step-83.h

Go to the documentation of this file.

 
)
{
  ar & values;
}
@endcode
 
In other words, the Tensor class stores an array of `dim` objects of a
tensor of one rank lower, and in order to serialize this array, the
function simply uses the overloaded `operator&` which recognizes that
`values` is a C-style array of fixed length, and what the data type of
the elements of this array are. It will then in turn call the
`serialize()` member function of `Tensor<rank-1,dim>`. (The recursion
has a stopping point which is not of importance for this discussion. The
actual implementation of class Tensor looks different, but you get the
general idea here.)
 
Depending on whether the `Archive` type used for the first argument of
this function is an input or output archive, `operator&` reads from or
writes into the `values` member variable. If the class had more member
variables, one would list them one after the other. The same is true
if a class has base classes. For example, here is the corresponding
function for the Quadrature class that stores quadrature points and
weights, and is derived from the Subscriptor base class:
@code
template <int dim>
template <class Archive>
inline void
Quadrature<dim>::serialize(Archive &ar, const unsigned int)
{
  // Forward to the (de)serialization function in the base class:
  ar & static_cast<Subscriptor &>(*this);
 
  // Then (de)serialize the member variables:
  ar & quadrature_points & weights;
}
@endcode
The beauty of having to write only one function for both serialization
and deserialization is that obviously one has to read data from the
archive in the same order as one writes into it, and that is easy
to get wrong if one had to write two separate functions -- but that
is automatically correct if one has the same function for both
purposes!
 
The BOOST serialization library also has mechanisms if one wants to
write separate save and load functions. This is useful if a class
stores a lot of data in one array and then has a cache of commonly
accessed data on the side for fast access; one would then only store
and load the large array, and in the load function re-build the cache
from the large array. (See also the discussion below on what one
actually wants to save.)
 
Based on these principles, let us consider how one would  (de)serialize
a program such as @ref step_19 "step-19". Recall that its principal class looked
essentially as follows:
@code
  template <int dim>
  class CathodeRaySimulator
  {
  public:
    CathodeRaySimulator();
 
    void run();
 
  private:
    void make_grid();
    void setup_system();
    void assemble_system();
    void solve_field();
    void refine_grid();
 
    void create_particles();
    void move_particles();
    void track_lost_particle(const typename Particles::ParticleIterator<dim> &        particle,
                             const typename Triangulation<dim>::active_cell_iterator &cell);
 
 
    void update_timestep_size();
    void output_results() const;
 
    Triangulation<dim>        triangulation;
    MappingQGeneric<dim>      mapping;
    FE_Q<dim>                 fe;
    DoFHandler<dim>           dof_handler;
    AffineConstraints<double> constraints;
 
    SparseMatrix<double>      system_matrix;
    SparsityPattern           sparsity_pattern;
 
    Vector<double>            solution;
    Vector<double>            system_rhs;
 
    Particles::ParticleHandler<dim> particle_handler;
    types::particle_index           next_unused_particle_id;
    types::particle_index           n_recently_lost_particles;
    types::particle_index           n_total_lost_particles;
    types::particle_index           n_particles_lost_through_anode;
 
    DiscreteTime time;
  };
@endcode
One would then first write a
function that allows (de)serialization of the data of this class
as follows:
@code
  template <int dim>
  template <class Archive>
  void CathodeRaySimulator<dim>::serialize(Archive &ar,
                                           const unsigned int /* version */)
  {
    ar & triangulation;
    ar & solution;
    ar & particle_handler;
    ar & next_unused_particle_id;
    ar & n_recently_lost_particles;
    ar & n_total_lost_particles;
    ar & n_particles_lost_through_anode;
    ar & time;
  }
@endcode
As discussed below, this is not the entire list of member variables: We
only want to serialize those that we cannot otherwise re-generate.
 
Next, we need a function that writes a checkpoint by creating an "output
archive" and using the usual `operator<<` to put an object into this archive. In
the code of this program, this will then look as follows (with some
minor modifications we will discuss later):
@code
  template <int dim>
  void CathodeRaySimulator<dim>::checkpoint()
  {
    std::ofstream                 checkpoint_file("checkpoint");
    boost::archive::text_oarchive archive(checkpoint_file,
                                          boost::archive::no_header);
 
    archive << *this;
  }
@endcode
 
What `operator<<` does here is to call the serialization functions of the right hand
operand (here, the `serialize()` function described above, with an output
archive as template argument), and create a serialized representation of the data. In the
end, serialization has put everything into the "archive" from which one
can extract a (sometimes very long) string that one can save in bulk
into a file, and that is exactly what happens when the destructor of the
`archive` variable is called.
 
BOOST serialization offers different archives, including ones that
store the data in text format (as we do above), in binary format, or
already compressed with something like gzip to minimize the amount of
space necessary. The specific type of archive to be used is selected
by the type of the `archive` variable above (and the corresponding
variable in the `restart()` function of course). This program uses a
text archive so that you can look at how a serialized representation
would actually look at, though a "real" program would of course try to
be more space efficient by using binary (and possibly compressed)
representations of the data.
 
In any case, the data we have thus created is read in very similarly
in the following function (again with some minor modifications):
@code
  template <int dim>
  void CathodeRaySimulator<dim>::restart()
  {
    std::ifstream checkpoint_file("checkpoint");
 
    boost::archive::text_iarchive archive(checkpoint_file,
                                          boost::archive::no_header);
 
    archive >> *this;
  }
@endcode
 
The magic of this approach is that one doesn't actually have to write very
much code to checkpoint or restart programs: It is all hidden in the
serialization functions of classes such as Triangulation,
Particles::ParticleHandler, etc., which the deal.II library provides for you.
 
 
<a name="step_83-Serializationofparallelprograms"></a><h4> Serialization of parallel programs </h4>
 
 
The program as provided here is sequential, i.e., it runs on a single
processor just like the original @ref step_19 "step-19" did. But what if you had a parallel
program -- say, something like @ref step_40 "step-40" -- that runs in parallel with MPI?
In that case, serialization and checkpoint/restart becomes more complicated.
While parallel execution is not something that is of concern to us
in this program, it is an issue that has influenced the design of how
deal.II does serialization; as a consequence, we need to talk through
what makes serialization of parallel programs more difficult in order
to understand why this program does things the way it does.
 
Intuitively, one might simply want to use the same idea as we used
here except that we let every MPI process serialize its own data, and
read its own data. This works, but there are some drawbacks:
- There is a certain subset of data that is replicated across all
  MPI processes and that would be then written by all processes. An example
  is the `time` data structure that stores the current time, time step
  size, and other information, and that better be the same on all processes.
  Typically, the replicated data isn't a large fraction of a program's
  overall memory usage, and perhaps writing it more than once isn't
  going to be too much of a problem, but it is unsatisfactory anyway
  to have this kind of data on disk more than once.
- One would have to think in detail how exactly one wants to represent
  the data on disk. One possibility would be for every MPI process to write
  its own file. On the other hand, checkpointing is most useful for large
  programs, using large numbers of processes -- it is not uncommon to use
  checkpointing on programs that run on 10,000 or more processors in parallel.
  This would then lead to 10,000 or more files on disk. That's unpleasant, and
  probably inefficient as well. We could address this by first letting all
  the processes serialize into a string in memory (using `std::ostringstream`)
  and then collating all of these strings into one file. The MPI I/O sub-system
  has facilities to make that happen, for example, but it will require a bit
  of thought not the least because the serialized data from each process
  will likely result in strings of different sizes.
- Perhaps the most important reason to rethink how one does things in parallel
  is because, with a little bit of thought, it is possible to checkpoint a
  program running with @f$N@f$ MPI processes and restart it with @f$M\neq N@f$
  processes. This may, at first, seem like a pointless exercise, but it is
  useful if one had, for example, a program that repeatedly refines the mesh
  and where it is inefficient to run the early refinement steps with a
  coarse mesh on too many processes, whereas it is too slow to run the later
  refinement steps with a fine mesh on too few processes.
 
In order to address these issues, in particular the last one, the right approach
is to deviate a bit from the simple scheme of having a `serialize()` function
that simply serializes/deserializes *everything* into an archive, and then have two
functions `checkpoint()` and `restart()` that for all practical purposes defer
all the work to the `serialize()` function. Instead, one splits all data into
two categories:
- Data that is tied to the cells of a triangulation. This includes the mesh itself,
  but also the particles in the Particles::ParticleHandler class, and most
  importantly the solution vector(s). The way to serialize such data is to
  attach the data to cells and then let the Triangulation class serialize the attached
  data along with its own data. If this is done in a way so that we can re-load
  a triangulation on a different number of processes than the data was written,
  then this automatically also ensures that we can restore solution vectors
  and Particles::ParticleHandler objects (and everything else we can attach
  to the cells of a triangulation) on a different number of processes.
- Other data. In finite element programs, this data is almost always replicated
  across processes, and so it is enough if the "root" process (typically the
  process with MPI rank zero) writes it to disk. Upon restart, the root
  process reads the data from disk, sends it to all other processes (however many
  of them there may be), and these then initialize their own copies of the
  replicated data structures.
 
These sorts of considerations have influenced the design of the Triangulation and
Particles::ParticleHandler classes. In particular, Particles::ParticleHandler's
`serialize()` function only serializes the "other data" category, but not the
actual particles; these can instead be attached to the triangulation by calling
Particles::ParticleHandler::prepare_for_serialization(), and then one can
call Triangulation::save() to actually write this information into a set of
files that become part of the checkpoint. Upon restart, we then first call
Triangulation::load(), followed by Particles::ParticleHandler::deserialize()
to retrieve the particles from the cells they are attached to.
 
(We could, with relatively minimal effort, use the same scheme for the solution
vector: The SolutionTransfer class can be used to attach the values of degrees
of freedom to cells, and then Triangulation::save() also writes these into
checkpoint files. SolutionTransfer would then be able to re-create the solution
vector upon restart in a similar way. However, in contrast to
Particles::ParticleHandler, the Vector class we use for the solution vector
can actually serialize itself completely, and so we will go with this
approach and save ourselves the dozen or so additional lines of code.)
 
Finally, even though we wrote the `serialize()` function above in such
a way that it also serializes the `triangulation` member variable, in practice
the call to Triangulation::save() we needed to deal with the particles *also*
saves the same kind of information, and Triangulation::load() reads it.
In other words, we are saving redundant information; in the actual
implementation of the program, we therefore skip the call to
@code
  ar & triangulation;
@endcode
We do still need to say
@code
  ar & particle_handler;
@endcode
because the information attached to the cells of the triangulation only contains
information about the particles themselves, whereas the previous line is
necessary to store information such as how many particles there are, what the
next unused particle index is, and other internal information about the class.
 
 
<a name="step_83-Checkpointingstrategies"></a><h3>Checkpointing strategies</h3>
 
 
Having discussed the general idea of checkpoint/restart, let us turn
to some more specific questions one has to answer: (i) What do we
actually want to save/restore? (ii) How often do we want to write
checkpoints?
 
 
<a name="step_83-Whattosaverestore"></a><h4>What to save/restore</h4>
 
 
We will base this tutorial on @ref step_19 "step-19", and so let us use it as an
example in this section. Recall that that program simulates an
electric field in which particles move from the electrode on one
side to the other side of the domain, i.e., we have both field-based
and particle-based information to store.
 
Recall the main class of @ref step_19 "step-19", which
had quite a lot of member variables one might want to
(de)serialize:
@code
  template <int dim>
  class CathodeRaySimulator
  {
  public:
    CathodeRaySimulator();
 
    void run();
 
  private:
    [... member functions ...]
 
    Triangulation<dim>        triangulation;
    MappingQGeneric<dim>      mapping;
    FE_Q<dim>                 fe;
    DoFHandler<dim>           dof_handler;
    AffineConstraints<double> constraints;
 
    SparseMatrix<double>      system_matrix;
    SparsityPattern           sparsity_pattern;
 
    Vector<double>            solution;
    Vector<double>            system_rhs;
 
    Particles::ParticleHandler<dim> particle_handler;
    types::particle_index           next_unused_particle_id;
    types::particle_index           n_recently_lost_particles;
    types::particle_index           n_total_lost_particles;
    types::particle_index           n_particles_lost_through_anode;
 
    DiscreteTime time;
  };
@endcode
Do we really need to save all of these to disk? That would presumably
lead to quite a lot of data that needs to be stored and, if necessary,
re-loaded.
 
In practice, one does not save all of this information, but only what
cannot be reasonably re-computed in different ways. What is saved
should also depend on also *where* in the program's algorithm one
currently is, and one generally finds a convenient point at which not
so much data needs to be stored. For the
current example of @ref step_19 "step-19", a time dependent problem, one could apply
the following considerations:
 
- The program runs with the same finite element every time, so there
  is no need to actually save the element: We know what polynomial
  degree we want, and can just re-generate the element upon
  restart. If the polynomial degree was a run-time parameter, then
  maybe we should serialize the polynomial degree rather than all of
  the myriad data structures that characterize a FE_Q object, given
  that we can always re-generate the object by just knowing its
  polynomial degree. This is the classical trade-off of space vs time:
  We can achieve the same outcome by saving far less data if we are
  willing to offer a bit of CPU time to regenerate all of the internal
  data structures of the FE_Q given the polynomial degree.
 
- We rebuild the matrix and sparsity pattern in each time step from
  the DoFHandler and the finite element. These are quite large data
  structures, but they are conceptually easy to re-create again as
  necessary. So they do not need to be saved to disk, and this is
  going to save quite a lot of space. Furthermore, we really only need
  the matrix for the linear solve; once we are done with the linear
  solve in the `solve_field()` function, the contents of the matrix
  are no longer used and are, indeed, overwritten in the next time
  step. As a consequence, there would really only be a point in saving
  the matrix if we did the checkpointing between the assembly and the
  linear solve -- but maybe that is just not a convenient point for
  this operation, and we should pick a better location. In practice,
  one generally puts the checkpointing at either the very end or the
  very beginning of the time stepping loop, given that this is the
  point where the number of variables whose values are currently
  active is minimal.
 
- We also do not need to save the DoFHandler object: If we know the
  triangulation, we can always just create a DoFHandler object during
  restart to enumerate degrees of freedom in the same way as we did
  the last time before a previous program run was checkpointed. In
  fact, the example implementation of the `checkpoint()` function
  shown above did not serialize the DoFHandler object for this very
  reason. On the other hand, we probably do want to save the
  Triangulation here given that the triangulation is not statically
  generated once at the beginning of the program and then never
  changed, but is dynamically adapted every few time steps. In order
  to re-generate the triangulation, we would therefore have to save
  which cells were refined/coarsened and when (i.e., the *history* of
  the triangulation), and this would likely cost substantially more
  disk space for long-running computations than just saving the
  triangulation itself.
 
Similar considerations can be applied to all member variables: Can we
re-generate their values with relatively little effort (in which case
they do not have to be saved) or is their state difficult or
impossible to re-generate if it is not saved to disk (in which case
the variable needs to be serialized)?
 
@note If you have carefully read @ref step_19 "step-19", you might now realize that
  strictly speaking, we do not *need* to checkpoint to solution vector.
  This is because the solution vector represents the electric field,
  which the program solves for at the beginning of each timestep and
  that this solve does not make reference to the electric field at
  previous time steps -- in other words, the electric field is not
  a "history variable": If we know the domain, the mesh, the finite
  element, and the positions of the particles, we can recompute the
  solution vector, and consequently we would not have to save it
  into the checkpoint file. However, this is perhaps more work than
  we want to do for checkpointing (which you will see is otherwise
  rather little code) and so, for pedagological purposes, we simply
  save the solution vector along with the other variables that
  actually do represent the history of the program.
 
 
<a name="step_83-Howpreciselyshouldwesavethedataofacheckpoint"></a><h4>How precisely should we save the data of a checkpoint</h4>
 
 
Recall that the goal of checkpointing is to end up with a safe copy of
where the program currently is in its computations. As a consequence,
we need to make sure that we do not end up in a situation where, for
example, we start overwriting the previous checkpoint file and
somewhere halfway through the serialization process, the machine
crashes and we end up with an aborted program and no functional
checkpoint file.
 
Instead, the procedure one generally follows to guard against this
kind of scenario is that checkpoints are written into a file that is
*separate* from the previous checkpoint file; only once we are past
the writing process and the file is safely on disk can we replace the
previous checkpoint file by the one just written -- that is, we *move*
the new file into place of the old one. You will see in the code how
this two-step process is implemented in practice.
 
The situation is made slightly more complicated by the fact that
in the program below, a "checkpoint" actually consists of a number
of files -- one file into which we write the program's member
variables, and several into which the triangulation puts its
information. We then would have to rename several files,
preferrably as a single, "atomic" operation that cannot be
interrupted. Implementing this is tricky and non-trivial (though
possible), and so we will not show this part and instead just
assume that nothing will happen between renaming the first
and the last of the files -- maybe not a great strategy in
general, but good enough for this tutorial program.
 
 
<a name="step_83-Howoftentosaverestore"></a><h4>How often to save/restore</h4>
 
 
Now that we know *what* we want to save and how we want to restore it,
we need to answer the question *how often* we want to checkpoint the
program. At least theoretically, this question has been answered many
decades ago already, see @cite Young1974 and @cite Daly2006. In
practice (as actually also in these theoretical derivations), it comes
down to (i) how long it takes to checkpoint data, and (ii) how
frequently we expect that the stored data will have to be used, i.e.,
how often the system crashes.
 
For example, if it takes five minutes to save the state of the
program, then we probably do not want to write a checkpoint every ten
minutes. On the other hand, if it only takes five seconds, then maybe
ten minutes is a reasonable frequency if we run on a modest 100 cores
and the machine does not crash very often, given that in that case the
overhead is only approximately 1%. Finally, if it takes five seconds
to save the state, but we are running on 100,000 processes (i.e., a
very expensive simulation) and the machine frequently crashes, then
maybe we are willing to offer a 5% penalty in the overall run time and
write a checkpoint every minute and a half given that we lose far less
work this way on average if the machine crashes at an unpredictable
moment in our computations. The papers cited above essentially just
formalize this sort of consideration.
 
In the program, we will not dwell on this and simply choose an ad-hoc
value of saving the state every ten time steps: That's too often in
practice, but is useful for experiencing how this works in
practice without having to run the program too long.
 *
 *
 * <a name="step_83-CommProg"></a>
 * <h1> The commented program</h1>
 * 
 * 
 * <a name="step_83-Includefiles"></a> 
 * <h3>Include files</h3>
 * 
 
 * 
 * This program, with the exception of the checkpointing component
 * is identical to @ref step_19 "step-19", and so the following include files are
 * all the same:
 * 
 * @code
 *   #include <deal.II/base/quadrature_lib.h>
 *   #include <deal.II/base/discrete_time.h>
 *   
 *   #include <deal.II/lac/dynamic_sparsity_pattern.h>
 *   #include <deal.II/lac/full_matrix.h>
 *   #include <deal.II/lac/precondition.h>
 *   #include <deal.II/lac/solver_cg.h>
 *   #include <deal.II/lac/sparse_matrix.h>
 *   #include <deal.II/lac/vector.h>
 *   #include <deal.II/lac/affine_constraints.h>
 *   
 *   #include <deal.II/grid/tria.h>
 *   #include <deal.II/grid/grid_refinement.h>
 *   #include <deal.II/grid/grid_tools.h>
 *   
 *   #include <deal.II/fe/mapping_q.h>
 *   #include <deal.II/matrix_free/fe_point_evaluation.h>
 *   #include <deal.II/fe/fe_q.h>
 *   #include <deal.II/fe/fe_values.h>
 *   
 *   #include <deal.II/dofs/dof_handler.h>
 *   #include <deal.II/dofs/dof_tools.h>
 *   
 *   #include <deal.II/numerics/data_out.h>
 *   #include <deal.II/numerics/vector_tools.h>
 *   #include <deal.II/numerics/error_estimator.h>
 *   
 *   #include <deal.II/particles/particle_handler.h>
 *   #include <deal.II/particles/data_out.h>
 *   
 * @endcode
 * 
 * The only thing new are the following two include files. They are the ones
 * that declare the classes we use as archives for reading (`iarchive` = input
 * archive) and writing (`oarchive` = output archive) serialized data:
 * 
 * @code
 *   #include <boost/archive/text_iarchive.hpp>
 *   #include <boost/archive/text_oarchive.hpp>
 *   
 *   #include <filesystem>
 *   #include <fstream>
 *   #include <string>
 *   
 * @endcode
 * 
 * 
 * <a name="step_83-Globaldefinitions"></a> 
 * <h3>Global definitions</h3>
 * 
 
 * 
 * As is customary, we put everything that corresponds to the details of the
 * program into a namespace of its own.
 * 
 * @code
 *   namespace Step83
 *   {
 *     using namespace dealii;
 *   
 *     namespace BoundaryIds
 *     {
 *       constexpr types::boundary_id open          = 101;
 *       constexpr types::boundary_id cathode       = 102;
 *       constexpr types::boundary_id focus_element = 103;
 *       constexpr types::boundary_id anode         = 104;
 *     } // namespace BoundaryIds
 *   
 *     namespace Constants
 *     {
 *       constexpr double electron_mass   = 9.1093837015e-31;
 *       constexpr double electron_charge = 1.602176634e-19;
 *   
 *       constexpr double V0 = 1;
 *   
 *       constexpr double E_threshold = 0.05;
 *   
 *       constexpr double electrons_per_particle = 3e15;
 *     } // namespace Constants
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_83-Themainclass"></a> 
 * <h3>The main class</h3>
 * 
 
 * 
 * The following is then the main class of this program. It is,
 * fundamentally, identical to @ref step_19 "step-19" with the exception of
 * the `checkpoint()` and `restart()` functions, along with the
 * `serialize()` function we use to serialize and deserialize the
 * data this class stores. The `serialize()` function is called
 * by the BOOST serialization framework, and consequently has to
 * have exactly the set of arguments used here. Furthermore, because
 * it is called by BOOST functions, it has to be `public`; the other
 * two new functions are as always made `private`.
 *   
 
 * 
 * The `run()` function has also been modified to enable simulation restart
 * via its new argument `do_restart` that indicates whether or not to
 * start the simulation from a checkpoint.
 * 
 * @code
 *     template <int dim>
 *     class CathodeRaySimulator
 *     {
 *     public:
 *       CathodeRaySimulator();
 *   
 *       void run(const bool do_restart);
 *   
 *       template <class Archive>
 *       void serialize(Archive &ar, const unsigned int version);
 *   
 *     private:
 *       void make_grid();
 *       void setup_system();
 *       void assemble_system();
 *       void solve_field();
 *       void refine_grid();
 *   
 *       void create_particles();
 *       void move_particles();
 *       void track_lost_particle(
 *         const Particles::ParticleIterator<dim>                  &particle,
 *         const typename Triangulation<dim>::active_cell_iterator &cell);
 *   
 *       void update_timestep_size();
 *       void output_results() const;
 *   
 *       void checkpoint();
 *       void restart();
 *   
 *       Triangulation<dim>        triangulation;
 *       const MappingQ<dim>       mapping;
 *       const FE_Q<dim>           fe;
 *       DoFHandler<dim>           dof_handler;
 *       AffineConstraints<double> constraints;
 *   
 *       SparseMatrix<double> system_matrix;
 *       SparsityPattern      sparsity_pattern;
 *   
 *       Vector<double> solution;
 *       Vector<double> system_rhs;
 *   
 *       Particles::ParticleHandler<dim> particle_handler;
 *       types::particle_index           next_unused_particle_id;
 *       types::particle_index           n_recently_lost_particles;
 *       types::particle_index           n_total_lost_particles;
 *       types::particle_index           n_particles_lost_through_anode;
 *   
 *       DiscreteTime time;
 *     };
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_83-ThecodeCathodeRaySimulatorcodeclassimplementation"></a> 
 * <h3>The <code>CathodeRaySimulator</code> class implementation</h3>
 * 
 
 * 
 * 
 * <a name="step_83-Theunchangedpartsoftheclass"></a> 
 * <h4>The unchanged parts of the class</h4>
 * 
 
 * 
 * Let us start with those parts of the class that are all unchanged
 * from @ref step_19 "step-19" and about which you can learn there. We will
 * then pick up with commentary again when we get to the two new
 * functions, `checkpoint()` and `restart()`, along with how the
 * `run()` function needs to be modified:
 * 
 * @code
 *     template <int dim>
 *     CathodeRaySimulator<dim>::CathodeRaySimulator()
 *       : mapping(1)
 *       , fe(2)
 *       , dof_handler(triangulation)
 *       , particle_handler(triangulation, mapping, /*n_properties=*/dim)
 *       , next_unused_particle_id(0)
 *       , n_recently_lost_particles(0)
 *       , n_total_lost_particles(0)
 *       , n_particles_lost_through_anode(0)
 *       , time(0, 1e-4)
 *     {
 *       particle_handler.signals.particle_lost.connect(
 *         [this](const typename Particles::ParticleIterator<dim>         &particle,
 *                const typename Triangulation<dim>::active_cell_iterator &cell) {
 *           this->track_lost_particle(particle, cell);
 *         });
 *     }
 *   
 *   
 *   
 *     template <int dim>
 *     void CathodeRaySimulator<dim>::make_grid()
 *     {
 *       static_assert(dim == 2,
 *                     "This function is currently only implemented for 2d.");
 *   
 *       const double       delta = 0.5;
 *       const unsigned int nx    = 5;
 *       const unsigned int ny    = 3;
 *   
 *       const std::vector<Point<dim>> vertices 
 *         = {{0, 0},
 *            {1, 0},
 *            {2, 0},
 *            {3, 0},
 *            {4, 0},
 *            {delta, 1},
 *            {1, 1},
 *            {2, 1},
 *            {3, 1},
 *            {4, 1},
 *            {0, 2},
 *            {1, 2},
 *            {2, 2},
 *            {3, 2},
 *            {4, 2}};
 *       AssertDimension(vertices.size(), nx * ny);
 *   
 *       const std::vector<unsigned int> cell_vertices[(nx - 1) * (ny - 1)] = {
 *         {0, 1, nx + 0, nx + 1},
 *         {1, 2, nx + 1, nx + 2},
 *         {2, 3, nx + 2, nx + 3},
 *         {3, 4, nx + 3, nx + 4},
 *   
 *         {5, nx + 1, 2 * nx + 0, 2 * nx + 1},
 *         {nx + 1, nx + 2, 2 * nx + 1, 2 * nx + 2},
 *         {nx + 2, nx + 3, 2 * nx + 2, 2 * nx + 3},
 *         {nx + 3, nx + 4, 2 * nx + 3, 2 * nx + 4}};
 *   
 *       std::vector<CellData<dim>> cells((nx - 1) * (ny - 1), CellData<dim>());
 *       for (unsigned int i = 0; i < cells.size(); ++i)
 *         {
 *           cells[i].vertices    = cell_vertices[i];
 *           cells[i].material_id = 0;
 *         }
 *   
 *       GridTools::consistently_order_cells(cells);
 *       triangulation.create_triangulation(
 *         vertices,
 *         cells,
 *         SubCellData()); // No boundary information
 *   
 *       triangulation.refine_global(2);
 *   
 *       for (auto &cell : triangulation.active_cell_iterators())
 *         for (auto &face : cell->face_iterators())
 *           if (face->at_boundary())
 *             {
 *               if ((face->center()[0] > 0) && (face->center()[0] < 0.5) &&
 *                   (face->center()[1] > 0) && (face->center()[1] < 2))
 *                 face->set_boundary_id(BoundaryIds::cathode);
 *               else if ((face->center()[0] > 0) && (face->center()[0] < 2))
 *                 face->set_boundary_id(BoundaryIds::focus_element);
 *               else if ((face->center()[0] > 4 - 1e-12) &&
 *                        ((face->center()[1] > 1.5) || (face->center()[1] < 0.5)))
 *                 face->set_boundary_id(BoundaryIds::anode);
 *               else
 *                 face->set_boundary_id(BoundaryIds::open);
 *             }
 *   
 *       triangulation.refine_global(1);
 *     }
 *   
 *   
 *     template <int dim>
 *     void CathodeRaySimulator<dim>::setup_system()
 *     {
 *       dof_handler.distribute_dofs(fe);
 *   
 *       solution.reinit(dof_handler.n_dofs());
 *       system_rhs.reinit(dof_handler.n_dofs());
 *   
 *       constraints.clear();
 *       DoFTools::make_hanging_node_constraints(dof_handler, constraints);
 *   
 *       VectorTools::interpolate_boundary_values(dof_handler,
 *                                                BoundaryIds::cathode,
 *                                                Functions::ConstantFunction<dim>(
 *                                                  -Constants::V0),
 *                                                constraints);
 *       VectorTools::interpolate_boundary_values(dof_handler,
 *                                                BoundaryIds::focus_element,
 *                                                Functions::ConstantFunction<dim>(
 *                                                  -Constants::V0),
 *                                                constraints);
 *       VectorTools::interpolate_boundary_values(dof_handler,
 *                                                BoundaryIds::anode,
 *                                                Functions::ConstantFunction<dim>(
 *                                                  +Constants::V0),
 *                                                constraints);
 *       constraints.close();
 *   
 *       DynamicSparsityPattern dsp(dof_handler.n_dofs());
 *       DoFTools::make_sparsity_pattern(dof_handler,
 *                                       dsp,
 *                                       constraints,
 *                                       /* keep_constrained_dofs = */ false);
 *       sparsity_pattern.copy_from(dsp);
 *   
 *       system_matrix.reinit(sparsity_pattern);
 *     }
 *   
 *   
 *   
 *     template <int dim>
 *     void CathodeRaySimulator<dim>::assemble_system()
 *     {
 *       system_matrix = 0;
 *       system_rhs    = 0;
 *   
 *       const QGauss<dim> quadrature_formula(fe.degree + 1);
 *   
 *       FEValues<dim> fe_values(fe,
 *                               quadrature_formula,
 *                               update_values | update_gradients |
 *                                 update_quadrature_points | update_JxW_values);
 *   
 *       const unsigned int dofs_per_cell = fe.dofs_per_cell;
 *   
 *       FullMatrix<double> cell_matrix(dofs_per_cell, dofs_per_cell);
 *       Vector<double>     cell_rhs(dofs_per_cell);
 *   
 *       std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
 *   
 *       for (const auto &cell : dof_handler.active_cell_iterators())
 *         {
 *           cell_matrix = 0;
 *           cell_rhs    = 0;
 *   
 *           fe_values.reinit(cell);
 *   
 *           for (const unsigned int q_index : fe_values.quadrature_point_indices())
 *             for (const unsigned int i : fe_values.dof_indices())
 *               {
 *                 for (const unsigned int j : fe_values.dof_indices())
 *                   cell_matrix(i, j) +=
 *                     (fe_values.shape_grad(i, q_index) * // grad phi_i(x_q)
 *                      fe_values.shape_grad(j, q_index) * // grad phi_j(x_q)
 *                      fe_values.JxW(q_index));           // dx
 *               }
 *   
 *           if (particle_handler.n_particles_in_cell(cell) > 0)
 *             for (const auto &particle : particle_handler.particles_in_cell(cell))
 *               {
 *                 const Point<dim> &reference_location =
 *                   particle.get_reference_location();
 *                 for (const unsigned int i : fe_values.dof_indices())
 *                   cell_rhs(i) +=
 *                     (fe.shape_value(i, reference_location) * // phi_i(x_p)
 *                      (-Constants::electrons_per_particle *   // N
 *                       Constants::electron_charge));          // e
 *               }
 *   
 *           cell->get_dof_indices(local_dof_indices);
 *           constraints.distribute_local_to_global(
 *             cell_matrix, cell_rhs, local_dof_indices, system_matrix, system_rhs);
 *         }
 *     }
 *   
 *   
 *   
 *     template <int dim>
 *     void CathodeRaySimulator<dim>::solve_field()
 *     {
 *       SolverControl            solver_control(1000, 1e-12);
 *       SolverCG<Vector<double>> solver(solver_control);
 *   
 *       PreconditionSSOR<SparseMatrix<double>> preconditioner;
 *       preconditioner.initialize(system_matrix, 1.2);
 *   
 *       solver.solve(system_matrix, solution, system_rhs, preconditioner);
 *   
 *       constraints.distribute(solution);
 *     }
 *   
 *   
 *   
 *     template <int dim>
 *     void CathodeRaySimulator<dim>::refine_grid()
 *     {
 *       Vector<float> estimated_error_per_cell(triangulation.n_active_cells());
 *   
 *       KellyErrorEstimator<dim>::estimate(dof_handler,
 *                                          QGauss<dim - 1>(fe.degree + 1),
 *                                          {},
 *                                          solution,
 *                                          estimated_error_per_cell);
 *   
 *       GridRefinement::refine_and_coarsen_fixed_number(triangulation,
 *                                                       estimated_error_per_cell,
 *                                                       0.1,
 *                                                       0.03);
 *   
 *       triangulation.execute_coarsening_and_refinement();
 *     }
 *   
 *   
 *   
 *     template <int dim>
 *     void CathodeRaySimulator<dim>::create_particles()
 *     {
 *       FEFaceValues<dim> fe_face_values(fe,
 *                                        QMidpoint<dim - 1>(),
 *                                        update_quadrature_points |
 *                                          update_gradients |
 *                                          update_normal_vectors);
 *   
 *       std::vector<Tensor<1, dim>> solution_gradients(
 *         fe_face_values.n_quadrature_points);
 *   
 *       for (const auto &cell : dof_handler.active_cell_iterators())
 *         for (const auto &face : cell->face_iterators())
 *           if (face->at_boundary() &&
 *               (face->boundary_id() == BoundaryIds::cathode))
 *             {
 *               fe_face_values.reinit(cell, face);
 *   
 *               const FEValuesExtractors::Scalar electric_potential(0);
 *               fe_face_values[electric_potential].get_function_gradients(
 *                 solution, solution_gradients);
 *               for (const unsigned int q_point :
 *                    fe_face_values.quadrature_point_indices())
 *                 {
 *                   const Tensor<1, dim> E = solution_gradients[q_point];
 *   
 *                   if ((E * fe_face_values.normal_vector(q_point) < 0) &&
 *                       (E.norm() > Constants::E_threshold))
 *                     {
 *                       const Point<dim> &location =
 *                         fe_face_values.quadrature_point(q_point);
 *   
 *                       Particles::Particle<dim> new_particle;
 *                       new_particle.set_location(location);
 *                       new_particle.set_reference_location(
 *                         mapping.transform_real_to_unit_cell(cell, location));
 *                       new_particle.set_id(next_unused_particle_id);
 *                       particle_handler.insert_particle(new_particle, cell);
 *   
 *                       ++next_unused_particle_id;
 *                     }
 *                 }
 *             }
 *   
 *       particle_handler.update_cached_numbers();
 *     }
 *   
 *   
 *   
 *     template <int dim>
 *     void CathodeRaySimulator<dim>::move_particles()
 *     {
 *       const double dt = time.get_next_step_size();
 *   
 *       Vector<double>            solution_values(fe.n_dofs_per_cell());
 *       FEPointEvaluation<1, dim> evaluator(mapping, fe, update_gradients);
 *   
 *       for (const auto &cell : dof_handler.active_cell_iterators())
 *         if (particle_handler.n_particles_in_cell(cell) > 0)
 *           {
 *             const typename Particles::ParticleHandler<
 *               dim>::particle_iterator_range particles_in_cell =
 *               particle_handler.particles_in_cell(cell);
 *   
 *             std::vector<Point<dim>> particle_positions;
 *             for (const auto &particle : particles_in_cell)
 *               particle_positions.push_back(particle.get_reference_location());
 *   
 *             cell->get_dof_values(solution, solution_values);
 *   
 *             evaluator.reinit(cell, particle_positions);
 *             evaluator.evaluate(make_array_view(solution_values),
 *                                EvaluationFlags::gradients);
 *   
 *             {
 *               typename Particles::ParticleHandler<dim>::particle_iterator
 *                 particle = particles_in_cell.begin();
 *               for (unsigned int particle_index = 0;
 *                    particle != particles_in_cell.end();
 *                    ++particle, ++particle_index)
 *                 {
 *                   const Tensor<1, dim> &E =
 *                     evaluator.get_gradient(particle_index);
 *   
 *                   const Tensor<1, dim> old_velocity(particle->get_properties());
 *   
 *                   const Tensor<1, dim> acceleration =
 *                     Constants::electron_charge / Constants::electron_mass * E;
 *   
 *                   const Tensor<1, dim> new_velocity =
 *                     old_velocity + acceleration * dt;
 *   
 *                   particle->set_properties(new_velocity);
 *   
 *                   const Point<dim> new_location =
 *                     particle->get_location() + dt * new_velocity;
 *                   particle->set_location(new_location);
 *                 }
 *             }
 *           }
 *   
 *       particle_handler.sort_particles_into_subdomains_and_cells();
 *     }
 *   
 *   
 *   
 *     template <int dim>
 *     void CathodeRaySimulator<dim>::track_lost_particle(
 *       const typename Particles::ParticleIterator<dim>         &particle,
 *       const typename Triangulation<dim>::active_cell_iterator &cell)
 *     {
 *       ++n_recently_lost_particles;
 *       ++n_total_lost_particles;
 *   
 *       const Point<dim> current_location              = particle->get_location();
 *       const Point<dim> approximate_previous_location = cell->center();
 *   
 *       if ((approximate_previous_location[0] < 4) && (current_location[0] > 4))
 *         {
 *           const Tensor<1, dim> direction =
 *             (current_location - approximate_previous_location) /
 *             (current_location[0] - approximate_previous_location[0]);
 *   
 *           const double right_boundary_intercept =
 *             approximate_previous_location[1] +
 *             (4 - approximate_previous_location[0]) * direction[1];
 *           if ((right_boundary_intercept > 0.5) &&
 *               (right_boundary_intercept < 1.5))
 *             ++n_particles_lost_through_anode;
 *         }
 *     }
 *   
 *   
 *   
 *     template <int dim>
 *     void CathodeRaySimulator<dim>::update_timestep_size()
 *     {
 *       if (time.get_step_number() > 0)
 *         {
 *           double min_cell_size_over_velocity = std::numeric_limits<double>::max();
 *   
 *           for (const auto &cell : dof_handler.active_cell_iterators())
 *             if (particle_handler.n_particles_in_cell(cell) > 0)
 *               {
 *                 const double cell_size = cell->minimum_vertex_distance();
 *   
 *                 double max_particle_velocity(0.0);
 *   
 *                 for (const auto &particle :
 *                      particle_handler.particles_in_cell(cell))
 *                   {
 *                     const Tensor<1, dim> velocity(particle.get_properties());
 *                     max_particle_velocity =
 *                       std::max(max_particle_velocity, velocity.norm());
 *                   }
 *   
 *                 if (max_particle_velocity > 0)
 *                   min_cell_size_over_velocity =
 *                     std::min(min_cell_size_over_velocity,
 *                              cell_size / max_particle_velocity);
 *               }
 *   
 *           constexpr double c_safety = 0.5;
 *           time.set_desired_next_step_size(c_safety * 0.5 *
 *                                           min_cell_size_over_velocity);
 *         }
 *       else
 *         {
 *           const QTrapezoid<dim> vertex_quadrature;
 *           FEValues<dim> fe_values(fe, vertex_quadrature, update_gradients);
 *   
 *           std::vector<Tensor<1, dim>> field_gradients(vertex_quadrature.size());
 *   
 *           double min_timestep = std::numeric_limits<double>::max();
 *   
 *           for (const auto &cell : dof_handler.active_cell_iterators())
 *             if (particle_handler.n_particles_in_cell(cell) > 0)
 *               {
 *                 const double cell_size = cell->minimum_vertex_distance();
 *   
 *                 fe_values.reinit(cell);
 *                 fe_values.get_function_gradients(solution, field_gradients);
 *   
 *                 double max_E = 0;
 *                 for (const auto q_point : fe_values.quadrature_point_indices())
 *                   max_E = std::max(max_E, field_gradients[q_point].norm());
 *   
 *                 if (max_E > 0)
 *                   min_timestep =
 *                     std::min(min_timestep,
 *                              std::sqrt(0.5 * cell_size *
 *                                        Constants::electron_mass /
 *                                        Constants::electron_charge / max_E));
 *               }
 *   
 *           time.set_desired_next_step_size(min_timestep);
 *         }
 *     }
 *   
 *   
 *   
 *     template <int dim>
 *     class ElectricFieldPostprocessor : public DataPostprocessorVector<dim>
 *     {
 *     public:
 *       ElectricFieldPostprocessor()
 *         : DataPostprocessorVector<dim>("electric_field", update_gradients)
 *       {}
 *   
 *       virtual void evaluate_scalar_field(
 *         const DataPostprocessorInputs::Scalar<dim> &input_data,
 *         std::vector<Vector<double>> &computed_quantities) const override
 *       {
 *         AssertDimension(input_data.solution_gradients.size(),
 *                         computed_quantities.size());
 *   
 *         for (unsigned int p = 0; p < input_data.solution_gradients.size(); ++p)
 *           {
 *             AssertDimension(computed_quantities[p].size(), dim);
 *             for (unsigned int d = 0; d < dim; ++d)
 *               computed_quantities[p][d] = input_data.solution_gradients[p][d];
 *           }
 *       }
 *     };
 *   
 *   
 *   
 *     template <int dim>
 *     void CathodeRaySimulator<dim>::output_results() const
 *     {
 *       {
 *         ElectricFieldPostprocessor<dim> electric_field;
 *         DataOut<dim>                    data_out;
 *         data_out.attach_dof_handler(dof_handler);
 *         data_out.add_data_vector(solution, "electric_potential");
 *         data_out.add_data_vector(solution, electric_field);
 *         data_out.build_patches();
 *   
 *         DataOutBase::VtkFlags output_flags;
 *         output_flags.time  = time.get_current_time();
 *         output_flags.cycle = time.get_step_number();
 *         output_flags.physical_units["electric_potential"] = "V";
 *         output_flags.physical_units["electric_field"]     = "V/m";
 *   
 *         data_out.set_flags(output_flags);
 *   
 *         std::ofstream output("solution-" +
 *                              Utilities::int_to_string(time.get_step_number(), 4) +
 *                              ".vtu");
 *         data_out.write_vtu(output);
 *       }
 *   
 *       {
 *         Particles::DataOut<dim> particle_out;
 *         particle_out.build_patches(
 *           particle_handler,
 *           std::vector<std::string>(dim, "velocity"),
 *           std::vector<DataComponentInterpretation::DataComponentInterpretation>(
 *             dim, DataComponentInterpretation::component_is_part_of_vector));
 *   
 *         DataOutBase::VtkFlags output_flags;
 *         output_flags.time                       = time.get_current_time();
 *         output_flags.cycle                      = time.get_step_number();
 *         output_flags.physical_units["velocity"] = "m/s";
 *   
 *         particle_out.set_flags(output_flags);
 *   
 *         std::ofstream output("particles-" +
 *                              Utilities::int_to_string(time.get_step_number(), 4) +
 *                              ".vtu");
 *         particle_out.write_vtu(output);
 *       }
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_83-CathodeRaySimulatorserialize"></a> 
 * <h4>CathodeRaySimulator::serialize()</h4>
 * 
 
 * 
 * The first of the new function is the one that is called by the
 * BOOST Serialization framework to serialize and deserialize the
 * data of this class. It has already been discussed in the introduction
 * to this program and so does not provide any surprises. All it does is
 * write those member variables of the current class that cannot be
 * re-created easily into an archive, or read these members from it.
 * (Whether `operator &` facilitates a write or read operation depends on
 * whether the `Archive` type is an output or input archive.)
 *   
 
 * 
 * The function takes a second argument, `version`, that can be used to
 * create checkpoints that have version numbers. This is useful if one
 * evolves programs by adding more member variables, but still wants to
 * retain the ability to read checkpoint files created with earlier
 * versions of the program. The `version` variable would, in that case,
 * be used to represent which version of the program wrote the file,
 * and if necessary to read only those variables that were written with
 * that past version, finding a different way to initialize the new member
 * variables that have been added since then. We will not make use of this
 * ability here.
 *   
 
 * 
 * Finally, while the program that indents all deal.II source files format
 * the following code as
 * <div class=CodeFragmentInTutorialComment>
 * @code
 *   ar &solution;
 * @endcode
 * </div>
 * as if we are taking the address of the `triangulation` variable, the
 * way you *should* read the code is as
 * <div class=CodeFragmentInTutorialComment>
 * @code
 *   ar & solution;
 * @endcode
 * </div>
 * where `operator &` is a binary operator that could either be interpreted
 * as `operator <<` for output or `operator >>` for input.
 *   
 
 * 
 * As discussed in the introduction, we do not serialize the `triangulation`
 * member variable, instead leaving that to separate calls in the
 * `checkpoint()` and `restart()` functions below.
 * 
 * @code
 *     template <int dim>
 *     template <class Archive>
 *     void CathodeRaySimulator<dim>::serialize(Archive &ar,
 *                                              const unsigned int /* version */)
 *     {
 *       ar &solution;
 *       ar &particle_handler;
 *       ar &next_unused_particle_id;
 *       ar &n_recently_lost_particles;
 *       ar &n_total_lost_particles;
 *       ar &n_particles_lost_through_anode;
 *       ar &time;
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_83-CathodeRaySimulatorcheckpoint"></a> 
 * <h4>CathodeRaySimulator::checkpoint()</h4>
 * 
 
 * 
 * The checkpoint function of the principal class of this program is then
 * quite straightforward: We create an output file (and check that it is
 * writable), create an output archive, and then move the serialized
 * contents of the current object (i.e., the `*this` object) into the
 * archive. The use of `operator<<` here calls the `serialize()` function
 * above with an output archive as argument. When the destructor of the
 * `archive` variable is called at the end of the code block within which
 * it lives, the entire archive is written into the output file stream it
 * is associated with.
 *   
 
 * 
 * As mentioned in the introduction, "real" applications would not use
 * text-based archives as provided by the `boost::archive::text_oarchive`
 * class, but use binary and potentially compressed versions. This can
 * easily be achieved by using differently named classes, and the BOOST
 * documentation explains how to do that.
 * 
 * @code
 *     template <int dim>
 *     void CathodeRaySimulator<dim>::checkpoint()
 *     {
 *       std::cout << "--- Writing checkpoint... ---" << std::endl << std::endl;
 *   
 *       {
 *         std::ofstream checkpoint_file("tmp.checkpoint_step_83");
 *         AssertThrow(checkpoint_file,
 *                     ExcMessage(
 *                       "Could not write to the <tmp.checkpoint_step_83> file."));
 *   
 *         boost::archive::text_oarchive archive(checkpoint_file);
 *   
 *         archive << *this;
 *       }
 *   
 * @endcode
 * 
 * The second part of the serialization is all of the data that we can
 * attach to cells -- see the discussion about this in the introduction.
 * Here, the only data we attach to cells are the particles. We then
 * let the triangulation save these into a set of files that all start
 * with the same prefix as we chose above, namely "tmp.checkpoint":
 * 
 * @code
 *       particle_handler.prepare_for_serialization();
 *       triangulation.save("tmp.checkpoint");
 *   
 *   
 * @endcode
 * 
 * At this point, the serialized data of this file has ended up in a number
 * of files that all start with `tmp.checkpoint` file. As mentioned in the
 * introduction, we do not want to directly overwrite the checkpointing
 * files from the previous checkpoint operation, for fear that the program
 * may be interrupted *while writing the checkpoint files*. This would
 * result in corrupted files, and defeat the whole purpose of checkpointing
 * because one cannot restart from such a file. On the other hand, if we got
 * here, we know that the "tmp.checkpoint*" files were successfully written,
 * and we can rename it to "checkpoint*", in the process replacing the old
 * file.
 *     
 
 * 
 * We do this move operation by calling the
 * [C++ function that does the renaming of
 * files](https://en.cppreference.com/w/cpp/filesystem/rename).
 * Note that it is documented as being for all practical purposes
 * "atomic", i.e., we do not need to worry that the program may be
 * interrupted somewhere within the renaming operation itself. Of
 * course, it is possible that we get interrupted somewhere between
 * renaming one file and the next, and that presents problems in
 * itself -- in essence, we want the entire renaming operation of all of
 * these files to be atomic. With a couple dozen lines of extra code, one
 * could address this issue (using strategies that databases use frequently:
 * if one operation fails, we need to
 * [rollback](https://en.wikipedia.org/wiki/Rollback_(data_management))
 * the entire transaction). For the purposes of this program, this is
 * perhaps too much, and we will simply hope that that doesn't happen,
 * perhaps based on the belief that renaming files is much faster than
 * writing them, and that unlike writing checkpoint files, renaming does not
 * require much memory or disk space and so does not risk running out of
 * either.
 *     
 
 * 
 * As a consequence, the following code first loops over all files in
 * the current directory, picks out those that start with the string
 * "tmp.checkpoint", and puts them into a list. In a second step,
 * we loop over the list and rename each of these files to one
 * whose name consists of the "tmp.checkpoint*" file but stripped off
 * its first four characters (i.e., only the "checkpoint*" part). We use
 * this approach, rather than listing the files we want to rename,
 * because we do not actually know the names of the files written by
 * the Triangulation::save() function, though we know how each of these
 * file names starts.
 * 
 * @code
 *       std::list<std::string> tmp_checkpoint_files;
 *       for (const auto &dir_entry : std::filesystem::directory_iterator("."))
 *         if (dir_entry.is_regular_file() &&
 *             (dir_entry.path().filename().string().find("tmp.checkpoint") == 0))
 *           tmp_checkpoint_files.push_back(dir_entry.path().filename().string());
 *   
 *       for (const std::string &filename : tmp_checkpoint_files)
 *         std::filesystem::rename(filename, filename.substr(4, std::string::npos));
 *     }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_83-CathodeRaySimulatorrestart"></a> 
 * <h4>CathodeRaySimulator::restart()</h4>
 * 
 
 * 
 * The restart function of this class then simply does the opposite:
 * It opens an input file (and triggers an error if that file cannot be
 * opened), associates an input archive with it, and then reads the
 * contents of the current object from it, again using the
 * `serialize()` function from above. Clearly, since we have written
 * data into a text-based archive above, we need to use the corresponding
 * `boost::archive::text_iarchive` class for reading.
 *   
 
 * 
 * In a second step, we ask the triangulation to read in cell-attached
 * data, and then tell the Particles::ParticleHandler object to re-create
 * its information about all of the particles from the data just read.
 *   
 
 * 
 * The function ends by printing a status message about having restarted:
 * 
 * @code
 *     template <int dim>
 *     void CathodeRaySimulator<dim>::restart()
 *     {
 *       {
 *         std::ifstream checkpoint_file("checkpoint_step_83");
 *         AssertThrow(checkpoint_file,
 *                     ExcMessage(
 *                       "Could not read from the <checkpoint_step_83> file."));
 *   
 *         boost::archive::text_iarchive archive(checkpoint_file);
 *         archive >> *this;
 *       }
 *   
 *       triangulation.load("checkpoint");
 *       particle_handler.deserialize();
 *   
 *       std::cout << "--- Restarting at t=" << time.get_current_time()
 *                 << ", dt=" << time.get_next_step_size() << std::endl
 *                 << std::endl;
 *     }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_83-CathodeRaySimulatorrun"></a> 
 * <h4>CathodeRaySimulator::run()</h4>
 * 
 
 * 
 * The last member function of the principal class of this program is then the
 * driver. The driver takes a single argument to indicate if the simulation
 * is a restart. If it is not a restart, the mesh is set up and the problem is
 * solved like in @ref step_19 "step-19". If it is a restart, then we read in everything that
 * is a history variable from the checkpoint file via the `restart()`
 * function. Recall that everything that is inside the `if` block at the top
 * of the function is exactly like in @ref step_19 "step-19", as is almost everything that
 * follows:
 * 
 * @code
 *     template <int dim>
 *     void CathodeRaySimulator<dim>::run(const bool do_restart)
 *     {
 *       if (do_restart == false)
 *         {
 *           make_grid();
 *   
 *           const unsigned int n_pre_refinement_cycles = 3;
 *           for (unsigned int refinement_cycle = 0;
 *                refinement_cycle < n_pre_refinement_cycles;
 *                ++refinement_cycle)
 *             {
 *               setup_system();
 *               assemble_system();
 *               solve_field();
 *               refine_grid();
 *             }
 *         }
 *       else
 *         {
 *           restart();
 *         }
 *   
 *       setup_system();
 *       do
 *         {
 *           std::cout << "Timestep " << time.get_step_number() + 1 << std::endl;
 *           std::cout << "  Field degrees of freedom:                 "
 *                     << dof_handler.n_dofs() << std::endl;
 *   
 *           assemble_system();
 *           solve_field();
 *   
 *           create_particles();
 *           std::cout << "  Total number of particles in simulation:  "
 *                     << particle_handler.n_global_particles() << std::endl;
 *   
 *           n_recently_lost_particles = 0;
 *           update_timestep_size();
 *           move_particles();
 *   
 *           time.advance_time();
 *   
 *           output_results();
 *   
 *           std::cout << "  Number of particles lost this time step:  "
 *                     << n_recently_lost_particles << std::endl;
 *           if (n_total_lost_particles > 0)
 *             std::cout << "  Fraction of particles lost through anode: "
 *                       << 1. * n_particles_lost_through_anode /
 *                            n_total_lost_particles
 *                       << std::endl;
 *   
 *           std::cout << std::endl
 *                     << "  Now at t=" << time.get_current_time()
 *                     << ", dt=" << time.get_previous_step_size() << '.'
 *                     << std::endl
 *                     << std::endl;
 *   
 * @endcode
 * 
 * The only other difference between this program and @ref step_19 "step-19" is that
 * we checkpoint the simulation every ten time steps:
 * 
 * @code
 *           if (time.get_step_number() % 10 == 0)
 *             checkpoint();
 *         }
 *       while (time.is_at_end() == false);
 *     }
 *   } // namespace Step83
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_83-Thecodemaincodefunction"></a> 
 * <h3>The <code>main</code> function</h3>
 * 
 
 * 
 * The final function of the program is then again the `main()` function. Its
 * overall structure is unchanged in all tutorial programs since @ref step_6 "step-6" and
 * so there is nothing new to discuss about this aspect.
 * 
 
 * 
 * The only difference is that we need to figure out whether a restart was
 * requested, or whether the program should simply start from scratch when
 * called. We do this using a command line argument: The `argc` argument to
 * `main()` indicates how many command line arguments were provided when
 * the program was called (counting the name of the program as the zeroth
 * argument), and `argv` is an array of strings with as many elements as
 * `argc` that contains these command line arguments. So if you call
 * the program as
 * <div class=CodeFragmentInTutorialComment>
 * @code
 *   ./step-83
 * @endcode
 * </div>
 * then `argc` will be 1, and `argv` will be the array with one element
 * and content `[ "./step-83" ]`. On the other hand, if you call the program
 * as
 * <div class=CodeFragmentInTutorialComment>
 * @code
 *   ./step-83 restart
 * @endcode
 * </div>
 * then `argc` will be 2, and `argv` will be the array with two elements
 * and content `[ "./step-83", "restart" ]`. Every other choice should be
 * flagged as an error. The following try block then does exactly this:
 * 
 * @code
 *   int main(int argc, char *argv[])
 *   {
 *     try
 *       {
 *         Step83::CathodeRaySimulator<2> cathode_ray_simulator;
 *   
 *         if (argc == 1)
 *           cathode_ray_simulator.run(false); // no restart
 *         else if ((argc == 2) && (std::string(argv[1]) == "restart"))
 *           cathode_ray_simulator.run(true); // yes restart
 *         else
 *           {
 *             std::cerr << "Error: The only allowed command line argument to this\n"
 *                          "       program is 'restart'."
 *                       << std::endl;
 *             return 1;
 *           }
 *       }
 *     catch (std::exception &exc)
 *       {
 *         std::cerr << std::endl
 *                   << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *         std::cerr << "Exception on processing: " << std::endl
 *                   << exc.what() << std::endl
 *                   << "Aborting!" << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *   
 *         return 1;
 *       }
 *     catch (...)
 *       {
 *         std::cerr << std::endl
 *                   << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *         std::cerr << "Unknown exception!" << std::endl
 *                   << "Aborting!" << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *         return 1;
 *       }
 *     return 0;
 *   }
 * @endcode
<a name="step_83-Results"></a><h1>Results</h1>
 
 
When you run this program, it produces the following output that is
almost exactly identical to what you get from @ref step_19 "step-19":
@code
Timestep 1
  Field degrees of freedom:                 4989
  Total number of particles in simulation:  20
  Number of particles lost this time step:  0
 
  Now at t=2.12647e-07, dt=2.12647e-07.
 
Timestep 2
  Field degrees of freedom:                 4989
  Total number of particles in simulation:  40
  Number of particles lost this time step:  0
 
  Now at t=4.14362e-07, dt=2.01715e-07.
 
Timestep 3
  Field degrees of freedom:                 4989
  Total number of particles in simulation:  60
  Number of particles lost this time step:  0
 
  Now at t=5.23066e-07, dt=1.08704e-07.
 
Timestep 4
  Field degrees of freedom:                 4989
  Total number of particles in simulation:  80
  Number of particles lost this time step:  0
 
  Now at t=6.08431e-07, dt=8.53649e-08.
 
Timestep 5
  Field degrees of freedom:                 4989
  Total number of particles in simulation:  100
  Number of particles lost this time step:  0
 
  Now at t=6.81935e-07, dt=7.35039e-08.
 
Timestep 6
  Field degrees of freedom:                 4989
  Total number of particles in simulation:  120
  Number of particles lost this time step:  0
 
  Now at t=7.47864e-07, dt=6.59294e-08.
 
Timestep 7
  Field degrees of freedom:                 4989
  Total number of particles in simulation:  140
  Number of particles lost this time step:  0
 
  Now at t=8.2516e-07, dt=7.72957e-08.
 
Timestep 8
  Field degrees of freedom:                 4989
  Total number of particles in simulation:  158
  Number of particles lost this time step:  0
 
  Now at t=8.95325e-07, dt=7.01652e-08.
 
Timestep 9
  Field degrees of freedom:                 4989
  Total number of particles in simulation:  172
  Number of particles lost this time step:  0
 
  Now at t=9.67852e-07, dt=7.25269e-08.
 
Timestep 10
  Field degrees of freedom:                 4989
  Total number of particles in simulation:  186
  Number of particles lost this time step:  0
 
  Now at t=1.03349e-06, dt=6.56398e-08.
 
--- Writing checkpoint... ---
 
Timestep 11
  Field degrees of freedom:                 4989
  Total number of particles in simulation:  198
  Number of particles lost this time step:  0
 
  Now at t=1.11482e-06, dt=8.13268e-08.
 
Timestep 12
  Field degrees of freedom:                 4989
  Total number of particles in simulation:  206
  Number of particles lost this time step:  0
 
  Now at t=1.18882e-06, dt=7.39967e-08.
 
Timestep 13
  Field degrees of freedom:                 4989
  Total number of particles in simulation:  212
  Number of particles lost this time step:  0
 
  Now at t=1.26049e-06, dt=7.16705e-08.
 
[...]
@endcode
The only thing that is different is the additional line after the tenth
output (which also appears after the 20th, 30th, etc., time step) indicating
that a checkpoint file has been written.
 
Because we chose to use a text-based format for the checkpoint file (which
you should not do in production codes because it uses quite a lot of disk space),
we can actually inspect this file. It will look like this, with many many more
lines:
@code
22 serialization::archive 18 0 0 0 0 0 7 0 0 3 1 0
4989 -1.00000000000000000e+00 -1.00000000000000000e+00 -1.00000000000000000e+00 -9.96108134982226390e-01 -1.00000000000000000e+00 -9.98361082867431748e-01
[...]
@endcode
What each of these numbers represents is hard to tell in practice, and also
entirely unimporant for our current purposes -- it's
a representation of the many objects that make up this program's state, and
from which one can restore its state. The point simply being that this is what
serialization produces: A long list (a sequence) of bits that we can put
into a file, and that we can later read again to recreate the state of the
program.
 
Now here's the fun part. Let's say you hit Control-C on the command
line at the point above (say, during time step 13 or 14). There's
a set of checkpoint files on disk that saved the state after ten time steps.
Based on the logic in `main()`, we should be able to restart from
this point if we run the program with
```
  ./step-83 restart
```
Indeed, this is what then happens:
@code
--- Restarting at t=1.03349e-06, dt=6.56398e-08
 
Timestep 11
  Field degrees of freedom:                 4989
  Total number of particles in simulation:  198
  Number of particles lost this time step:  0
 
  Now at t=1.11482e-06, dt=8.13268e-08.
 
Timestep 12
  Field degrees of freedom:                 4989
  Total number of particles in simulation:  206
  Number of particles lost this time step:  0
 
  Now at t=1.18882e-06, dt=7.39967e-08.
 
[...]
@endcode
After the status message that shows that we are restarting, this is
indeed *the exact same output* for the following time steps we had
gotten previously -- in other words, saving the complete state
seems to have worked, and the program has continued as if it had
never been interrupted!
 
 
 
<a name="extensions"></a>
<a name="step_83-Possibilitiesforextensions"></a><h3>Possibilities for extensions</h3>
 
 
<a name="step_83-Makingefficiencyapriority"></a><h4> Making efficiency a priority </h4>
 
 
While the techniques we have shown here are fully sufficient for what
we want to do, namely checkpoint and restart computations, and are in
fact also fully sufficient for much larger code bases such as
[ASPECT](https://aspect.geodynamics.org), one could go beyond what is
still a relatively simple scheme.
 
Specifically, among the things we need to recognize is that writing
large amounts of data to disk is expensive and can take a good long
time to finish -- for example, for large parallel computations with,
say, a billion unknowns, checkpoints can run into the hundred gigabyte
range or beyond. One may ask whether that could be avoided, or at
least whether we can mitigate the cost.
 
One way to do that is to first serialize the state of the program into
a buffer in memory (like the `Archive` objects the `serialize()`
functions write to and read from), and once that is done, start a *separate*
thread do the writing while the rest of the program continues with computations.
This is useful because writing the data to disk often takes a
long time but not a lot of CPU power: It just takes time to move the
data through the network to the file server, and from there onto the
actual disks. This is something that might as well happen while we are
doing something useful again (namely, solving more time steps). Should
the machine crash during this phase, nothing is lost: As discussed in
the introduction, we are writing the checkpoint into a temporary file
(which will be lost in the case of a machine failure), but we have
kept the previous checkpoint around until we know that the temporary
file is complete and can be moved over the old one.
 
The only thing we have to pay attention in this background-writing
scheme is that we cannot start with creating a new checkpoint while
the previous one is still being written in the background.
 
Doing this all is not technically very difficult; the code just
requires more explanation than lines of code, and so we omit doing
this in the program here. But you can take a look at the
`MainLoop::output()` function of @ref step_69 "step-69" to see how such a code looks
like.
 
A variation of this general approach is that each process writes its
data immediately, but into files that are held on fast file systems --
say, a node-local SSD rather than a file server shared by the entire
cluster. One would then just tell the operating system to move this
file to the centeral file server in a second step, and this step can
happen in the background at whatever speed the operating system can
provide. Or perhaps one leaves *most* of these files on the fast local
file system in hopes that the restart happens before these files are
purged (say, by a script that runs nightly) and only moves these files
to the permanent file system every tenth time we create a checkpoint.
 
In all of these cases, the logic quickly becomes quite complicated. As
usual, the solution is not to re-invent the wheel: Libraries such as
[VeloC](https://www.anl.gov/mcs/veloc-very-low-overhead-transparent-multilevel-checkpointrestart),
developed by the Exascale Computing Project (ECP) already do all of
this and more, for codes that are orders of magnitude more complex
than the little example here.
 
Separately, one might want to try to reduce the amount of time it
takes to serialize objects into a buffer in memory. As mentioned
above, we use the
[BOOST serialization
library](https://www.boost.org/doc/libs/1_74_0/libs/serialization/doc/index.html)
for this task, but it is not the only player in town. One could, for
example, use the [bitsery](https://github.com/fraillt/bitsery), [cereal](https://github.com/USCiLab/cereal), or
[zpp](https://github.com/eyalz800/serializer) projects instead, which can be substantially faster than BOOST.
 *
 *
<a name="step_83-PlainProg"></a>
<h1> The plain program</h1>
@include "step-83.cc"
*/