.. _ch_performance:

Hints for Performance Tuning
----------------------------

This chapter provides hints on how to get to achieve best performance
with PETSc, particularly on distributed-memory machines with multiple
CPU sockets per node. We focus on machine-related performance
optimization here; algorithmic aspects like preconditioner selection are
not the focus of this section.

Maximizing Memory Bandwidth
~~~~~~~~~~~~~~~~~~~~~~~~~~~

Most operations in PETSc deal with large datasets (typically vectors and
sparse matrices) and perform relatively few arithmetic operations for
each byte loaded or stored from global memory. Therefore, the
*arithmetic intensity* expressed as the ratio of floating point
operations to the number of bytes loaded and stored is usually well
below unity for typical PETSc operations. On the other hand, modern CPUs
are able to execute on the order of 10 floating point operations for
each byte loaded or stored. As a consequence, almost all PETSc
operations are limited by the rate at which data can be loaded or stored
(*memory bandwidth limited*) rather than by the rate of floating point
operations.

This section discusses ways to maximize the memory bandwidth achieved by
applications based on PETSc. Where appropriate, we include benchmark
results in order to provide quantitative results on typical performance
gains one can achieve through parallelization, both on a single compute
node and across nodes. In particular, we start with the answer to the
common question of why performance generally does not increase 20-fold
with a 20-core CPU.

.. _subsec:bandwidth-vs-processes:

Memory Bandwidth vs. Processes
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Consider the addition of two large vectors, with the result written to a
third vector. Because there are no dependencies across the different
entries of each vector, the operation is embarrasingly parallel.

.. figure:: /images/docs/manual/stream-results-intel.*
   :alt: Memory bandwidth obtained on Intel hardware (dual socket except KNL) over the number of processes used. One can get close to peak memory bandwidth with only a few processes.
   :name: fig_stream_intel
   :width: 80.0%

   Memory bandwidth obtained on Intel hardware (dual socket except KNL)
   over the number of processes used. One can get close to peak memory
   bandwidth with only a few processes.

As :numref`fig_stream_intel` shows, the performance gains due to
parallelization on different multi- and many-core CPUs quickly
saturates. The reason is that only a fraction of the total number of CPU
cores is required to saturate the memory channels. For example, a
dual-socket system equipped with Haswell 12-core Xeon CPUs achieves more
than 80 percent of achievable peak memory bandwidth with only four
processes per socket (8 total), cf. :numref:`fig_stream_intel`.
Consequently, running with more than 8 MPI ranks on such a system will
not increase performance substantially. For the same reason, PETSc-based
applications usually do not benefit from hyper-threading.

PETSc provides a simple way to measure memory bandwidth for different
numbers of processes via the target ``make streams`` executed from
``$PETSC_DIR``. The output provides an overview of the possible speedup
one can obtain on the given machine (not necessarily a shared memory
system). For example, the following is the most relevant output obtained
on a dual-socket system equipped with two six-core-CPUs with
hyperthreading:

.. code-block:: none

   np  speedup
   1 1.0
   2 1.58
   3 2.19
   4 2.42
   5 2.63
   6 2.69
   ...
   21 3.82
   22 3.49
   23 3.79
   24 3.71
   Estimation of possible speedup of MPI programs based on Streams benchmark.
   It appears you have 1 node(s)

On this machine, one should expect a speed-up of typical memory
bandwidth-bound PETSc applications of at most 4x when running multiple
MPI ranks on the node. Most of the gains are already obtained when
running with only 4-6 ranks. Because a smaller number of MPI ranks
usually implies better preconditioners and better performance for
smaller problems, the best performance for PETSc applications may be
obtained with fewer ranks than there are physical CPU cores available.

Following the results from the above run of ``make streams``, we
recommend to use additional nodes instead of placing additional MPI
ranks on the nodes. In particular, weak scaling (i.e. constant load per
process, increasing the number of processes) and strong scaling
(i.e. constant total work, increasing the number of processes) studies
should keep the number of processes per node constant.

Non-Uniform Memory Access (NUMA) and Process Placement
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

CPUs in nodes with more than one CPU socket are internally connected via
a high-speed fabric, cf. :numref:`fig_numa`, to enable data
exchange as well as cache coherency. Because main memory on modern
systems is connected via the integrated memory controllers on each CPU,
memory is accessed in a non-uniform way: A process running on one socket
has direct access to the memory channels of the respective CPU, whereas
requests for memory attached to a different CPU socket need to go
through the high-speed fabric. Consequently, best aggregate memory
bandwidth on the node is obtained when the memory controllers on each
CPU are fully saturated. However, full saturation of memory channels is
only possible if the data is distributed across the different memory
channels.

.. figure:: /images/docs/manual/numa.*
   :alt: Schematic of a two-socket NUMA system. Processes should be spread across both CPUs to obtain full bandwidth.
   :name: fig_numa
   :width: 90.0%

   Schematic of a two-socket NUMA system. Processes should be spread
   across both CPUs to obtain full bandwidth.

Data in memory on modern machines is allocated by the operating system
based on a first-touch policy. That is, memory is not allocated at the
point of issuing ``malloc()``, but at the point when the respective
memory segment is actually touched (read or write). Upon first-touch,
memory is allocated on the memory channel associated with the respective
CPU the process is running on. Only if all memory on the respective CPU
is already in use (either allocated or as IO cache), memory available
through other sockets is considered.

Maximum memory bandwidth can be achieved by ensuring that processes are
spread over all sockets in the respective node. For example, the
recommended placement of a 8-way parallel run on a four-socket machine
is to assign two processes to each CPU socket. To do so, one needs to
know the enumeration of cores and pass the requested information to
``mpirun``. Consider the hardware topology information returned by
``lstopo`` (part of the hwloc package) for the following two-socket
machine, in which each CPU consists of six cores and supports
hyperthreading:

.. code-block:: none

   Machine (126GB total)
     NUMANode L#0 (P#0 63GB)
       Package L#0 + L3 L#0 (15MB)
         L2 L#0 (256KB) + L1d L#0 (32KB) + L1i L#0 (32KB) + Core L#0
           PU L#0 (P#0)
           PU L#1 (P#12)
         L2 L#1 (256KB) + L1d L#1 (32KB) + L1i L#1 (32KB) + Core L#1
           PU L#2 (P#1)
           PU L#3 (P#13)
         L2 L#2 (256KB) + L1d L#2 (32KB) + L1i L#2 (32KB) + Core L#2
           PU L#4 (P#2)
           PU L#5 (P#14)
         L2 L#3 (256KB) + L1d L#3 (32KB) + L1i L#3 (32KB) + Core L#3
           PU L#6 (P#3)
           PU L#7 (P#15)
         L2 L#4 (256KB) + L1d L#4 (32KB) + L1i L#4 (32KB) + Core L#4
           PU L#8 (P#4)
           PU L#9 (P#16)
         L2 L#5 (256KB) + L1d L#5 (32KB) + L1i L#5 (32KB) + Core L#5
           PU L#10 (P#5)
           PU L#11 (P#17)
     NUMANode L#1 (P#1 63GB)
       Package L#1 + L3 L#1 (15MB)
         L2 L#6 (256KB) + L1d L#6 (32KB) + L1i L#6 (32KB) + Core L#6
           PU L#12 (P#6)
           PU L#13 (P#18)
         L2 L#7 (256KB) + L1d L#7 (32KB) + L1i L#7 (32KB) + Core L#7
           PU L#14 (P#7)
           PU L#15 (P#19)
         L2 L#8 (256KB) + L1d L#8 (32KB) + L1i L#8 (32KB) + Core L#8
           PU L#16 (P#8)
           PU L#17 (P#20)
         L2 L#9 (256KB) + L1d L#9 (32KB) + L1i L#9 (32KB) + Core L#9
           PU L#18 (P#9)
           PU L#19 (P#21)
         L2 L#10 (256KB) + L1d L#10 (32KB) + L1i L#10 (32KB) + Core L#10
           PU L#20 (P#10)
           PU L#21 (P#22)
         L2 L#11 (256KB) + L1d L#11 (32KB) + L1i L#11 (32KB) + Core L#11
           PU L#22 (P#11)
           PU L#23 (P#23)

The relevant physical processor IDs are shown in parentheses prefixed by
``P#``. Here, IDs 0 and 12 share the same physical core and have a
common L2 cache. IDs 0, 12, 1, 13, 2, 14, 3, 15, 4, 16, 5, 17 share the
same socket and have a common L3 cache.

A good placement for a run with six processes is to locate three
processes on the first socket and three processes on the second socket.
Unfortunately, mechanisms for process placement vary across MPI
implementations, so make sure to consult the manual of your MPI
implementation. The following discussion is based on how processor
placement is done with MPICH and OpenMPI, where one needs to pass
``--bind-to core --map-by socket`` to ``mpirun``:

.. code-block:: none

   $> mpirun -n 6 --bind-to core --map-by socket ./stream
   process 0 binding: 100000000000100000000000
   process 1 binding: 000000100000000000100000
   process 2 binding: 010000000000010000000000
   process 3 binding: 000000010000000000010000
   process 4 binding: 001000000000001000000000
   process 5 binding: 000000001000000000001000
   Triad:        45403.1949   Rate (MB/s)

In this configuration, process 0 is bound to the first physical core on
the first socket (with IDs 0 and 12), process 1 is bound to the first
core on the second socket (IDs 6 and 18), and similarly for the
remaining processes. The achieved bandwidth of 45 GB/sec is close to the
practical peak of about 50 GB/sec available on the machine. If, however,
all MPI processes are located on the same socket, memory bandwidth drops
significantly:

.. code-block:: none

   $> mpirun -n 6 --bind-to core --map-by core ./stream
   process 0 binding: 100000000000100000000000
   process 1 binding: 010000000000010000000000
   process 2 binding: 001000000000001000000000
   process 3 binding: 000100000000000100000000
   process 4 binding: 000010000000000010000000
   process 5 binding: 000001000000000001000000
   Triad:        25510.7507   Rate (MB/s)

All processes are now mapped to cores on the same socket. As a result,
only the first memory channel is fully saturated at 25.5 GB/sec.

| One must not assume that ``mpirun`` uses good defaults. To
  demonstrate, compare the full output of ``make streams`` from
  :any:`subsec:bandwidth-vs-processes` on the left with
  the results on the right obtained by passing
  ``--bind-to core --map-by socket``:

.. code-block:: none

   $> make streams
   np  speedup
   1 1.0
   2 1.58
   3 2.19
   4 2.42
   5 2.63
   6 2.69
   7 2.31
   8 2.42
   9 2.37
   10 2.65
   11 2.3
   12 2.53
   13 2.43
   14 2.63
   15 2.74
   16 2.7
   17 3.28
   18 3.66
   19 3.95
   20 3.07
   21 3.82
   22 3.49
   23 3.79
   24 3.71

.. code-block:: none

   $> make streams MPI_BINDING="--bind-to core --map-by socket"
   np  speedup
   1 1.0
   2 1.59
   3 2.66
   4 3.5
   5 3.56
   6 4.23
   7 3.95
   8 4.39
   9 4.09
   10 4.46
   11 4.15
   12 4.42
   13 3.71
   14 3.83
   15 4.08
   16 4.22
   17 4.18
   18 4.31
   19 4.22
   20 4.28
   21 4.25
   22 4.23
   23 4.28
   24 4.22

|
| For the non-optimized version on the left, the speedup obtained when
  using any number of processes between 3 and 13 is essentially constant
  up to fluctuations, indicating that all processes were by default
  executed on the same socket. Only with 14 or more processes, the
  speedup number increases again. In contrast, the results of
  ``make streams`` with proper processor placement shown on the right
  resulted in slightly higher overall parallel speedup (identical
  baselines), in smaller performance fluctuations, and more than 90
  percent of peak bandwidth with only six processes.

Machines with job submission systems such as SLURM usually provide
similar mechanisms for processor placements through options specified in
job submission scripts. Please consult the respective manuals.

Additional Process Placement Considerations and Details
'''''''''''''''''''''''''''''''''''''''''''''''''''''''

For a typical, memory bandwidth-limited PETSc application, the primary
consideration in placing MPI processes is ensuring that processes are
evenly distributed among sockets, and hence using all available memory
channels. Increasingly complex processor designs and cache hierarchies,
however, mean that performance may also be sensitive to how processes
are bound to the resources within each socket. Performance on the two
processor machine in the preceding example may be relatively insensitive
to such placement decisions, because one L3 cache is shared by all cores
within a NUMA domain, and each core has its own L2 and L1 caches.
However, processors that are less “flat”, with more complex hierarchies,
may be more sensitive. In many AMD Opterons or the second-generation
“Knights Landing” Intel Xeon Phi, for instance, L2 caches are shared
between two cores. On these processors, placing consecutive MPI ranks on
cores that share the same L2 cache may benefit performance if the two
ranks communicate frequently with each other, because the latency
between cores sharing an L2 cache may be roughly half that of two cores
not sharing one. There may be benefit, however, in placing consecutive
ranks on cores that do not share an L2 cache, because (if there are
fewer MPI ranks than cores) this increases the total L2 cache capacity
and bandwidth available to the application. There is a trade-off to be
considered between placing processes close together (in terms of shared
resources) to optimize for efficient communication and synchronization
vs. farther apart to maximize available resources (memory channels,
caches, I/O channels, etc.), and the best strategy will depend on the
application and the software and hardware stack.

Different process placement strategies can affect performance at least
as much as some commonly explored settings, such as compiler
optimization levels. Unfortunately, exploration of this space is
complicated by two factors: First, processor and core numberings may be
completely arbitrary, changing with BIOS version, etc., and second—as
already noted—there is no standard mechanism used by MPI implementations
(or job schedulers) to specify process affinity. To overcome the first
issue, we recommend using the ``lstopo`` utility of the Portable
Hardware Locality (``hwloc``) software package (which can be installed
by configuring PETSc with ``–download-hwloc``) to understand the
processor topology of your machine. We cannot fully address the second
issue—consult the documenation for your MPI implementation and/or job
scheduler—but we offer some general observations on understanding
placement options:

-  An MPI implementation may support a notion of *domains* in which a
   process may be pinned. A domain may simply correspond to a single
   core; however, the MPI implementation may allow a deal of flexibility
   in specifying domains that encompass multiple cores, span sockets,
   etc. Some implementations, such as Intel MPI, provide means to
   specify whether domains should be “compact”—composed of cores sharing
   resources such as caches—or “scatter”-ed, with little resource
   sharing (possibly even spanning sockets).

-  Separate from the specification of domains, MPI implementations often
   support different *orderings* in which MPI ranks should be bound to
   these domains. Intel MPI, for instance, supports “compact” ordering
   to place consecutive ranks close in terms of shared resources,
   “scatter” to place them far apart, and “bunch” to map proportionally
   to sockets while placing ranks as close together as possible within
   the sockets.

-  An MPI implemenation that supports process pinning should offer some
   way to view the rank assignments. Use this output in conjunction with
   the topology obtained via ``lstopo`` or a similar tool to determine
   if the placements correspond to something you believe is reasonable
   for your application. Do not assume that the MPI implementation is
   doing something sensible by default!

Performance Pitfalls and Advice
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This section looks into a potpourri of performance pitfalls encountered
by users in the past. Many of these pitfalls require a deeper
understanding of the system and experience to detect. The purpose of
this section is to summarize and share our experience so that these
pitfalls can be avoided in the future.

Debug vs. Optimized Builds
^^^^^^^^^^^^^^^^^^^^^^^^^^

PETSc’s ``configure`` defaults to building PETSc with debug mode
enabled. Any code development should be done in this mode, because it
provides handy debugging facilities such as accurate stack traces,
memory leak checks, or memory corruption checks. Note that PETSc has no
reliable way of knowing whether a particular run is a production or
debug run. In the case that a user requests profiling information via
``-log_view``, a debug build of PETSc issues the following warning:

.. code-block:: none

         ##########################################################
         #                                                        #
         #                          WARNING!!!                    #
         #                                                        #
         #   This code was compiled with a debugging option,      #
         #   To get timing results run configure                  #
         #   using --with-debugging=no, the performance will      #
         #   be generally two or three times faster.              #
         #                                                        #
         ##########################################################

Conversely, one way of checking whether a particular build of PETSc has
debugging enabled is to inspect the output of ``-log_view``.

Debug mode will generally be most useful for code development if
appropriate compiler options are set to faciliate debugging. The
compiler should be instructed to generate binaries with debug symbols
(command line option ``-g`` for most compilers), and the optimization
level chosen should either completely disable optimizations (``-O0`` for
most compilers) or enable only optimizations that do not interfere with
debugging (GCC, for instance, supports a ``-Og`` optimization level that
does this).

Only once the new code is thoroughly tested and ready for production,
one should disable debugging facilities by passing
``--with-debugging=no`` to

``configure``. One should also ensure that an appropriate compiler
optimization level is set. Note that some compilers (e.g., Intel)
default to fairly comprehensive optimization levels, while others (e.g.,
GCC) default to no optimization at all. The best optimization flags will
depend on your code, the compiler, and the target architecture, but we
offer a few guidelines for finding those that will offer the best
performance:

-  Most compilers have a number of optimization levels (with level n
   usually specified via ``-On``) that provide a quick way to enable
   sets of several optimization flags. We suggest trying the higher
   optimization levels (the highest level is not guaranteed to produce
   the fastest executable, so some experimentation may be merited). With
   most recent processors now supporting some form of SIMD or vector
   instructions, it is important to choose a level that enables the
   compiler’s auto-vectorizer; many compilers do not enable
   auto-vectorization at lower optimization levels (e.g., GCC does not
   enable it below ``-O3`` and the Intel compiler does not enable it
   below ``-O2``).

-  For processors supporting newer vector instruction sets, such as
   Intel AVX2 and AVX-512, it is also important to direct the compiler
   to generate code that targets these processors (.e.g., ``-march=knl``
   to generate AVX-512 binaries targeting the Intel “Knights Landing”
   Xeon Phi processor); otherwise, the executables built will not
   utilize the newer instructions sets and will not take advantage of
   the vector processing units.

-  Beyond choosing the optimization levels, some value-unsafe
   optimizations (such as using reciprocals of values instead of
   dividing by those values, or allowing re-association of operands in a
   series of calculations) for floating point calculations may yield
   significant performance gains. Compilers often provide flags (e.g.,
   ``-ffast-math`` in GCC) to enable a set of these optimizations, and
   they may be turned on when using options for very aggressive
   optimization (``-fast`` or ``-Ofast`` in many compilers). These are
   worth exploring to maximize performance, but, if employed, it
   important to verify that these do not cause erroneous results with
   your code, since calculations may violate the IEEE standard for
   floating-point arithmetic.

Profiling
^^^^^^^^^

Users should not spend time optimizing a code until after having
determined where it spends the bulk of its time on realistically sized
problems. As discussed in detail in :any:`ch_profiling`, the
PETSc routines automatically log performance data if certain runtime
options are specified.

To obtain a summary of where and how much time is spent in different
sections of the code, use one of the following options:

-  Run the code with the option ``-log_view`` to print a performance
   summary for various phases of the code.

-  Run the code with the option ``-log_mpe`` ``[logfilename]``, which
   creates a logfile of events suitable for viewing with Jumpshot (part
   of MPICH).

Then, focus on the sections where most of the time is spent. If you
provided your own callback routines, e.g. for residual evaluations,
search the profiling output for routines such as ``SNESFunctionEval`` or
``SNESJacobianEval``. If their relative time is significant (say, more
than 30 percent), consider optimizing these routines first. Generic
instructions on how to optimize your callback functions are difficult;
you may start by reading performance optimization guides for your
system’s hardware.

Aggregation
^^^^^^^^^^^

Performing operations on chunks of data rather than a single element at
a time can significantly enhance performance because of cache reuse or
lower data motion. Typical examples are:

-  Insert several (many) elements of a matrix or vector at once, rather
   than looping and inserting a single value at a time. In order to
   access elements in of vector repeatedly, employ ``VecGetArray()`` to
   allow direct manipulation of the vector elements.

-  When possible, use ``VecMDot()`` rather than a series of calls to
   ``VecDot()``.

-  If you require a sequence of matrix-vector products with the same
   matrix, consider packing your vectors into a single matrix and use
   matrix-matrix multiplications.

-  Users should employ a reasonable number of ``PetscMalloc()`` calls in
   their codes. Hundreds or thousands of memory allocations may be
   appropriate; however, if tens of thousands are being used, then
   reducing the number of ``PetscMalloc()`` calls may be warranted. For
   example, reusing space or allocating large chunks and dividing it
   into pieces can produce a significant savings in allocation overhead.
   :any:`sec_dsreuse` gives details.

Aggressive aggregation of data may result in inflexible datastructures
and code that is hard to maintain. We advise users to keep these
competing goals in mind and not blindly optimize for performance only.

.. _sec_perf_memory:

Memory Allocation for Sparse Matrix Assembly
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Since the process of dynamic memory allocation for sparse matrices is
inherently very expensive, accurate preallocation of memory is crucial
for efficient sparse matrix assembly. One should use the matrix creation
routines for particular data structures, such as ``MatCreateSeqAIJ()``
and ``MatCreateAIJ()`` for compressed, sparse row formats, instead of
the generic ``MatCreate()`` routine. For problems with multiple degrees
of freedom per node, the block, compressed, sparse row formats, created
by ``MatCreateSeqBAIJ()`` and ``MatCreateBAIJ()``, can significantly
enhance performance. :any:`sec_matsparse` includes
extensive details and examples regarding preallocation.

.. _sec_symbolfactor:

Memory Allocation for Sparse Matrix Factorization
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

When symbolically factoring an AIJ matrix, PETSc has to guess how much
fill there will be. Careful use of the fill parameter in the
``MatILUInfo`` structure when calling ``MatLUFactorSymbolic()`` or
``MatILUFactorSymbolic()`` can reduce greatly the number of mallocs and
copies required, and thus greatly improve the performance of the
factorization. One way to determine a good value for the fill parameter
is to run a program with the option ``-info``. The symbolic
factorization phase will then print information such as

.. code-block:: none

   Info:MatILUFactorSymbolic_AIJ:Realloc 12 Fill ratio:given 1 needed 2.16423

This indicates that the user should have used a fill estimate factor of
about 2.17 (instead of 1) to prevent the 12 required mallocs and copies.
The command line option

.. code-block:: none

   -pc_ilu_fill 2.17

will cause PETSc to preallocate the correct amount of space for
incomplete (ILU) factorization. The corresponding option for direct (LU)
factorization is ``-pc_factor_fill <fill_amount>``.

.. _detecting-memory-problems:

Detecting Memory Allocation Problems
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

PETSc provides a number of tools to aid in detection of problems with
memory allocation, including leaks and use of uninitialized space. We
briefly describe these below.

-  The PETSc memory allocation (which collects statistics and performs
   error checking), is employed by default for codes compiled in a
   debug-mode (configured with ``--with-debugging=1``). PETSc memory
   allocation can be activated for optimized-mode (configured with
   ``--with-debugging=0``) using the option ``-malloc``. The option
   ``-malloc=0`` forces the use of conventional memory allocation when
   debugging is enabled. When running timing tests, one should build
   libraries in optimized mode.

-  When the PETSc memory allocation routines are used, the option
   ``-malloc_dump`` will print a list of unfreed memory at the
   conclusion of a program. If all memory has been freed, only a message
   stating the maximum allocated space will be printed. However, if some
   memory remains unfreed, this information will be printed. Note that
   the option ``-malloc_dump`` merely activates a call to
   ``PetscMallocDump()`` during ``PetscFinalize()`` the user can also
   call ``PetscMallocDump()`` elsewhere in a program.

-  Another useful option for use with PETSc memory allocation routines
   is ``-malloc_view``, which activates logging of all calls to malloc
   and reports memory usage, including all Fortran arrays. This option
   provides a more complete picture than ``-malloc_dump`` for codes that
   employ Fortran with hardwired arrays. The option ``-malloc_view``
   activates logging by calling ``PetscMallocViewSet()`` in
   ``PetscInitialize()`` and then prints the log by calling
   ``PetscMallocView()`` in ``PetscFinalize()``. The user can also call
   these routines elsewhere in a program. When finer granularity is
   desired, the user should call ``PetscMallocGetCurrentUsage()`` and
   ``PetscMallocGetMaximumUsage()`` for memory allocated by PETSc, or
   ``PetscMemoryGetCurrentUsage()`` and ``PetscMemoryGetMaximumUsage()``
   for the total memory used by the program. Note that
   ``PetscMemorySetGetMaximumUsage()`` must be called before
   ``PetscMemoryGetMaximumUsage()`` (typically at the beginning of the
   program).

.. _sec_dsreuse:

Data Structure Reuse
^^^^^^^^^^^^^^^^^^^^

Data structures should be reused whenever possible. For example, if a
code often creates new matrices or vectors, there often may be a way to
reuse some of them. Very significant performance improvements can be
achieved by reusing matrix data structures with the same nonzero
pattern. If a code creates thousands of matrix or vector objects,
performance will be degraded. For example, when solving a nonlinear
problem or timestepping, reusing the matrices and their nonzero
structure for many steps when appropriate can make the code run
significantly faster.

A simple technique for saving work vectors, matrices, etc. is employing
a user-defined context. In C and C++ such a context is merely a
structure in which various objects can be stashed; in Fortran a user
context can be an integer array that contains both parameters and
pointers to PETSc objects. See
`SNES Tutorial ex5 <../../src/snes/tutorials/ex5.c.html>`__
and
`SNES Tutorial ex5f <../../src/snes/tutorials/ex5f.F90.html>`__
for examples of user-defined application contexts in C and Fortran,
respectively.

Numerical Experiments
^^^^^^^^^^^^^^^^^^^^^

PETSc users should run a variety of tests. For example, there are a
large number of options for the linear and nonlinear equation solvers in
PETSc, and different choices can make a *very* big difference in
convergence rates and execution times. PETSc employs defaults that are
generally reasonable for a wide range of problems, but clearly these
defaults cannot be best for all cases. Users should experiment with many
combinations to determine what is best for a given problem and customize
the solvers accordingly.

-  Use the options ``-snes_view``, ``-ksp_view``, etc. (or the routines
   ``KSPView()``, ``SNESView()``, etc.) to view the options that have
   been used for a particular solver.

-  Run the code with the option ``-help`` for a list of the available
   runtime commands.

-  Use the option ``-info`` to print details about the solvers’
   operation.

-  Use the PETSc monitoring discussed in :any:`ch_profiling`
   to evaluate the performance of various numerical methods.

.. _sec_slestips:

Tips for Efficient Use of Linear Solvers
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

As discussed in :any:`chapter_ksp`, the default linear
solvers are

-  | uniprocess: GMRES(30) with ILU(0) preconditioning

-  | multiprocess: GMRES(30) with block Jacobi preconditioning, where
     there is 1 block per process, and each block is solved with ILU(0)

One should experiment to determine alternatives that may be better for
various applications. Recall that one can specify the ``KSP`` methods
and preconditioners at runtime via the options:

.. code-block:: none

   -ksp_type <ksp_name> -pc_type <pc_name>

One can also specify a variety of runtime customizations for the
solvers, as discussed throughout the manual.

In particular, note that the default restart parameter for GMRES is 30,
which may be too small for some large-scale problems. One can alter this
parameter with the option ``-ksp_gmres_restar <restart>`` or by calling
``KSPGMRESSetRestart()``. :any:`sec_ksp` gives
information on setting alternative GMRES orthogonalization routines,
which may provide much better parallel performance.

For elliptic problems one often obtains good performance and scalability
with multigrid solvers. Consult :any:`sec_amg` for
available options. Our experience is that GAMG works particularly well
for elasticity problems, whereas hypre does well for scalar problems.

System-Related Problems
^^^^^^^^^^^^^^^^^^^^^^^

The performance of a code can be affected by a variety of factors,
including the cache behavior, other users on the machine, etc. Below we
briefly describe some common problems and possibilities for overcoming
them.

-  **Problem too large for physical memory size**: When timing a
   program, one should always leave at least a ten percent margin
   between the total memory a process is using and the physical size of
   the machine’s memory. One way to estimate the amount of memory used
   by given process is with the UNIX ``getrusage`` system routine. Also,
   the PETSc option ``-log_view`` prints the amount of memory used by
   the basic PETSc objects, thus providing a lower bound on the memory
   used. Another useful option is ``-malloc_view`` which reports all
   memory, including any Fortran arrays in an application code.

-  **Effects of other users**: If other users are running jobs on the
   same physical processor nodes on which a program is being profiled,
   the timing results are essentially meaningless.

-  **Overhead of timing routines on certain machines**: On certain
   machines, even calling the system clock in order to time routines is
   slow; this skews all of the flop rates and timing results. The file
   ``$PETSC_DIR/src/benchmarks/PetscTime.c`` (`source <../../src/benchmarks/PetscTime.c.html>`__)
   contains a simple test problem that will approximate the amount of
   time required to get the current time in a running program. On good
   systems it will on the order of :math:`10^{-6}` seconds or less.

-  **Problem too large for good cache performance**: Certain machines
   with lower memory bandwidths (slow memory access) attempt to
   compensate by having a very large cache. Thus, if a significant
   portion of an application fits within the cache, the program will
   achieve very good performance; if the code is too large, the
   performance can degrade markedly. To analyze whether this situation
   affects a particular code, one can try plotting the total flop rate
   as a function of problem size. If the flop rate decreases rapidly at
   some point, then the problem may likely be too large for the cache
   size.

-  **Inconsistent timings**: Inconsistent timings are likely due to
   other users on the machine, thrashing (using more virtual memory than
   available physical memory), or paging in of the initial executable.
   :any:`sec_profaccuracy` provides information on
   overcoming paging overhead when profiling a code. We have found on
   all systems that if you follow all the advise above your timings will
   be consistent within a variation of less than five percent.
