quantities(const std::vector<string> &names)
{
    vector<unsigned int> indices;
    for (auto name : names)
        indices.push_back(add_quantity(name));
    return indices;
}
 
 
void Balance::allocate(unsigned int n_loc_dofs,
        unsigned int max_dofs_per_boundary)
{
    ASSERT(! allocation_done_);
    n_loc_dofs_ = n_loc_dofs;
    max_dofs_per_boundary_ = max_dofs_per_boundary;
}
 
void Balance::lazy_initialize()
{
    if (allocation_done_) return;
 
    auto &marks = TimeGovernor::marks();
    if (add_output_times_)
        marks.add_to_type_all(output_mark_type_, balance_output_type_);
    // if there are no balance marks turn balance off
    if (marks.begin(balance_output_type_) == marks.end(balance_output_type_) )
    {
        balance_on_ = false;
        cumulative_ = false;
        return;
    }
 
    // Max. number of regions to which a single dof can contribute.
    // TODO: estimate or compute this number directly (from mesh or dof handler).
    const int n_bulk_regs_per_dof = min(10, (int)mesh_->region_db().bulk_size());
    const unsigned int n_quant = quantities_.size();
    const unsigned int n_bdr_reg = mesh_->region_db().boundary_size();
    const unsigned int n_blk_reg = mesh_->region_db().bulk_size();
 
 
    // enumerate boundary edges by unique id and
    // create map: (local  bulk ele idx, side idx) -> boundary edge id
    // create vector that maps: boundary edge id -> region boundary index
    unsigned int be_id = 0;
    for (unsigned int loc_el = 0; loc_el < mesh_->get_el_ds()->lsize(); loc_el++)
    {
       ElementAccessor<3> elm = mesh_->element_accessor( mesh_->get_el_4_loc()[loc_el] );
        if (elm->boundary_idx_ != nullptr)
        {
            for(unsigned int si=0; si<elm->n_sides(); si++)
            {
                Boundary *b = elm.side(si)->cond();
                if (b != nullptr){
                    LongIdx ele_side_uid = get_boundary_edge_uid(elm.side(si));
                    be_id_map_[ele_side_uid] = be_id;
                    be_regions_.push_back(b->region().boundary_idx());
                    be_id++;
                }
            }
        }
    }
 
 
    fluxes_    .resize(n_quant, vector<double>(n_bdr_reg, 0));
    fluxes_in_ .resize(n_quant, vector<double>(n_bdr_reg, 0));
    fluxes_out_.resize(n_quant, vector<double>(n_bdr_reg, 0));
 
    masses_     .resize(n_quant, vector<double>(n_blk_reg, 0));
    sources_    .resize(n_quant, vector<double>(n_blk_reg, 0));
    sources_in_ .resize(n_quant, vector<double>(n_blk_reg, 0));
    sources_out_.resize(n_quant, vector<double>(n_blk_reg, 0));
 
    sum_fluxes_     .resize(n_quant, 0);
    sum_fluxes_in_  .resize(n_quant, 0);
    sum_fluxes_out_ .resize(n_quant, 0);
    sum_masses_     .resize(n_quant, 0);
    sum_sources_    .resize(n_quant, 0);
    sum_sources_in_ .resize(n_quant, 0);
    sum_sources_out_.resize(n_quant, 0);
 
    if (cumulative_)
    {
        initial_mass_      .resize(n_quant, 0);
        integrated_fluxes_ .resize(n_quant, 0);
        integrated_sources_.resize(n_quant, 0);
        increment_sources_.resize(n_quant, 0);
        increment_fluxes_.resize(n_quant, 0);
    }
 
 
 
    region_mass_matrix_ = new Mat[n_quant];
    be_flux_matrix_ = new Mat[n_quant];
    region_source_matrix_ = new Mat[n_quant];
    region_source_rhs_ = new Mat[n_quant];
    region_mass_vec_ = new Vec[n_quant];
    be_flux_vec_ = new Vec[n_quant];
    region_source_vec_ = new Vec[n_quant];
 
    for (unsigned int c=0; c<n_quant; ++c)
    {
        chkerr(MatCreateAIJ(PETSC_COMM_WORLD,
                n_loc_dofs_,
                (rank_==0)?mesh_->region_db().bulk_size():0,
                PETSC_DECIDE,
                PETSC_DECIDE,
                (rank_==0)?n_bulk_regs_per_dof:0,
                0,
                (rank_==0)?0:n_bulk_regs_per_dof,
                0,
                &(region_mass_matrix_[c])));
 
        chkerr(MatCreateAIJ(PETSC_COMM_WORLD,
               n_loc_dofs_,
               (rank_==0)?mesh_->region_db().bulk_size():0,
               PETSC_DECIDE,
               PETSC_DECIDE,
               (rank_==0)?n_bulk_regs_per_dof:0,
               0,
               (rank_==0)?0:n_bulk_regs_per_dof,
               0,
               &(region_source_matrix_[c])));
 
       chkerr(MatCreateAIJ(PETSC_COMM_WORLD,
               n_loc_dofs_,
               (rank_==0)?mesh_->region_db().bulk_size():0,
               PETSC_DECIDE,
               PETSC_DECIDE,
               (rank_==0)?n_bulk_regs_per_dof:0,
               0,
               (rank_==0)?0:n_bulk_regs_per_dof,
               0,
               &(region_source_rhs_[c])));
 
       chkerr(MatCreateAIJ(PETSC_COMM_WORLD,
               be_regions_.size(),  // n local rows, number of local boundary edges
               n_loc_dofs_,         // n local cols (local rows of multiplying vector)
               PETSC_DECIDE,        // n global rows
               PETSC_DECIDE,        // n global cols
               max_dofs_per_boundary_,  // allocation, local poriton
               0,
               0,
               0,
               &(be_flux_matrix_[c])));
        
        chkerr(VecCreateMPI(PETSC_COMM_WORLD,
                (rank_==0)?mesh_->region_db().bulk_size():0,
                PETSC_DECIDE,
                &(region_mass_vec_[c])));
 
        chkerr(VecCreateMPI(PETSC_COMM_WORLD,
                be_regions_.size(),
                PETSC_DECIDE,
                &(be_flux_vec_[c])));
 
        chkerr(VecCreateMPI(PETSC_COMM_WORLD,
                (rank_==0)?mesh_->region_db().bulk_size():0,
                PETSC_DECIDE,
                &(region_source_vec_[c])));
    }
 
    chkerr(MatCreateAIJ(PETSC_COMM_WORLD,
            be_regions_.size(),
            (rank_==0)?mesh_->region_db().boundary_size():0,
            PETSC_DECIDE,
            PETSC_DECIDE,
            (rank_==0)?1:0,
            0,
            (rank_==0)?0:1,
            0,
            &region_be_matrix_
            ));
    chkerr(VecGetOwnershipRange(be_flux_vec_[0], &be_offset_, NULL));
    for (unsigned int loc_el=0; loc_el<be_regions_.size(); ++loc_el)
    {
        chkerr(MatSetValue(region_be_matrix_,
                be_offset_+loc_el,
                be_regions_[loc_el],
                1,
                INSERT_VALUES));
    }
    chkerr(MatAssemblyBegin(region_be_matrix_, MAT_FINAL_ASSEMBLY));
    chkerr(MatAssemblyEnd(region_be_matrix_, MAT_FINAL_ASSEMBLY));
 
    double *ones_array;
    chkerr(VecCreateMPI(PETSC_COMM_WORLD,
            n_loc_dofs_,
            PETSC_DECIDE,
            &ones_));
    chkerr(VecGetArray(ones_, &ones_array));
    fill_n(ones_array, n_loc_dofs_, 1);
    chkerr(VecRestoreArray(ones_, &ones_array));
 
    chkerr(VecCreateMPI(PETSC_COMM_WORLD,
            be_regions_.size(),
            PETSC_DECIDE,
            &ones_be_));
    chkerr(VecGetArray(ones_be_, &ones_array));
    fill_n(ones_array, be_regions_.size(), 1);
    chkerr(VecRestoreArray(ones_be_, &ones_array));
 
 
    if (rank_ == 0) {
        // set default value by output_format_
        std::string default_file_name;
        switch (output_format_)
        {
        case txt:
            default_file_name = file_prefix_ + "_balance.txt";
            break;
        case gnuplot:
            default_file_name = file_prefix_ + "_balance.dat";
            break;
        case legacy:
            default_file_name = file_prefix_ + "_balance.txt";
            break;
        }
 
        balance_output_file_ = input_record_.val<FilePath>("file", FilePath(default_file_name, FilePath::output_file));
        try {
            balance_output_file_.open_stream(output_);
        } INPUT_CATCH(FilePath::ExcFileOpen, FilePath::EI_Address_String, input_record_)
 
 
        // set file name of YAML output
        if (do_yaml_output_) {
            string yaml_file_name = file_prefix_ + "_balance.yaml";
            FilePath(yaml_file_name, FilePath::output_file).open_stream(output_yaml_);
        }
    }
 
    allocation_done_ = true;
}
 
 
 
bool Balance::is_current()
{
    if (! balance_on_) return false;
 
    auto &marks = TimeGovernor::marks();
    bool res =  marks.current(time_->step(), balance_output_type_) != marks.end(balance_output_type_);
 
    //cout << "flag: " << res << " time: " << step.end() << " type:" << hex << balance_output_type_ << endl;
    return res;
}
 
void Balance::start_mass_assembly(unsigned int quantity_idx)
{
    lazy_initialize();
    if (! balance_on_) return;
    chkerr(MatZeroEntries(region_mass_matrix_[quantity_idx]));
    chkerr(VecZeroEntries(region_mass_vec_[quantity_idx]));
}
 
 
void Balance::start_flux_assembly(unsigned int quantity_idx)
{
    lazy_initialize();
    if (! balance_on_) return;
    chkerr(MatZeroEntries(be_flux_matrix_[quantity_idx]));
    chkerr(VecZeroEntries(be_flux_vec_[quantity_idx]));
}
 
 
void Balance::start_source_assembly(unsigned int quantity_idx)
{
    lazy_initialize();
    if (! balance_on_) return;
    chkerr(MatZeroEntries(region_source_matrix_[quantity_idx]));
    chkerr(MatZeroEntries(region_source_rhs_[quantity_idx]));
    chkerr(VecZeroEntries(region_source_vec_[quantity_idx]));
}
 
 
void Balance::finish_mass_assembly(unsigned int quantity_idx)
{
    ASSERT(allocation_done_);
    if (! balance_on_) return;
 
    chkerr(MatAssemblyBegin(region_mass_matrix_[quantity_idx], MAT_FINAL_ASSEMBLY));
    chkerr(MatAssemblyEnd(region_mass_matrix_[quantity_idx], MAT_FINAL_ASSEMBLY));
    chkerr(VecAssemblyBegin(region_mass_vec_[quantity_idx]));
    chkerr(VecAssemblyEnd(region_mass_vec_[quantity_idx]));
}
 
void Balance::finish_flux_assembly(unsigned int quantity_idx)
{
    ASSERT(allocation_done_);
    if (! balance_on_) return;
 
    chkerr(MatAssemblyBegin(be_flux_matrix_[quantity_idx], MAT_FINAL_ASSEMBLY));
    chkerr(MatAssemblyEnd(be_flux_matrix_[quantity_idx], MAT_FINAL_ASSEMBLY));
    chkerr(VecAssemblyBegin(be_flux_vec_[quantity_idx]));
    chkerr(VecAssemblyEnd(be_flux_vec_[quantity_idx]));
}
 
void Balance::finish_source_assembly(unsigned int quantity_idx)
{
    ASSERT(allocation_done_);
    if (! balance_on_) return;
 
    chkerr(MatAssemblyBegin(region_source_matrix_[quantity_idx], MAT_FINAL_ASSEMBLY));
    chkerr(MatAssemblyEnd(region_source_matrix_[quantity_idx], MAT_FINAL_ASSEMBLY));
    chkerr(MatAssemblyBegin(region_source_rhs_[quantity_idx], MAT_FINAL_ASSEMBLY));
    chkerr(MatAssemblyEnd(region_source_rhs_[quantity_idx], MAT_FINAL_ASSEMBLY));
    chkerr(MatMultTranspose(region_source_rhs_[quantity_idx], ones_, region_source_vec_[quantity_idx]));
}
 
 
 
 
void Balance::add_mass_matrix_values(unsigned int quantity_idx,
        unsigned int region_idx,
        const vector<LongIdx> &dof_indices,
        const vector<double> &values)
{
    ASSERT_DBG(allocation_done_);
    if (! balance_on_) return;
 
    PetscInt reg_array[1] = { (int)region_idx };
 
    chkerr_assert(MatSetValues(region_mass_matrix_[quantity_idx],
            dof_indices.size(),
            &(dof_indices[0]),
            1,
            reg_array,
            &(values[0]),
            ADD_VALUES));
}
 
 
void Balance::add_flux_matrix_values(unsigned int quantity_idx,
        SideIter side,
        const vector<LongIdx> &dof_indices,
        const vector<double> &values)
{
    ASSERT_DBG(allocation_done_);
    if (! balance_on_) return;
 
    PetscInt elem_array[1] = { int(be_offset_ + be_id_map_[get_boundary_edge_uid(side)]) };
    chkerr_assert(MatSetValues(be_flux_matrix_[quantity_idx],
            1,
            elem_array,
            dof_indices.size(),
            &(dof_indices[0]),
            &(values[0]),
            ADD_VALUES));
}
 
 
void Balance::add_source_matrix_values(unsigned int quantity_idx,
        unsigned int region_idx,
        const vector<LongIdx> &dof_indices,
        const vector<double> &values)
{
    ASSERT_DBG(allocation_done_);
    if (! balance_on_) return;
 
    PetscInt reg_array[1] = { (int)region_idx };
 
    chkerr_assert(MatSetValues(region_source_matrix_[quantity_idx],
            dof_indices.size(),
            &(dof_indices[0]),
            1,
            reg_array,
            &(values[0]),
            ADD_VALUES));
}
 
 
void Balance::add_mass_vec_value(unsigned int quantity_idx,
        unsigned int region_idx,
        double value)
{
  chkerr_assert(VecSetValue(region_mass_vec_[quantity_idx],
            region_idx,
            value,
            ADD_VALUES));
}
 
 
void Balance::add_flux_vec_value(unsigned int quantity_idx,
        SideIter side,
        double value)
{
    ASSERT_DBG(allocation_done_);
    if (! balance_on_) return;
 
    chkerr_assert(VecSetValue(be_flux_vec_[quantity_idx],
            be_offset_ + be_id_map_[get_boundary_edge_uid(side)],
            value,
            ADD_VALUES));
}
 
 
void Balance::add_source_vec_values(unsigned int quantity_idx,
        unsigned int region_idx,
        const vector<LongIdx> &dof_indices,
        const vector<double> &values)
{
    ASSERT_DBG(allocation_done_);
    if (! balance_on_) return;
 
    PetscInt reg_array[1] = { (int)region_idx };
 
    chkerr_assert(MatSetValues(region_source_rhs_[quantity_idx],
            dof_indices.size(),
            &(dof_indices[0]),
            1,
            reg_array,
            &(values[0]),
            ADD_VALUES));
}
 
 
void Balance::add_cumulative_source(unsigned int quantity_idx, double source)
{
    ASSERT_DBG(allocation_done_);
    if (!cumulative_) return;
 
    if (rank_ == 0)
        increment_sources_[quantity_idx] += source;
}
 
 
void Balance::calculate_cumulative(unsigned int quantity_idx,
        const Vec &solution)
{
    ASSERT_DBG(allocation_done_);
    if (!cumulative_) return;
    if (time_->tlevel() <= 0) return;
 
    // sources
    Vec bulk_vec;
 
    chkerr(VecCreateMPIWithArray(PETSC_COMM_WORLD,
            1,
            (rank_==0)?mesh_->region_db().bulk_size():0,
            PETSC_DECIDE,
            &(sources_[quantity_idx][0]),
            &bulk_vec));
 
    // compute sources on bulk regions: S'.u + s
    chkerr(VecZeroEntries(bulk_vec));
    chkerr(MatMultTransposeAdd(region_source_matrix_[quantity_idx],
            solution, region_source_vec_[quantity_idx], bulk_vec));
 
    double sum_sources;
    chkerr(VecSum(bulk_vec, &sum_sources));
    chkerr(VecDestroy(&bulk_vec));
 
    if (rank_ == 0)
        // sum sources in one step
        increment_sources_[quantity_idx] += sum_sources*time_->dt();
 
    
    // fluxes 
    Vec boundary_vec;
 
    chkerr(VecCreateMPIWithArray(PETSC_COMM_WORLD,
            1,
            (rank_==0)?mesh_->region_db().boundary_size():0,
            PETSC_DECIDE,
            &(fluxes_[quantity_idx][0]),
            &boundary_vec));
 
    // compute fluxes on boundary regions: R'.(F.u + f)
    chkerr(VecZeroEntries(boundary_vec));
    Vec temp;
    chkerr(VecDuplicate(ones_be_, &temp));
    chkerr(MatMultAdd(be_flux_matrix_[quantity_idx], solution, be_flux_vec_[quantity_idx], temp));
    // Since internally we keep outgoing fluxes, we change sign
    // to write to output _incoming_ fluxes.
    chkerr(VecScale(temp, -1));
    chkerr(MatMultTranspose(region_be_matrix_, temp, boundary_vec));
    chkerr(VecDestroy(&temp));
 
    double sum_fluxes;
    chkerr(VecSum(boundary_vec, &sum_fluxes));
    chkerr(VecDestroy(&boundary_vec));
 
    if (rank_ == 0)
        // sum fluxes in one step
        increment_fluxes_[quantity_idx] += sum_fluxes*time_->dt();
}
 
 
void Balance::calculate_mass(unsigned int quantity_idx,
        const Vec &solution,
        vector<double> &output_array)
{
    ASSERT_DBG(allocation_done_);
    if (! balance_on_) return;
 
    Vec bulk_vec;
 
    chkerr(VecCreateMPIWithArray(PETSC_COMM_WORLD,
            1,
            (rank_==0)?mesh_->region_db().bulk_size():0,
            PETSC_DECIDE,
            &(output_array[0]),
            &bulk_vec));
 
    // compute mass on regions: M'.u
    chkerr(VecZeroEntries(bulk_vec));
    chkerr(MatMultTransposeAdd(region_mass_matrix_[quantity_idx], 
                               solution, 
                               region_mass_vec_[quantity_idx], 
                               bulk_vec));
    chkerr(VecDestroy(&bulk_vec));
}
 
void Balance::calculate_instant(unsigned int quantity_idx, const Vec& solution)
{
    if ( !is_current() ) return;
    
    calculate_mass(quantity_idx, solution, masses_[quantity_idx]);
    
    // calculate source
    Vec bulk_vec;
 
    chkerr(VecCreateMPIWithArray(PETSC_COMM_WORLD,
            1,
            (rank_==0)?mesh_->region_db().bulk_size():0,
            PETSC_DECIDE,
            &(sources_[quantity_idx][0]),
            &bulk_vec));
 
    // compute sources on bulk regions: S'.u + s
    chkerr(VecZeroEntries(bulk_vec));
    chkerr(MatMultTransposeAdd(region_source_matrix_[quantity_idx],
            solution,
            region_source_vec_[quantity_idx],
            bulk_vec));
 
    // compute positive/negative sources
    int lsize;
    Vec mat_r, rhs_r;
    const double *sol_array, *mat_array, *rhs_array;
    chkerr(VecGetLocalSize(solution, &lsize));
    chkerr(VecDuplicate(solution, &mat_r));
    chkerr(VecDuplicate(solution, &rhs_r));
    chkerr(VecGetArrayRead(solution, &sol_array));
    for (unsigned int r=0; r<mesh_->region_db().bulk_size(); ++r)
    {
        MatGetColumnVector(region_source_matrix_[quantity_idx], mat_r, r);
        MatGetColumnVector(region_source_rhs_[quantity_idx], rhs_r, r);
 
        VecGetArrayRead(mat_r, &mat_array);
        VecGetArrayRead(rhs_r, &rhs_array);
 
        sources_in_[quantity_idx][r] = 0;
        sources_out_[quantity_idx][r] = 0;
        for (int i=0; i<lsize; ++i)
        {
            double f = mat_array[i]*sol_array[i] + rhs_array[i];
            if (f > 0) sources_in_[quantity_idx][r] += f;
            else sources_out_[quantity_idx][r] += f;
        }
 
        VecRestoreArrayRead(mat_r, &mat_array);
        VecRestoreArrayRead(rhs_r, &rhs_array);
    }
    chkerr(VecRestoreArrayRead(solution, &sol_array));
    chkerr(VecDestroy(&rhs_r));
    chkerr(VecDestroy(&mat_r));
    chkerr(VecDestroy(&bulk_vec));
 
    
    // calculate flux
    Vec boundary_vec;
 
    chkerr(VecCreateMPIWithArray(PETSC_COMM_WORLD, 1,
            (rank_==0)?mesh_->region_db().boundary_size():0,
            PETSC_DECIDE, &(fluxes_[quantity_idx][0]), &boundary_vec));
 
    // compute fluxes on boundary regions: R'.(F.u + f)
    chkerr(VecZeroEntries(boundary_vec));
    Vec temp;
    chkerr(VecDuplicate(ones_be_, &temp));
    chkerr(MatMultAdd(be_flux_matrix_[quantity_idx], solution, be_flux_vec_[quantity_idx], temp));
    // Since internally we keep outgoing fluxes, we change sign
    // to write to output _incoming_ fluxes.
    chkerr(VecScale(temp, -1));
    chkerr(MatMultTranspose(region_be_matrix_, temp, boundary_vec));
 
    // compute positive/negative fluxes
    fluxes_in_[quantity_idx].assign(mesh_->region_db().boundary_size(), 0);
    fluxes_out_[quantity_idx].assign(mesh_->region_db().boundary_size(), 0);
    const double *flux_array;
//  int lsize;
    chkerr(VecGetArrayRead(temp, &flux_array));
    chkerr(VecGetLocalSize(temp, &lsize));
    for (int e=0; e<lsize; ++e)
    {
        if (flux_array[e] < 0)
            fluxes_out_[quantity_idx][be_regions_[e]] += flux_array[e];
        else
            fluxes_in_[quantity_idx][be_regions_[e]] += flux_array[e];
    }
    chkerr(VecRestoreArrayRead(temp, &flux_array));
    chkerr(VecDestroy(&temp));
    chkerr(VecDestroy(&boundary_vec));
}
 
 
 
 
 
void Balance::output()
{
    ASSERT_DBG(allocation_done_);
    if (! balance_on_) return;
    if (! is_current() ) return;
    
    // gather results from processes and sum them up
    const unsigned int n_quant = quantities_.size();
    const unsigned int n_blk_reg = mesh_->region_db().bulk_size();
    const unsigned int n_bdr_reg = mesh_->region_db().boundary_size();
    const int buf_size = n_quant*2*n_blk_reg + n_quant*2*n_bdr_reg;
    double sendbuffer[buf_size], recvbuffer[buf_size];
    for (unsigned int qi=0; qi<n_quant; qi++)
    {
        for (unsigned int ri=0; ri<n_blk_reg; ri++)
        {
            sendbuffer[qi*2*n_blk_reg +           + ri] = sources_in_[qi][ri];
            sendbuffer[qi*2*n_blk_reg + n_blk_reg + ri] = sources_out_[qi][ri];
        }
        for (unsigned int ri=0; ri<n_bdr_reg; ri++)
        {
            sendbuffer[n_quant*2*n_blk_reg + qi*2*n_bdr_reg +           + ri] = fluxes_in_[qi][ri];
            sendbuffer[n_quant*2*n_blk_reg + qi*2*n_bdr_reg + n_bdr_reg + ri] = fluxes_out_[qi][ri];
        }
    }
    MPI_Reduce(&sendbuffer,recvbuffer,buf_size,MPI_DOUBLE,MPI_SUM,0,PETSC_COMM_WORLD);
    // for other than 0th process update last_time and finish,
    // on process #0 sum balances over all regions and calculate
    // cumulative balance over time.
    if (rank_ == 0)
    {
        // update balance vectors
        for (unsigned int qi=0; qi<n_quant; qi++)
        {
            for (unsigned int ri=0; ri<n_blk_reg; ri++)
            {
                sources_in_[qi][ri]  = recvbuffer[qi*2*n_blk_reg +           + ri];
                sources_out_[qi][ri] = recvbuffer[qi*2*n_blk_reg + n_blk_reg + ri];
            }
            for (unsigned int ri=0; ri<n_bdr_reg; ri++)
            {
                fluxes_in_[qi][ri]  = recvbuffer[n_quant*2*n_blk_reg + qi*2*n_bdr_reg +           + ri];
                fluxes_out_[qi][ri] = recvbuffer[n_quant*2*n_blk_reg + qi*2*n_bdr_reg + n_bdr_reg + ri];
            }
        }
    }
 
 
    // The convention for input/output of fluxes is that positive means inward.
    // Therefore in the following code we switch sign of fluxes.
    if (rank_ == 0)
    {
        sum_fluxes_.assign(n_quant, 0);
        sum_fluxes_in_.assign(n_quant, 0);
        sum_fluxes_out_.assign(n_quant, 0);
        sum_masses_.assign(n_quant, 0);
        sum_sources_.assign(n_quant, 0);
        sum_sources_in_.assign(n_quant, 0);
        sum_sources_out_.assign(n_quant, 0);
 
        // sum all boundary fluxes
        const RegionSet & b_set = mesh_->region_db().get_region_set(".BOUNDARY");
        for( RegionSet::const_iterator reg = b_set.begin(); reg != b_set.end(); ++reg)
        {
            for (unsigned int qi=0; qi<n_quant; qi++)
            {
                sum_fluxes_[qi]     += fluxes_    [qi][reg->boundary_idx()];
                sum_fluxes_in_[qi]  += fluxes_in_ [qi][reg->boundary_idx()];
                sum_fluxes_out_[qi] += fluxes_out_[qi][reg->boundary_idx()];
            }
        }
 
        // sum all volume sources
        const RegionSet & bulk_set = mesh_->region_db().get_region_set("BULK");
        for( RegionSet::const_iterator reg = bulk_set.begin(); reg != bulk_set.end(); ++reg)
        {
            for (unsigned int qi=0; qi<n_quant; qi++)
            {
                sum_masses_[qi] += masses_[qi][reg->bulk_idx()];
                sum_sources_[qi] += sources_[qi][reg->bulk_idx()];
                sum_sources_in_[qi] += sources_in_[qi][reg->bulk_idx()];
                sum_sources_out_[qi] += sources_out_[qi][reg->bulk_idx()];
            }
        }
 
        // cumulative balance over time
        if (cumulative_)
        {
            // save initial time and mass
            if (initial_)
            {
                last_time_ = time_->init_time();
                for (unsigned int qi=0; qi<n_quant; qi++)
                    initial_mass_[qi] = sum_masses_[qi];
                initial_ = false;
            }
 
            for (unsigned int qi=0; qi<n_quant; qi++)
            {
                integrated_fluxes_[qi] += increment_fluxes_[qi];
                integrated_sources_[qi] += increment_sources_[qi];
            }
        }
    }
 
    last_time_ = time_->t();
 
 
    // perform actual output
    switch (output_format_)
    {
    case txt:
        output_csv(time_->t(), '\t', "");
        break;
    case gnuplot:
        output_csv(time_->t(), ' ', "#", 30);
        break;
    case legacy:
        output_legacy(time_->t());
        break;
    }
    // output in YAML format
    output_yaml(time_->t());
 
    if (rank_ == 0)
    {
        sum_fluxes_.assign(n_quant, 0);
        sum_sources_.assign(n_quant, 0);
        increment_fluxes_.assign(n_quant, 0);
        increment_sources_.assign(n_quant, 0);
    }
}
 
 
void Balance::output_legacy(double time)
{
    // write output only on process #0
    if (rank_ != 0) return;
 
    const unsigned int n_quant = quantities_.size();
 
    // print the head of mass balance file
    unsigned int c = 6; //column number without label
    unsigned int w = 14;  //column width
    unsigned int wl = 2*(w-5)+7;  //label column width
    string bc_head_format = "# %-*s%-*s%-*s%-*s%-*s%-*s\n",
           bc_format = "%*s%-*d%-*s%-*s%-*g%-*g%-*g\n",
           bc_total_format = "# %-*s%-*s%-*g%-*g%-*g\n";
 
    output_ << "# " << setw((w*c+wl-14)/2) << setfill('-') << "--"
            << " MASS BALANCE "
            << setw((w*c+wl-14)/2) << setfill('-') << "" << endl
            << "# Time: " << (time / time_->get_coef()) << "[" << time_->get_unit_string() << "]\n\n\n";
 
    // header for table of boundary fluxes
    output_ << "# Mass flux through boundary [M/T]:\n# "
            << setiosflags(ios::left) << setfill(' ')
            << setw(w)  << "[boundary_id]"
            << setw(wl) << "[label]"
            << setw(w)  << "[substance]"
            << setw(w)  << "[total flux]"
            << setw(w)  << "[outward flux]"
            << setw(w)  << "[inward flux]"
            << endl;
 
    // draw long line
    output_ << "# " << setw(w*c+wl) << setfill('-') << "" << setfill(' ') << endl;
 
    // print mass fluxes over boundaries
    const RegionSet & b_set = mesh_->region_db().get_region_set(".BOUNDARY");
    for( RegionSet::const_iterator reg = b_set.begin(); reg != b_set.end(); ++reg) {
        for (unsigned int qi=0; qi<n_quant; qi++) {
            output_ << setw(2)  << ""
                    << setw(w)  << (int)reg->id()
                    << setw(wl) << reg->label().c_str()
                    << setw(w)  << quantities_[qi].name_.c_str()
                    << setw(w)  << fluxes_[qi][reg->boundary_idx()]
                    << setw(w)  << fluxes_out_[qi][reg->boundary_idx()]
                    << setw(w)  << fluxes_in_[qi][reg->boundary_idx()]
                    << endl;
        }
    }
 
    // draw long line
    output_ << "# " << setw(w*c+wl) << setfill('-') << "" << setfill(' ') << endl;
 
    // total boundary balance
    for (unsigned int qi=0; qi<n_quant; qi++)
        output_ << "# " << setiosflags(ios::left)
                << setw(w+wl) << "Total mass flux of substance [M/T]"
                << setw(w)    << quantities_[qi].name_.c_str()
                << setw(w)    << sum_fluxes_[qi]
                << setw(w)    << sum_fluxes_out_[qi]
                << setw(w)    << sum_fluxes_in_[qi]
                << endl;
    output_ << "\n\n";
 
 
    // header for table of volume sources and masses
    string src_head_format = "# %-*s%-*s%-*s%-*s%-*s\n",
           src_format = "%*s%-*d%-*s%-*s%-*g%-*g\n",
           src_total_format = "# %-*s%-*s%-*g%-*g\n";
    output_ << "# Mass [M] and sources [M/T] on regions:\n"
            << "# " << setiosflags(ios::left)
            << setw(w)  << "[region_id]"
            << setw(wl) << "[label]"
            << setw(w)  << "[substance]"
            << setw(w)  << "[total_mass]"
            << setw(w)  << "[total_source]"
            << endl;
 
    // draw long line
    output_ << "# " << setw(w*c+wl) << setfill('-') << "" << setfill(' ') << endl;
 
    // print  balance of volume sources and masses
    const RegionSet & bulk_set = mesh_->region_db().get_region_set("BULK");
    for( RegionSet::const_iterator reg = bulk_set.begin(); reg != bulk_set.end(); ++reg)
    {
        for (unsigned int qi=0; qi<n_quant; qi++)
        {
            output_ << setw(2)  << ""
                    << setw(w)  << (int)reg->id()
                    << setw(wl) << reg->label().c_str()
                    << setw(w)  << quantities_[qi].name_.c_str()
                    << setw(w)  << masses_[qi][reg->bulk_idx()]
                    << setw(w)  << sources_[qi][reg->bulk_idx()]
                    << endl;
        }
    }
 
    // draw long line
    output_ << "# " << setw(w*c+wl) << setfill('-') << "" << setfill(' ') << endl;
 
    // total sources balance
    for (unsigned int qi=0; qi<n_quant; qi++)
        output_ << "# " << setiosflags(ios::left) << setw(w+wl) << "Total mass [M] and sources [M/T]"
                << setw(w) << quantities_[qi].name_.c_str()
                << setw(w) << sum_masses_[qi]
                << setw(w) << sum_sources_[qi]
                << endl;
 
    if (cumulative_)
    {
        // Print cumulative sources
        output_ << "\n\n# Cumulative mass balance on time interval ["
                << setiosflags(ios::left) << time_->init_time() << ","
                << setiosflags(ios::left) << time << "]\n"
                << "# Initial mass [M] + sources integrated over time [M] - flux integrated over time [M] = current mass [M]\n"
                << "# " << setiosflags(ios::left)
                << setw(w) << "[substance]"
                << setw(w) << "[A=init. mass]"
                << setw(w) << "[B=source]"
                << setw(w) << "[C=flux]"
                << setw(w) << "[A+B-C]"
                << setw(w) << "[D=curr. mass]"
                << setw(w) << "[A+B-C-D=err.]"
                << setw(w) << "[rel. error]"
                << endl;
 
        for (unsigned int qi=0; qi<n_quant; qi++)
        {
            double denominator = max(fabs(initial_mass_[qi]+integrated_sources_[qi]-integrated_fluxes_[qi]),fabs(sum_masses_[qi]));
            output_ << "  " << setiosflags(ios::left)
                    << setw(w) << quantities_[qi].name_.c_str()
                    << setw(w) << initial_mass_[qi]
                    << setw(w) << integrated_sources_[qi]
                    << setw(w) << integrated_fluxes_[qi]
                    << setw(w) << initial_mass_[qi]+integrated_sources_[qi]-integrated_fluxes_[qi]
                    << setw(w) << sum_masses_[qi]
                    << setw(w) << initial_mass_[qi]+integrated_sources_[qi]-integrated_fluxes_[qi]-sum_masses_[qi]
                    << setw(w) << fabs(initial_mass_[qi]+integrated_sources_[qi]+integrated_fluxes_[qi]-sum_masses_[qi])/(denominator==0?1:denominator)
                    << endl;
        }
    }
 
    output_ << endl << endl;
}
 
 
std::string Balance::csv_zero_vals(unsigned int cnt, char delimiter)
{
    std::stringstream ss;
    for (unsigned int i=0; i<cnt; i++) ss << format_csv_val(0, delimiter);
    return ss.str();
}
 
 
void Balance::output_csv(double time, char delimiter, const std::string& comment_string, unsigned int repeat)
{
    // write output only on process #0
    if (rank_ != 0) return;
 
    const unsigned int n_quant = quantities_.size();
 
    // print data header only on first line
    if (repeat==0 && output_line_counter_==0) format_csv_output_header(delimiter, comment_string);
 
    // print sources and masses over bulk regions
    const RegionSet & bulk_set = mesh_->region_db().get_region_set("BULK");
    for( RegionSet::const_iterator reg = bulk_set.begin(); reg != bulk_set.end(); ++reg)
    {
        for (unsigned int qi=0; qi<n_quant; qi++)
        {
            // print data header (repeat header after every "repeat" lines)
            if (repeat && (output_line_counter_%repeat == 0)) format_csv_output_header(delimiter, comment_string);
 
            output_ << format_csv_val(time / time_->get_coef(), delimiter, true)
                    << format_csv_val(reg->label(), delimiter)
                    << format_csv_val(quantities_[qi].name_, delimiter)
                    << csv_zero_vals(3, delimiter)
                    << format_csv_val(masses_[qi][reg->bulk_idx()], delimiter)
                    << format_csv_val(sources_[qi][reg->bulk_idx()], delimiter)
                    << format_csv_val(sources_in_[qi][reg->bulk_idx()], delimiter)
                    << format_csv_val(sources_out_[qi][reg->bulk_idx()], delimiter)
                    << csv_zero_vals(5, delimiter) << endl;
            ++output_line_counter_;
        }
    }
 
    // print mass fluxes over boundaries
    const RegionSet & b_set = mesh_->region_db().get_region_set(".BOUNDARY");
    for( RegionSet::const_iterator reg = b_set.begin(); reg != b_set.end(); ++reg)
    {
        for (unsigned int qi=0; qi<n_quant; qi++) {
            // print data header (repeat header after every "repeat" lines)
            if (repeat && (output_line_counter_%repeat == 0)) format_csv_output_header(delimiter, comment_string);
 
            output_ << format_csv_val(time / time_->get_coef(), delimiter, true)
                    << format_csv_val(reg->label(), delimiter)
                    << format_csv_val(quantities_[qi].name_, delimiter)
                    << format_csv_val(fluxes_[qi][reg->boundary_idx()], delimiter)
                    << format_csv_val(fluxes_in_[qi][reg->boundary_idx()], delimiter)
                    << format_csv_val(fluxes_out_[qi][reg->boundary_idx()], delimiter)
                    << csv_zero_vals(9, delimiter) << endl;
            ++output_line_counter_;
        }
    }
 
    if (cumulative_)
    {
        for (unsigned int qi=0; qi<n_quant; qi++)
        {
            // print data header (repeat header after every "repeat" lines)
            if (repeat && (output_line_counter_%repeat == 0)) format_csv_output_header(delimiter, comment_string);
 
            double error = sum_masses_[qi] - (initial_mass_[qi] + integrated_sources_[qi] + integrated_fluxes_[qi]);
            output_ << format_csv_val(time / time_->get_coef(), delimiter, true)
                    << format_csv_val("ALL", delimiter)
                    << format_csv_val(quantities_[qi].name_, delimiter)
                    << format_csv_val(sum_fluxes_[qi], delimiter)
                    << format_csv_val(sum_fluxes_in_[qi], delimiter)
                    << format_csv_val(sum_fluxes_out_[qi], delimiter)
                    << format_csv_val(sum_masses_[qi], delimiter)
                    << format_csv_val(sum_sources_[qi], delimiter)
                    << format_csv_val(sum_sources_in_[qi], delimiter)
                    << format_csv_val(sum_sources_out_[qi], delimiter)
                    << format_csv_val(increment_fluxes_[qi], delimiter)
                    << format_csv_val(increment_sources_[qi], delimiter)
                    << format_csv_val(integrated_fluxes_[qi], delimiter)
                    << format_csv_val(integrated_sources_[qi], delimiter)
                    << format_csv_val(error, delimiter) << endl;
            ++output_line_counter_;
        }
    }
 
}
 
 
void Balance::format_csv_output_header(char delimiter, const std::string& comment_string)
{
    std::stringstream ss;
    if (delimiter == ' ') {
        ss << setw(output_column_width-comment_string.size()) << "\"time [" << time_->get_unit_string() << "]\"";
    } else {
        ss << "\"time [" << time_->get_unit_string() << "]\"";
    }
 
    output_ << comment_string << ss.str()
            << format_csv_val("region", delimiter)
            << format_csv_val("quantity [" + units_.format_text() + "]", delimiter)
            << format_csv_val("flux", delimiter)
            << format_csv_val("flux_in", delimiter)
            << format_csv_val("flux_out", delimiter)
            << format_csv_val("mass", delimiter)
            << format_csv_val("source", delimiter)
            << format_csv_val("source_in", delimiter)
            << format_csv_val("source_out", delimiter)
            << format_csv_val("flux_increment", delimiter)
            << format_csv_val("source_increment", delimiter)
            << format_csv_val("flux_cumulative", delimiter)
            << format_csv_val("source_cumulative", delimiter)
            << format_csv_val("error", delimiter)
            << endl;
}
 
std::string Balance::format_csv_val(std::string val, char delimiter, bool initial)
{
    std::stringstream ss;
    std::replace( val.begin(), val.end(), '\"', '\'');
 
    if (!initial) ss << delimiter;
    if (delimiter == ' ') {
        std::stringstream sval;
        sval << "\"" << val << "\"";
        ss << " " << setw(output_column_width-1) << sval.str();
    } else {
        ss << "\"" << val << "\"";
    }
 
    return ss.str();
}
 
std::string Balance::format_csv_val(double val, char delimiter, bool initial)
{
    std::stringstream ss;
 
    if (!initial) ss << delimiter;
    if (delimiter == ' ') {
        ss << " " << setw(output_column_width-1) << val;
    } else {
        ss << val;
    }
    return ss.str();
}
 
void Balance::output_yaml(double time)
{
 
    // write output only on process #0
    if (!do_yaml_output_  || rank_ != 0) return;
 
    const unsigned int n_quant = quantities_.size();
 
    // print data header only once
    if (!output_yaml_header_) {
        output_yaml_ << "column_names: [ flux,  flux_in,  flux_out,  mass,  source,  source_in,  source_out,  flux_increment,  "
            << "source_increment,  flux_cumulative,  source_cumulative,  error ]"  << endl;
        output_yaml_ << "data:" << endl;
        output_yaml_header_ = true;
    }
 
    output_yaml_ << setfill(' ');
 
    // print sources and masses over bulk regions
    const RegionSet & bulk_set = mesh_->region_db().get_region_set("BULK");
    for( RegionSet::const_iterator reg = bulk_set.begin(); reg != bulk_set.end(); ++reg)
    {
        for (unsigned int qi=0; qi<n_quant; qi++)
        {
            output_yaml_ << "  - time: " << (time / time_->get_coef()) << endl;
            output_yaml_ << setw(4) << "" << "region: " << reg->label() << endl;
            output_yaml_ << setw(4) << "" << "quantity: " << quantities_[qi].name_ << endl;
            output_yaml_ << setw(4) << "" << "data: " << "[ 0, 0, 0, " << masses_[qi][reg->bulk_idx()] << ", "
                         << sources_[qi][reg->bulk_idx()] << ", " << sources_in_[qi][reg->bulk_idx()] << ", "
                         << sources_out_[qi][reg->bulk_idx()] << ", 0, 0, 0, 0, 0 ]" << endl;
        }
    }
 
    // print mass fluxes over boundaries
    const RegionSet & b_set = mesh_->region_db().get_region_set(".BOUNDARY");
    for( RegionSet::const_iterator reg = b_set.begin(); reg != b_set.end(); ++reg)
    {
        for (unsigned int qi=0; qi<n_quant; qi++) {
            output_yaml_ << "  - time: " << (time / time_->get_coef()) << endl;
            output_yaml_ << setw(4) << "" << "region: " << reg->label() << endl;
            output_yaml_ << setw(4) << "" << "quantity: " << quantities_[qi].name_ << endl;
            output_yaml_ << setw(4) << "" << "data: " << "[ " << fluxes_[qi][reg->boundary_idx()] << ", "
                         << fluxes_in_[qi][reg->boundary_idx()] << ", " << fluxes_out_[qi][reg->boundary_idx()]
                         << ", 0, 0, 0, 0, 0, 0, 0, 0, 0 ]" << endl;
        }
    }
 
    if (cumulative_)
    {
        for (unsigned int qi=0; qi<n_quant; qi++)
        {
            double error = sum_masses_[qi] - (initial_mass_[qi] + integrated_sources_[qi] + integrated_fluxes_[qi]);
            output_yaml_ << "  - time: " << (time / time_->get_coef()) << endl;
            output_yaml_ << setw(4) << "" << "region: ALL" << endl;
            output_yaml_ << setw(4) << "" << "quantity: " << quantities_[qi].name_ << endl;
            output_yaml_ << setw(4) << "" << "data: " << "[ " << sum_fluxes_[qi] << ", "
                         << sum_fluxes_in_[qi] << ", " << sum_fluxes_out_[qi] << ", "
                         << sum_masses_[qi] << ", " << sum_sources_[qi] << ", "
                         << sum_sources_in_[qi] << ", " << sum_sources_out_[qi] << ", "
                         << increment_fluxes_[qi] << ", " << increment_sources_[qi] << ", "
                         << integrated_fluxes_[qi] << ", " << integrated_sources_[qi] << ", "
                         << error << " ]" << endl;
        }
    }
}