.. _chapter_ksp:

KSP: Linear System Solvers
--------------------------

The ``KSP`` object is the heart of PETSc, because it provides uniform
and efficient access to all of the package’s linear system solvers,
including parallel and sequential, direct and iterative. ``KSP`` is
intended for solving systems of the form

.. math::
  :label: eq_axeqb

   A x = b,

where :math:`A` denotes the matrix representation of a linear operator,
:math:`b` is the right-hand-side vector, and :math:`x` is the solution
vector. ``KSP`` uses the same calling sequence for both direct and
iterative solution of a linear system. In addition, particular solution
techniques and their associated options can be selected at runtime.

The combination of a Krylov subspace method and a preconditioner is at
the center of most modern numerical codes for the iterative solution of
linear systems. Many textbooks (e.g. :cite:`fgn` :cite:`VanDerVorst2003`, or :cite:`saad2003`) provide an
overview of the theory of such methods.
The ``KSP`` package, discussed in
:any:`sec_ksp`, provides many popular Krylov subspace
iterative methods; the ``PC`` module, described in
:any:`sec_pc`, includes a variety of preconditioners.

.. _sec_usingksp:

Using KSP
~~~~~~~~~

To solve a linear system with ``KSP``, one must first create a solver
context with the command

::

   KSPCreate(MPI_Comm comm,KSP *ksp);

Here ``comm`` is the MPI communicator and ``ksp`` is the newly formed
solver context. Before actually solving a linear system with ``KSP``,
the user must call the following routine to set the matrices associated
with the linear system:

::

   KSPSetOperators(KSP ksp,Mat Amat,Mat Pmat);

The argument ``Amat``, representing the matrix that defines the linear
system, is a symbolic placeholder for any kind of matrix or operator. In
particular, ``KSP`` *does* support matrix-free methods. The routine
``MatCreateShell()`` in :any:`sec_matrixfree`
provides further information regarding matrix-free methods. Typically,
the matrix from which the preconditioner is to be constructed, ``Pmat``,
is the same as the matrix that defines the linear system, ``Amat``;
however, occasionally these matrices differ (for instance, when a
preconditioning matrix is obtained from a lower order method than that
employed to form the linear system matrix).

Much of the power of ``KSP`` can be accessed through the single routine

::

   KSPSetFromOptions(KSP ksp);

This routine accepts the option ``-help`` as well as any of
the ``KSP`` and ``PC`` options discussed below. To solve a linear
system, one sets the rhs and solution vectors using and executes the
command

::

   KSPSolve(KSP ksp,Vec b,Vec x);

where ``b`` and ``x`` respectively denote the right-hand-side and
solution vectors. On return, the iteration number at which the iterative
process stopped can be obtained using

::

   KSPGetIterationNumber(KSP ksp, PetscInt *its);

Note that this does not state that the method converged at this
iteration: it can also have reached the maximum number of iterations, or
have diverged.

:any:`sec_convergencetests` gives more details
regarding convergence testing. Note that multiple linear solves can be
performed by the same ``KSP`` context. Once the ``KSP`` context is no
longer needed, it should be destroyed with the command

::

   KSPDestroy(KSP *ksp);

The above procedure is sufficient for general use of the ``KSP``
package. One additional step is required for users who wish to customize
certain preconditioners (e.g., see :any:`sec_bjacobi`) or
to log certain performance data using the PETSc profiling facilities (as
discussed in :any:`ch_profiling`). In this case, the user can
optionally explicitly call

::

   KSPSetUp(KSP ksp);

before calling ``KSPSolve()`` to perform any setup required for the
linear solvers. The explicit call of this routine enables the separate
monitoring of any computations performed during the set up phase, such
as incomplete factorization for the ILU preconditioner.

The default solver within ``KSP`` is restarted GMRES, preconditioned for
the uniprocess case with ILU(0), and for the multiprocess case with the
block Jacobi method (with one block per process, each of which is solved
with ILU(0)). A variety of other solvers and options are also available.
To allow application programmers to set any of the preconditioner or
Krylov subspace options directly within the code, we provide routines
that extract the ``PC`` and ``KSP`` contexts,

::

   KSPGetPC(KSP ksp,PC *pc);

The application programmer can then directly call any of the ``PC`` or
``KSP`` routines to modify the corresponding default options.

To solve a linear system with a direct solver (currently supported by
PETSc for sequential matrices, and by several external solvers through
PETSc interfaces (see :any:`sec_externalsol`)) one may use
the options ``-ksp_type`` ``preonly`` ``-pc_type`` ``lu`` (see below).

By default, if a direct solver is used, the factorization is *not* done
in-place. This approach prevents the user from the unexpected surprise
of having a corrupted matrix after a linear solve. The routine
``PCFactorSetUseInPlace()``, discussed below, causes factorization to be
done in-place.

Solving Successive Linear Systems
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

When solving multiple linear systems of the same size with the same
method, several options are available. To solve successive linear
systems having the *same* preconditioner matrix (i.e., the same data
structure with exactly the same matrix elements) but different
right-hand-side vectors, the user should simply call ``KSPSolve()``,
multiple times. The preconditioner setup operations (e.g., factorization
for ILU) will be done during the first call to ``KSPSolve()`` only; such
operations will *not* be repeated for successive solves.

To solve successive linear systems that have *different* preconditioner
matrices (i.e., the matrix elements and/or the matrix data structure
change), the user *must* call ``KSPSetOperators()`` and ``KSPSolve()``
for each solve. See :any:`sec_usingksp` for a description
of various flags for ``KSPSetOperators()`` that can save work for such
cases.

.. _sec_ksp:

Krylov Methods
~~~~~~~~~~~~~~

The Krylov subspace methods accept a number of options, many of which
are discussed below. First, to set the Krylov subspace method that is to
be used, one calls the command

::

   KSPSetType(KSP ksp,KSPType method);

The type can be one of ``KSPRICHARDSON``, ``KSPCHEBYSHEV``, ``KSPCG``,
``KSPGMRES``, ``KSPTCQMR``, ``KSPBCGS``, ``KSPCGS``, ``KSPTFQMR``,
``KSPCR``, ``KSPLSQR``, ``KSPBICG``, ``KSPPREONLY``. or others; see
:any:`tab-kspdefaults` or the ``KSPType`` man page for more.
The ``KSP`` method can also be set with the options database command
``-ksp_type``, followed by one of the options ``richardson``,
``chebyshev``, ``cg``, ``gmres``, ``tcqmr``, ``bcgs``, ``cgs``,
``tfqmr``, ``cr``, ``lsqr``, ``bicg``, ``preonly.``, or others (see
:any:`tab-kspdefaults` or the ``KSPType`` man page) There are
method-specific options. For instance, for the Richardson, Chebyshev, and
GMRES methods:

::

   KSPRichardsonSetScale(KSP ksp,PetscReal scale);
   KSPChebyshevSetEigenvalues(KSP ksp,PetscReal emax,PetscReal emin);
   KSPGMRESSetRestart(KSP ksp,PetscInt max_steps);

The default parameter values are
``damping_factor=1.0, emax=0.01, emin=100.0``, and ``max_steps=30``. The
GMRES restart and Richardson damping factor can also be set with the
options ``-ksp_gmres_restart <n>`` and
``-ksp_richardson_scale <factor>``.

The default technique for orthogonalization of the Hessenberg matrix in
GMRES is the unmodified (classical) Gram-Schmidt method, which can be
set with

::

   KSPGMRESSetOrthogonalization(KSP ksp,KSPGMRESClassicalGramSchmidtOrthogonalization);

or the options database command ``-ksp_gmres_classicalgramschmidt``. By
default this will *not* use iterative refinement to improve the
stability of the orthogonalization. This can be changed with the option

::

   KSPGMRESSetCGSRefinementType(KSP ksp,KSPGMRESCGSRefinementType type)

or via the options database with

::

   -ksp_gmres_cgs_refinement_type none,ifneeded,always

The values for ``KSPGMRESCGSRefinementType()`` are
``KSP_GMRES_CGS_REFINEMENT_NONE``, ``KSP_GMRES_CGS_REFINEMENT_IFNEEDED``
and ``KSP_GMRES_CGS_REFINEMENT_ALWAYS``.

One can also use modified Gram-Schmidt, by using the orthogonalization
routine ``KSPGMRESModifiedGramSchmidtOrthogonalization()`` or by using
the command line option ``-ksp_gmres_modifiedgramschmidt``.

For the conjugate gradient method with complex numbers, there are two
slightly different algorithms depending on whether the matrix is
Hermitian symmetric or truly symmetric (the default is to assume that it
is Hermitian symmetric). To indicate that it is symmetric, one uses the
command

::

   KSPCGSetType(KSP ksp,KSPCGType KSP_CG_SYMMETRIC);

Note that this option is not valid for all matrices.

The LSQR algorithm does not involve a preconditioner; any preconditioner
set to work with the ``KSP`` object is ignored if ``KSPLSQR`` was
selected.

By default, ``KSP`` assumes an initial guess of zero by zeroing the
initial value for the solution vector that is given; this zeroing is
done at the call to ``KSPSolve()``. To use a nonzero initial guess, the
user *must* call

::

   KSPSetInitialGuessNonzero(KSP ksp,PetscBool flg);

.. _sec_ksppc:

Preconditioning within KSP
^^^^^^^^^^^^^^^^^^^^^^^^^^

Since the rate of convergence of Krylov projection methods for a
particular linear system is strongly dependent on its spectrum,
preconditioning is typically used to alter the spectrum and hence
accelerate the convergence rate of iterative techniques. Preconditioning
can be applied to the system :any:`eq_axeqb` by

.. math::
  :label: eq_prec

   (M_L^{-1} A M_R^{-1}) \, (M_R x) = M_L^{-1} b,

where :math:`M_L` and :math:`M_R` indicate preconditioning matrices (or,
matrices from which the preconditioner is to be constructed). If
:math:`M_L = I` in :any:`eq_prec`, right preconditioning
results, and the residual of :any:`eq_axeqb`,

.. math:: r \equiv b - Ax = b - A M_R^{-1} \, M_R x,

is preserved. In contrast, the residual is altered for left
(:math:`M_R = I`) and symmetric preconditioning, as given by

.. math:: r_L \equiv M_L^{-1} b - M_L^{-1} A x = M_L^{-1} r.

By default, most KSP implementations use left preconditioning. Some more
naturally use other options, though. For instance, ``KSPQCG`` defaults
to use symmetric preconditioning and ``KSPFGMRES`` uses right
preconditioning by default. Right preconditioning can be activated for
some methods by using the options database command
``-ksp_pc_side right`` or calling the routine

::

   KSPSetPCSide(KSP ksp,PCSide PC_RIGHT);

Attempting to use right preconditioning for a method that does not
currently support it results in an error message of the form

.. code-block:: none

   KSPSetUp_Richardson:No right preconditioning for KSPRICHARDSON

We summarize the defaults for the residuals used in KSP convergence
monitoring within :any:`tab-kspdefaults`. Details regarding
specific convergence tests and monitoring routines are presented in the
following sections. The preconditioned residual is used by default for
convergence testing of all left-preconditioned ``KSP`` methods. For the
conjugate gradient, Richardson, and Chebyshev methods the true residual
can be used by the options database command
``ksp_norm_type unpreconditioned`` or by calling the routine

::

   KSPSetNormType(KSP ksp,KSP_NORM_UNPRECONDITIONED);


.. list-table:: KSP Objects
  :name: tab-kspdefaults
  :header-rows: 1

  * - Method
    - KSPType
    - Options Database Name
  * - Richardson
    - ``KSPRICHARDSON``
    - ``richardson``
  * - Chebyshev
    - ``KSPCHEBYSHEV``
    - ``chebyshev``
  * - Conjugate Gradient :cite:`hs:52`
    - ``KSPCG``
    - ``cg``
  * - Pipelined Conjugate Gradients :cite:`GhyselsVanroose2014`
    - ``KSPPIPECG``
    - ``pipecg``
  * - Pipelined Conjugate Gradients (Gropp)
    - ``KSPGROPPCG``
    - ``groppcg``
  * - Pipelined Conjugate Gradients with Residual Replacement
    - ``KSPPIPECGRR``
    - ``pipecgrr``
  * - Conjugate Gradients for the Normal Equations
    - ``KSPCGNE``
    - ``cgne``
  * - Flexible Conjugate Gradients :cite:`flexibleCG`
    - ``KSPFCG``
    - ``fcg``
  * -  Pipelined, Flexible Conjugate Gradients :cite:`SananSchneppMay2016`
    - ``KSPPIPEFCG``
    - ``pipefcg``
  * - Conjugate Gradients for Least Squares
    - ``KSPCGLS``
    - ``cgls``
  * - Conjugate Gradients with Constraint (1)
    - ``KSPNASH``
    - ``nash``
  * - Conjugate Gradients with Constraint (2)
    - ``KSPSTCG``
    - ``stcg``
  * - Conjugate Gradients with Constraint (3)
    - ``KSPGLTR``
    - ``gltr``
  * - Conjugate Gradients with Constraint (4)
    - ``KSPQCG``
    - ``qcg``
  * - BiConjugate Gradient
    - ``KSPBICG``
    - ``bicg``
  * - BiCGSTAB :cite:`v:92`
    - ``KSPBCGS``
    - ``bcgs``
  * - Improved BiCGSTAB
    - ``KSPIBCGS``
    - ``ibcgs``
  * - Flexible BiCGSTAB
    - ``KSPFBCGS``
    - ``fbcgs``
  * - Flexible BiCGSTAB (variant)
    - ``KSPFBCGSR``
    - ``fbcgsr``
  * - Enhanced BiCGSTAB(L)
    - ``KSPBCGSL``
    - ``bcgsl``
  * - Minimal Residual Method :cite:`PaigeSaunders1975`
    - ``KSPMINRES``
    - ``minres``
  * - Generalized Minimal Residual :cite:`ss:86`
    - ``KSPGMRES``
    - ``gmres``
  * - Flexible Generalized Minimal Residual :cite:`Saad1993`
    - ``KSPFGMRES``
    - ``fgmres``
  * - Deflated Generalized Minimal Residual
    - ``KSPDGMRES``
    - ``dgmres``
  * - Pipelined Generalized Minimal Residual :cite:`GhyselsAshbyMeerbergenVanroose2013`
    - ``KSPPGMRES``
    - ``pgmres``
  * - Pipelined, Flexible Generalized Minimal Residual :cite:`SananSchneppMay2016`
    - ``KSPPIPEFGMRES``
    - ``pipefgmres``
  * - Generalized Minimal Residual with Accelerated Restart
    - ``KSPLGMRES``
    - ``lgmres``
  * - Conjugate Residual :cite:`eisenstat1983variational`
    - ``KSPCR``
    - ``cr``
  * - Generalized Conjugate Residual
    - ``KSPGCR``
    - ``gcr``
  * - Pipelined Conjugate Residual
    - ``KSPPIPECR``
    - ``pipecr``
  * - Pipelined, Flexible Conjugate Residual :cite:`SananSchneppMay2016`
    - ``KSPPIPEGCR``
    - ``pipegcr``
  * - FETI-DP
    - ``KSPFETIDP``
    - ``fetidp``
  * - Conjugate Gradient Squared :cite:`so:89`
    - ``KSPCGS``
    - ``cgs``
  * - Transpose-Free Quasi-Minimal Residual (1) :cite:`f:93`
    - ``KSPTFQMR``
    - ``tfqmr``
  * - Transpose-Free Quasi-Minimal Residual (2)
    - ``KSPTCQMR``
    - ``tcqmr``
  * - Least Squares Method
    - ``KSPLSQR``
    - ``lsqr``
  * - Symmetric LQ Method :cite:`PaigeSaunders1975`
    - ``KSPSYMMLQ``
    - ``symmlq``
  * - TSIRM
    - ``KSPTSIRM``
    - ``tsirm``
  * - Python Shell
    - ``KSPPYTHON``
    - ``python``
  * - Shell for no ``KSP`` method
    - ``KSPPREONLY``
    - ``preonly``


Note: the bi-conjugate gradient method requires application of both the
matrix and its transpose plus the preconditioner and its transpose.
Currently not all matrices and preconditioners provide this support and
thus the ``KSPBICG`` cannot always be used.

Note: PETSc implements the FETI-DP (Finite Element Tearing and
Interconnecting Dual-Primal) method as an implementation of ``KSP`` since it recasts the
original problem into a contstrained minimization one with Lagrange
multipliers. The only matrix type supported is ``MATIS``. Support for
saddle point problems is provided. See the man page for ``KSPFETIDP`` for
further details.

.. _sec_convergencetests:

Convergence Tests
^^^^^^^^^^^^^^^^^

The default convergence test, ``KSPConvergedDefault()``, is based on the
:math:`l_2`-norm of the residual. Convergence (or divergence) is decided
by three quantities: the decrease of the residual norm relative to the
norm of the right hand side, ``rtol``, the absolute size of the residual
norm, ``atol``, and the relative increase in the residual, ``dtol``.
Convergence is detected at iteration :math:`k` if

.. math:: \| r_k \|_2 < {\rm max} ( \text{rtol} * \| b \|_2, \text{atol}),

where :math:`r_k = b - A x_k`. Divergence is detected if

.. math:: \| r_k \|_2 > \text{dtol} * \| b \|_2.

These parameters, as well as the maximum number of allowable iterations,
can be set with the routine

::

   KSPSetTolerances(KSP ksp,PetscReal rtol,PetscReal atol,PetscReal dtol,PetscInt maxits);

The user can retain the default value of any of these parameters by
specifying ``PETSC_DEFAULT`` as the corresponding tolerance; the
defaults are ``rtol=1e-5``, ``atol=1e-50``, ``dtol=1e5``, and
``maxits=1e4``. These parameters can also be set from the options
database with the commands ``-ksp_rtol`` ``<rtol>``, ``-ksp_atol``
``<atol>``, ``-ksp_divtol`` ``<dtol>``, and ``-ksp_max_it`` ``<its>``.

In addition to providing an interface to a simple convergence test,
``KSP`` allows the application programmer the flexibility to provide
customized convergence-testing routines. The user can specify a
customized routine with the command

::

   KSPSetConvergenceTest(KSP ksp,PetscErrorCode (*test)(KSP ksp,PetscInt it,PetscReal rnorm, KSPConvergedReason *reason,void *ctx),void *ctx,PetscErrorCode (*destroy)(void *ctx));

The final routine argument, ``ctx``, is an optional context for private
data for the user-defined convergence routine, ``test``. Other ``test``
routine arguments are the iteration number, ``it``, and the residual’s
:math:`l_2` norm, ``rnorm``. The routine for detecting convergence,
``test``, should set ``reason`` to positive for convergence, 0 for no
convergence, and negative for failure to converge. A full list of
possible values for ``KSPConvergedReason`` is given in
``include/petscksp.h``. You can use ``KSPGetConvergedReason()`` after
``KSPSolve()`` to see why convergence/divergence was detected.

.. _sec_kspmonitor:

Convergence Monitoring
^^^^^^^^^^^^^^^^^^^^^^

By default, the Krylov solvers run silently without displaying
information about the iterations. The user can indicate that the norms
of the residuals should be displayed by using ``-ksp_monitor`` within
the options database. To display the residual norms in a graphical
window (running under X Windows), one should use
``-ksp_monitor draw::draw_lg``. Application programmers can also
provide their own routines to perform the monitoring by using the
command

::

   KSPMonitorSet(KSP ksp,PetscErrorCode (*mon)(KSP ksp,PetscInt it,PetscReal rnorm,void *ctx),void *ctx,PetscErrorCode (*mondestroy)(void**));

The final routine argument, ``ctx``, is an optional context for private
data for the user-defined monitoring routine, ``mon``. Other ``mon``
routine arguments are the iteration number (``it``) and the residual’s
:math:`l_2` norm (``rnorm``). A helpful routine within user-defined
monitors is ``PetscObjectGetComm((PetscObject)ksp,MPI_Comm *comm)``,
which returns in ``comm`` the MPI communicator for the ``KSP`` context.
See :any:`sec_writing` for more discussion of the use of
MPI communicators within PETSc.

Several monitoring routines are supplied with PETSc, including

::

   KSPMonitorResidual(KSP,PetscInt,PetscReal, void *);
   KSPMonitorSingularValue(KSP,PetscInt,PetscReal,void *);
   KSPMonitorTrueResidual(KSP,PetscInt,PetscReal, void *);

The default monitor simply prints an estimate of the :math:`l_2`-norm of
the residual at each iteration. The routine
``KSPMonitorSingularValue()`` is appropriate only for use with the
conjugate gradient method or GMRES, since it prints estimates of the
extreme singular values of the preconditioned operator at each
iteration. Since ``KSPMonitorTrueResidualNorm()`` prints the true
residual at each iteration by actually computing the residual using the
formula :math:`r = b - Ax`, the routine is slow and should be used only
for testing or convergence studies, not for timing. These monitors may
be accessed with the command line options ``-ksp_monitor``,
``-ksp_monitor_singular_value``, and ``-ksp_monitor_true_residual``.

To employ the default graphical monitor, one should use the command
``-ksp_monitor draw::draw_lg``.

One can cancel hardwired monitoring routines for KSP at runtime with
``-ksp_monitor_cancel``.

Unless the Krylov method converges so that the residual norm is small,
say :math:`10^{-10}`, many of the final digits printed with the
``-ksp_monitor`` option are meaningless. Worse, they are different on
different machines; due to different round-off rules used by, say, the
IBM RS6000 and the Sun SPARC. This makes testing between different
machines difficult. The option ``-ksp_monitor_short`` causes PETSc to
print fewer of the digits of the residual norm as it gets smaller; thus
on most of the machines it will always print the same numbers making
cross system testing easier.

Understanding the Operator’s Spectrum
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Since the convergence of Krylov subspace methods depends strongly on the
spectrum (eigenvalues) of the preconditioned operator, PETSc has
specific routines for eigenvalue approximation via the Arnoldi or
Lanczos iteration. First, before the linear solve one must call

::

   KSPSetComputeEigenvalues(KSP ksp,PETSC_TRUE);

Then after the ``KSP`` solve one calls

::

   KSPComputeEigenvalues(KSP ksp,PetscInt n,PetscReal *realpart,PetscReal *complexpart,PetscInt *neig);

Here, ``n`` is the size of the two arrays and the eigenvalues are
inserted into those two arrays. ``neig`` is the number of eigenvalues
computed; this number depends on the size of the Krylov space generated
during the linear system solution, for GMRES it is never larger than the
restart parameter. There is an additional routine

::

   KSPComputeEigenvaluesExplicitly(KSP ksp, PetscInt n,PetscReal *realpart,PetscReal *complexpart);

that is useful only for very small problems. It explicitly computes the
full representation of the preconditioned operator and calls LAPACK to
compute its eigenvalues. It should be only used for matrices of size up
to a couple hundred. The ``PetscDrawSP*()`` routines are very useful for
drawing scatter plots of the eigenvalues.

The eigenvalues may also be computed and displayed graphically with the
options data base commands ``-ksp_view_eigenvalues draw`` and
``-ksp_view_eigenvalues_explicitly draw``. Or they can be dumped to the
screen in ASCII text via ``-ksp_view_eigenvalues`` and
``-ksp_view_eigenvalues_explicitly``.

Other KSP Options
^^^^^^^^^^^^^^^^^

To obtain the solution vector and right hand side from a ``KSP``
context, one uses

::

   KSPGetSolution(KSP ksp,Vec *x);
   KSPGetRhs(KSP ksp,Vec *rhs);

During the iterative process the solution may not yet have been
calculated or it may be stored in a different location. To access the
approximate solution during the iterative process, one uses the command

::

   KSPBuildSolution(KSP ksp,Vec w,Vec *v);

where the solution is returned in ``v``. The user can optionally provide
a vector in ``w`` as the location to store the vector; however, if ``w``
is ``NULL``, space allocated by PETSc in the ``KSP`` context is used.
One should not destroy this vector. For certain ``KSP`` methods, (e.g.,
GMRES), the construction of the solution is expensive, while for many
others it doesn’t evenrequire a vector copy.

Access to the residual is done in a similar way with the command

::

   KSPBuildResidual(KSP ksp,Vec t,Vec w,Vec *v);

Again, for GMRES and certain other methods this is an expensive
operation.

.. _sec_pc:

Preconditioners
~~~~~~~~~~~~~~~

As discussed in :any:`sec_ksppc`, Krylov subspace methods
are typically used in conjunction with a preconditioner. To employ a
particular preconditioning method, the user can either select it from
the options database using input of the form ``-pc_type <methodname>``
or set the method with the command

::

   PCSetType(PC pc,PCType method);

In :any:`tab-pcdefaults` we summarize the basic
preconditioning methods supported in PETSc. See the ``PCType`` manual
page for a complete list. The ``PCSHELL`` preconditioner uses a
specific, application-provided preconditioner. The direct
preconditioner, ``PCLU`` , is, in fact, a direct solver for the linear
system that uses LU factorization. ``PCLU`` is included as a
preconditioner so that PETSc has a consistent interface among direct and
iterative linear solvers.

.. list-table:: PETSc Preconditioners (partial list)
   :name: tab-pcdefaults
   :header-rows: 1

   * - Method
     - PCType
     - Options Database Name
   * - Jacobi
     - ``PCJACOBI``
     - ``jacobi``
   * - Block Jacobi
     - ``PCBJACOBI``
     - ``bjacobi``
   * - SOR (and SSOR)
     - ``PCSOR``
     - ``sor``
   * - SOR with Eisenstat trick
     - ``PCEISENSTAT``
     - ``eisenstat``
   * - Incomplete Cholesky
     - ``PCICC``
     - ``icc``
   * - Incomplete LU
     - ``PCILU``
     - ``ilu``
   * - Additive Schwarz
     - ``PCASM``
     - ``asm``
   * - Generalized Additive Schwarz
     - ``PCGASM``
     - ``gasm``
   * - Algebraic Multigrid
     - ``PCGAMG``
     - ``gamg``
   * - Balancing Domain Decomposition by Constraints
     - ``PCBDDC``
     - ``bddc``
   * - Linear solver
     - ``PCKSP``
     - ``ksp``
   * - Combination of preconditioners
     - ``PCCOMPOSITE``
     - ``composite``
   * - LU
     - ``PCLU``
     - ``lu``
   * - Cholesky
     - ``PCCHOLESKY``
     - ``cholesky``
   * - No preconditioning
     - ``PCNONE``
     - ``none``
   * - Shell for user-defined ``PC``
     - ``PCSHELL``
     - ``shell``

Each preconditioner may have associated with it a set of options, which
can be set with routines and options database commands provided for this
purpose. Such routine names and commands are all of the form
``PC<TYPE><Option>`` and ``-pc_<type>_<option> [value]``. A complete
list can be found by consulting the ``PCType`` manual page; we discuss
just a few in the sections below.

.. _sec_ilu_icc:

ILU and ICC Preconditioners
^^^^^^^^^^^^^^^^^^^^^^^^^^^

Some of the options for ILU preconditioner are

::

   PCFactorSetLevels(PC pc,PetscInt levels);
   PCFactorSetReuseOrdering(PC pc,PetscBool  flag);
   PCFactorSetDropTolerance(PC pc,PetscReal dt,PetscReal dtcol,PetscInt dtcount);
   PCFactorSetReuseFill(PC pc,PetscBool  flag);
   PCFactorSetUseInPlace(PC pc,PetscBool flg);
   PCFactorSetAllowDiagonalFill(PC pc,PetscBool flg);

When repeatedly solving linear systems with the same ``KSP`` context,
one can reuse some information computed during the first linear solve.
In particular, ``PCFactorSetReuseOrdering()`` causes the ordering (for
example, set with ``-pc_factor_mat_ordering_type`` ``order``) computed
in the first factorization to be reused for later factorizations.
``PCFactorSetUseInPlace()`` is often used with ``PCASM`` or
``PCBJACOBI`` when zero fill is used, since it reuses the matrix space
to store the incomplete factorization it saves memory and copying time.
Note that in-place factorization is not appropriate with any ordering
besides natural and cannot be used with the drop tolerance
factorization. These options may be set in the database with

-  ``-pc_factor_levels <levels>``

-  ``-pc_factor_reuse_ordering``

-  ``-pc_factor_reuse_fill``

-  ``-pc_factor_in_place``

-  ``-pc_factor_nonzeros_along_diagonal``

-  ``-pc_factor_diagonal_fill``

See :any:`sec_symbolfactor` for information on
preallocation of memory for anticipated fill during factorization. By
alleviating the considerable overhead for dynamic memory allocation,
such tuning can significantly enhance performance.

PETSc supports incomplete factorization preconditioners
for several matrix types for sequential matrices (for example
``MATSEQAIJ``, ``MATSEQBAIJ``, and ``MATSEQSBAIJ``).

SOR and SSOR Preconditioners
^^^^^^^^^^^^^^^^^^^^^^^^^^^^

PETSc only provides only a sequential SOR preconditioner; it can only be
used with sequential matrices or as the subblock preconditioner when
using block Jacobi or ASM preconditioning (see below).

The options for SOR preconditioning with ``PCSOR`` are

::

   PCSORSetOmega(PC pc,PetscReal omega);
   PCSORSetIterations(PC pc,PetscInt its,PetscInt lits);
   PCSORSetSymmetric(PC pc,MatSORType type);

The first of these commands sets the relaxation factor for successive
over (under) relaxation. The second command sets the number of inner
iterations ``its`` and local iterations ``lits`` (the number of
smoothing sweeps on a process before doing a ghost point update from the
other processes) to use between steps of the Krylov space method. The
total number of SOR sweeps is given by ``its*lits``. The third command
sets the kind of SOR sweep, where the argument ``type`` can be one of
``SOR_FORWARD_SWEEP``, ``SOR_BACKWARD_SWEEP`` or
``SOR_SYMMETRIC_SWEEP``, the default being ``SOR_FORWARD_SWEEP``.
Setting the type to be ``SOR_SYMMETRIC_SWEEP`` produces the SSOR method.
In addition, each process can locally and independently perform the
specified variant of SOR with the types ``SOR_LOCAL_FORWARD_SWEEP``,
``SOR_LOCAL_BACKWARD_SWEEP``, and ``SOR_LOCAL_SYMMETRIC_SWEEP``. These
variants can also be set with the options ``-pc_sor_omega <omega>``,
``-pc_sor_its <its>``, ``-pc_sor_lits <lits>``, ``-pc_sor_backward``,
``-pc_sor_symmetric``, ``-pc_sor_local_forward``,
``-pc_sor_local_backward``, and ``-pc_sor_local_symmetric``.

The Eisenstat trick :cite:`eisenstat81` for SSOR
preconditioning can be employed with the method ``PCEISENSTAT``
(``-pc_type`` ``eisenstat``). By using both left and right
preconditioning of the linear system, this variant of SSOR requires
about half of the floating-point operations for conventional SSOR. The
option ``-pc_eisenstat_no_diagonal_scaling``) (or the routine
``PCEisenstatSetNoDiagonalScaling()``) turns off diagonal scaling in
conjunction with Eisenstat SSOR method, while the option
``-pc_eisenstat_omega <omega>`` (or the routine
``PCEisenstatSetOmega(PC pc,PetscReal omega)``) sets the SSOR relaxation
coefficient, ``omega``, as discussed above.

.. _sec_factorization:

LU Factorization
^^^^^^^^^^^^^^^^

The LU preconditioner provides several options. The first, given by the
command

::

   PCFactorSetUseInPlace(PC pc,PetscBool flg);

causes the factorization to be performed in-place and hence destroys the
original matrix. The options database variant of this command is
``-pc_factor_in_place``. Another direct preconditioner option is
selecting the ordering of equations with the command
``-pc_factor_mat_ordering_type <ordering>``. The possible orderings are

-  ``MATORDERINGNATURAL`` - Natural

-  ``MATORDERINGND`` - Nested Dissection

-  ``MATORDERING1WD`` - One-way Dissection

-  ``MATORDERINGRCM`` - Reverse Cuthill-McKee

-  ``MATORDERINGQMD`` - Quotient Minimum Degree

These orderings can also be set through the options database by
specifying one of the following: ``-pc_factor_mat_ordering_type``
``natural``, or ``nd``, or ``1wd``, or ``rcm``, or ``qmd``. In addition,
see ``MatGetOrdering()``, discussed in :any:`sec_matfactor`.

The sparse LU factorization provided in PETSc does not perform pivoting
for numerical stability (since they are designed to preserve nonzero
structure), and thus occasionally a LU factorization will fail with a
zero pivot when, in fact, the matrix is non-singular. The option
``-pc_factor_nonzeros_along_diagonal <tol>`` will often help eliminate
the zero pivot, by preprocessing the column ordering to remove small
values from the diagonal. Here, ``tol`` is an optional tolerance to
decide if a value is nonzero; by default it is ``1.e-10``.

In addition, :any:`sec_symbolfactor` provides information
on preallocation of memory for anticipated fill during factorization.
Such tuning can significantly enhance performance, since it eliminates
the considerable overhead for dynamic memory allocation.

.. _sec_bjacobi:

Block Jacobi and Overlapping Additive Schwarz Preconditioners
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The block Jacobi and overlapping additive Schwarz methods in PETSc are
supported in parallel; however, only the uniprocess version of the block
Gauss-Seidel method is currently in place. By default, the PETSc
implementations of these methods employ ILU(0) factorization on each
individual block (that is, the default solver on each subblock is
``PCType=PCILU``, ``KSPType=KSPPREONLY``); the user can set alternative
linear solvers via the options ``-sub_ksp_type`` and ``-sub_pc_type``.
In fact, all of the ``KSP`` and ``PC`` options can be applied to the
subproblems by inserting the prefix ``-sub_`` at the beginning of the
option name. These options database commands set the particular options
for *all* of the blocks within the global problem. In addition, the
routines

::

   PCBJacobiGetSubKSP(PC pc,PetscInt *n_local,PetscInt *first_local,KSP **subksp);
   PCASMGetSubKSP(PC pc,PetscInt *n_local,PetscInt *first_local,KSP **subksp);

extract the ``KSP`` context for each local block. The argument
``n_local`` is the number of blocks on the calling process, and
``first_local`` indicates the global number of the first block on the
process. The blocks are numbered successively by processes from zero
through :math:`b_g-1`, where :math:`b_g` is the number of global blocks.
The array of ``KSP`` contexts for the local blocks is given by
``subksp``. This mechanism enables the user to set different solvers for
the various blocks. To set the appropriate data structures, the user
*must* explicitly call ``KSPSetUp()`` before calling
``PCBJacobiGetSubKSP()`` or ``PCASMGetSubKSP(``). For further details,
see
`KSP Tutorial ex7 <https://www.mcs.anl.gov/petsc/petsc-current/src/ksp/ksp/tutorials/ex7.c.html>`__
or
`KSP Tutorial ex8 <https://www.mcs.anl.gov/petsc/petsc-current/src/ksp/ksp/tutorials/ex8.c.html>`__.

The block Jacobi, block Gauss-Seidel, and additive Schwarz
preconditioners allow the user to set the number of blocks into which
the problem is divided. The options database commands to set this value
are ``-pc_bjacobi_blocks`` ``n`` and ``-pc_bgs_blocks`` ``n``, and,
within a program, the corresponding routines are

::

   PCBJacobiSetTotalBlocks(PC pc,PetscInt blocks,PetscInt *size);
   PCASMSetTotalSubdomains(PC pc,PetscInt n,IS *is,IS *islocal);
   PCASMSetType(PC pc,PCASMType type);

The optional argument ``size`` is an array indicating the size of each
block. Currently, for certain parallel matrix formats, only a single
block per process is supported. However, the ``MATMPIAIJ`` and
``MATMPIBAIJ`` formats support the use of general blocks as long as no
blocks are shared among processes. The ``is`` argument contains the
index sets that define the subdomains.

The object ``PCASMType`` is one of ``PC_ASM_BASIC``,
``PC_ASM_INTERPOLATE``, ``PC_ASM_RESTRICT``, or\ ``PC_ASM_NONE`` and may
also be set with the options database ``-pc_asm_type`` ``[basic``,
``interpolate``, ``restrict``, ``none]``. The type ``PC_ASM_BASIC`` (or
``-pc_asm_type`` ``basic``) corresponds to the standard additive Schwarz
method that uses the full restriction and interpolation operators. The
type ``PC_ASM_RESTRICT`` (or ``-pc_asm_type`` ``restrict``) uses a full
restriction operator, but during the interpolation process ignores the
off-process values. Similarly, ``PC_ASM_INTERPOLATE`` (or
``-pc_asm_type`` ``interpolate``) uses a limited restriction process in
conjunction with a full interpolation, while ``PC_ASM_NONE`` (or
``-pc_asm_type`` ``none``) ignores off-process values for both
restriction and interpolation. The ASM types with limited restriction or
interpolation were suggested by Xiao-Chuan Cai and Marcus Sarkis
:cite:`cs97a`. ``PC_ASM_RESTRICT`` is the PETSc default, as
it saves substantial communication and for many problems has the added
benefit of requiring fewer iterations for convergence than the standard
additive Schwarz method.

The user can also set the number of blocks and sizes on a per-process
basis with the commands

::

   PCBJacobiSetLocalBlocks(PC pc,PetscInt blocks,PetscInt *size);
   PCASMSetLocalSubdomains(PC pc,PetscInt N,IS *is,IS *islocal);

For the ASM preconditioner one can use the following command to set the
overlap to compute in constructing the subdomains.

::

   PCASMSetOverlap(PC pc,PetscInt overlap);

The overlap defaults to 1, so if one desires that no additional overlap
be computed beyond what may have been set with a call to
``PCASMSetTotalSubdomains()`` or ``PCASMSetLocalSubdomains()``, then
``overlap`` must be set to be 0. In particular, if one does *not*
explicitly set the subdomains in an application code, then all overlap
would be computed internally by PETSc, and using an overlap of 0 would
result in an ASM variant that is equivalent to the block Jacobi
preconditioner. Note that one can define initial index sets ``is`` with
*any* overlap via ``PCASMSetTotalSubdomains()`` or
``PCASMSetLocalSubdomains()``; the routine ``PCASMSetOverlap()`` merely
allows PETSc to extend that overlap further if desired.

``PCGASM`` is an experimental generalization of ``PCASM`` that allows
the user to specify subdomains that span multiple MPI ranks. This can be
useful for problems where small subdomains result in poor convergence.
To be effective, the multirank subproblems must be solved using a
sufficient strong subsolver, such as LU, for which ``SuperLU_DIST`` or a
similar parallel direct solver could be used; other choices may include
a multigrid solver on the subdomains.

The interface for ``PCGASM`` is similar to that of ``PCASM``. In
particular, ``PCGASMType`` is one of ``PC_GASM_BASIC``,
``PC_GASM_INTERPOLATE``, ``PC_GASM_RESTRICT``, ``PC_GASM_NONE``. These
options have the same meaning as with ``PCASM`` and may also be set with
the options database ``-pc_gasm_type`` ``[basic``, ``interpolate``,
``restrict``, ``none]``.

Unlike ``PCASM``, however, ``PCGASM`` allows the user to define
subdomains that span multiple MPI ranks. The simplest way to do this is
using a call to ``PCGASMSetTotalSubdomains(PC pc,PetscPetscInt N)`` with
the total number of subdomains ``N`` that is smaller than the MPI
communicator ``size``. In this case ``PCGASM`` will coalesce ``size/N``
concecutive single-rank subdomains into a single multi-rank subdomain.
The single-rank subdomains contain the degrees of freedom corresponding
to the locally-owned rows of the ``PCGASM`` preconditioning matrix –
these are the subdomains ``PCASM`` and ``PCGASM`` use by default.

Each of the multirank subdomain subproblems is defined on the
subcommunicator that contains the coalesced ``PCGASM`` ranks. In general
this might not result in a very good subproblem if the single-rank
problems corresponding to the coalesced ranks are not very strongly
connected. In the future this will be addressed with a hierarchical
partitioner that generates well-connected coarse subdomains first before
subpartitioning them into the single-rank subdomains.

In the meantime the user can provide his or her own multi-rank
subdomains by calling\ ``PCGASMSetSubdomains(PC,IS[],IS[])`` where each
of the ``IS`` objects on the list defines the inner (without the
overlap) or the outer (including the overlap) subdomain on the
subcommunicator of the ``IS`` object. A helper subroutine
``PCGASMCreateSubdomains2D()`` is similar to PCASM’s but is capable of
constructing multi-rank subdomains that can be then used with
``PCGASMSetSubdomains()``. An alternative way of creating multi-rank
subdomains is by using the underlying DM object, if it is capable of
generating such decompositions via ``DMCreateDomainDecomposition()``.
Ordinarily the decomposition specified by the user via
``PCGASMSetSubdomains()`` takes precedence, unless
``PCGASMSetUseDMSubdomains()`` instructs ``PCGASM`` to prefer
``DM``-created decompositions.

Currently there is no support for increasing the overlap of multi-rank
subdomains via\ ``PCGASMSetOverlap()`` – this functionality works only
for subdomains that fit within a single MPI rank, exactly as in
``PCASM``.

Examples of the described PCGASM usage can be found in
`KSP Tutorial ex62 <https://www.mcs.anl.gov/petsc/petsc-current/src/ksp/ksp/tutorials/ex62.c.html>`__.
In particular, ``runex62_superlu_dist`` illustrates the use of
``SuperLU_DIST`` as the subdomain solver on coalesced multi-rank
subdomains. The ``runex62_2D_*`` examples illustrate the use of
``PCGASMCreateSubdomains2D()``.

.. _sec_amg:

Algebraic Multigrid (AMG) Preconditioners
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

PETSc has a native algebraic multigrid preconditioner ``PCGAMG`` –
*gamg* – and interfaces to two external AMG packages: *hypre* and *ML*.
*Hypre* is relatively monolithic in that a PETSc matrix is into a hypre
matrix and then *hypre* is called to do the entire solve. *ML* is more
modular in that PETSc only has *ML* generate the coarse grid spaces
(columns of the prolongation operator), which is core of an AMG method,
and then constructs a ``PCMG`` with Galerkin coarse grid operator
construction. GAMG is designed from the beginning to be modular, to
allow for new components to be added easily and also populates a
multigrid preconditioner ``PCMG`` so generic multigrid parameters are
used. PETSc provides a fully supported (smoothed) aggregation AMG,
(``-pc_type gamg -pc_gamg_type agg`` or ``PCSetType(pc,PCGAMG)`` and
``PCGAMGSetType(pc,PCGAMGAGG)``, as well as reference implementations of
a classical AMG method (``-pc_gamg_type classical``), a hybrid geometric
AMG method (``-pc_gamg_type geo``), and a 2.5D AMG method DofColumns
:cite:`IsaacStadlerGhattas2015`. GAMG does require the use
of (MPI)AIJ matrices. For instance, BAIJ matrices are not supported. One
can use AIJ instead of BAIJ without changing any code other than the
constructor (or the ``-mat_type`` from the command line). For instance,
``MatSetValuesBlocked`` works with AIJ matrices.

GAMG provides unsmoothed aggregation (``-pc_gamg_agg_nsmooths 0``) and
smoothed aggregation (``-pc_gamg_agg_nsmooths 1`` or
``PCGAMGSetNSmooths(pc,1)``). Smoothed aggregation (SA) is recommended
for symmetric positive definite systems. Unsmoothed aggregation can be
useful for asymmetric problems and problems where highest eigen
estimates are problematic. If poor convergence rates are observed using
the smoothed version one can test unsmoothed aggregation.

**Eigenvalue estimates:** The parameters for the KSP eigen estimator,
use for SA, can be set with ``-pc_gamg_esteig_ksp_max_it`` and
``-pc_gamg_esteig_ksp_type``. For example CG generally converges to the
highest eigenvalue fast than GMRES (the default for KSP) if your problem
is symmetric positive definite. One can specify CG with
``-pc_gamg_esteig_ksp_type cg``. The default for
``-pc_gamg_esteig_ksp_max_it`` is 10, which we have found is pretty safe
with a (default) safety factor of 1.1. One can specify the range of real
eigenvalues, in the same way that one can for Chebyshev KSP solvers
(smoothers), with ``-pc_gamg_eigenvalues <emin,emax>``. GAMG sets the MG
smoother type to chebyshev by default. By default, GAMG uses its eigen
estimate, if it has one, for Chebyshev smoothers if the smoother uses
Jacobi preconditioning. This can be overridden with
``-pc_gamg_use_sa_esteig  <true,false>``.

AMG methods requires knowledge of the number of degrees of freedom per
vertex, the default is one (a scalar problem). Vector problems like
elasticity should set the block size of the matrix appropriately with
``-mat_block_size bs`` or ``MatSetBlockSize(mat,bs)``. Equations must be
ordered in “vertex-major” ordering (e.g.,
:math:`x_1,y_1,z_1,x_2,y_2,...`).

**Near null space:** Smoothed aggregation requires an explicit
representation of the (near) null space of the operator for optimal
performance. One can provide an orthonormal set of null space vectors
with ``MatSetNearNullSpace()``. The vector of all ones is the default,
for each variable given by the block size (e.g., the translational rigid
body modes). For elasticity, where rotational rigid body modes are
required to complete the near null space you can use
``MatNullSpaceCreateRigidBody()`` to create the null space vectors and
then ``MatSetNearNullSpace()``.

**Coarse grid data model:** The GAMG framework provides for reducing the
number of active processes on coarse grids to reduce communication costs
when there is not enough parallelism to keep relative communication
costs down. Most AMG solver reduce to just one active process on the
coarsest grid (the PETSc MG framework also supports redundantly solving
the coarse grid on all processes to potentially reduce communication
costs), although this forcing to one process can be overridden if one
wishes to use a parallel coarse grid solver. GAMG generalizes this by
reducing the active number of processes on other coarse grids as well.
GAMG will select the number of active processors by fitting the desired
number of equation per process (set with
``-pc_gamg_process_eq_limit <50>,``) at each level given that size of
each level. If :math:`P_i < P` processors are desired on a level
:math:`i` then the first :math:`P_i` ranks are populated with the grid
and the remaining are empty on that grid. One can, and probably should,
repartition the coarse grids with ``-pc_gamg_repartition <true>,``,
otherwise an integer process reduction factor (:math:`q`) is selected
and the equations on the first :math:`q` processes are move to process
0, and so on. As mentioned multigrid generally coarsens the problem
until it is small enough to be solved with an exact solver (eg, LU or
SVD) in a relatively small time. GAMG will stop coarsening when the
number of equation on a grid falls below at threshold give by
``-pc_gamg_coarse_eq_limit <50>,``.

**Coarse grid parameters:** There are several options to provide
parameters to the coarsening algorithm and parallel data layout. Run a
code that uses GAMG with ``-help`` to get full listing of GAMG
parameters with short parameter descriptions. The rate of coarsening is
critical in AMG performance – too slow of coarsening will result in an
overly expensive solver per iteration and too fast coarsening will
result in decrease in the convergence rate. ``-pc_gamg_threshold <0>``
and ``-pc_gamg_square_graph <1>,`` are the primary parameters that
control coarsening rates, which is very important for AMG performance. A
greedy maximal independent set (MIS) algorithm is used in coarsening.
Squaring the graph implements so called MIS-2, the root vertex in an
aggregate is more than two edges away from another root vertex, instead
of more than one in MIS. The threshold parameter sets a normalized
threshold for which edges are removed from the MIS graph, thereby
coarsening slower. Zero will keep all non-zero edges, a negative number
will keep zero edges, a positive number will drop small edges. Typical
finite threshold values are in the range of :math:`0.01 - 0.05`. There
are additional parameters for changing the weights on coarse grids.
Note, the parallel algorithm requires symmetric weights/matrix. You must
use ``-pc_gamg_sym_graph <true>`` to symmetrize the graph if your
problem is not symmetric.

**Trouble shooting algebraic multigrid methods:** If *GAMG*, *ML*, or
*hypre* does not perform well the first thing to try is one of the other
methods. Often the default parameters or just the strengths of different
algorithms can fix performance problems or provide useful information to
guide further debugging. There are several sources of poor performance
of AMG solvers and often special purpose methods must be developed to
achieve the full potential of multigrid. To name just a few sources of
performance degradation that may not be fixed with parameters in PETSc
currently: non-elliptic operators, curl/curl operators, highly stretched
grids or highly anisotropic problems, large jumps in material
coefficients with complex geometry (AMG is particularly well suited to
jumps in coefficients but it is not a perfect solution), highly
incompressible elasticity, not to mention ill-posed problems, and many
others. For Grad-Div and Curl-Curl operators, you may want to try the
Auxiliary-space Maxwell Solver (AMS,
``-pc_type hypre -pc_hypre_type ams``) or the Auxiliary-space Divergence
Solver (ADS, ``-pc_type hypre -pc_hypre_type ads``) solvers. These
solvers need some additional information on the underlying mesh;
specifically, AMS needs the discrete gradient operator, which can be
specified via ``PCHYPRESetDiscreteGradient()``. In addition to the
discrete gradient, ADS also needs the specification of the discrete curl
operator, which can be set using ``PCHYPRESetDiscreteCurl()``.

**I am converging slowly, what do I do?** AMG methods are sensitive to
coarsening rates and methods; for GAMG use ``-pc_gamg_threshold <x>`` to
regulate coarsening rates and PCGAMGSetThreshold, higher values decrease
coarsening rate. Squaring the graph is the second mechanism for
increasing coarsening rate. Use ``-pc_gamg_square_graph <N>,``, or
``PCGAMGSetSquareGraph(pc,N)``, to square the graph on the finest N
levels. A high threshold (e.g., :math:`x=0.08`) will result in an
expensive but potentially powerful preconditioner, and a low threshold
(e.g., :math:`x=0.0`) will result in faster coarsening, fewer levels,
cheaper solves, and generally worse convergence rates.

One can run with ``-info`` and grep for “GAMG” to get some statistics on
each level, which can be used to see if you are coarsening at an
appropriate rate. With smoothed aggregation you generally want to coarse
at about a rate of 3:1 in each dimension. Coarsening too slow will
result in large numbers of non-zeros per row on coarse grids (this is
reported). The number of non-zeros can go up very high, say about 300
(times the degrees-of-freedom per vertex) on a 3D hex mesh. One can also
look at the grid complexity, which is also reported (the ration of the
total number of matrix entries for all levels to the number of matrix
entries on the fine level). Grid complexity should be well under 2.0 and
preferably around :math:`1.3` or lower. If convergence is poor and the
Galerkin coarse grid construction is much smaller than the time for each
solve then one can safely decrease the coarsening rate.
``-pc_gamg_threshold`` :math:`0.0` is the simplest and most robust
option, and is recommended if poor convergence rates are observed, at
least until the source of the problem is discovered. In conclusion, if
convergence is slow then decreasing the coarsening rate (increasing the
threshold) should be tried.

**A note on Chebyshev smoothers.** Chebyshev solvers are attractive as
multigrid smoothers because they can target a specific interval of the
spectrum which is the purpose of a smoother. The spectral bounds for
Chebyshev solvers are simple to compute because they rely on the highest
eigenvalue of your (diagonally preconditioned) operator, which is
conceptually simple to compute. However, if this highest eigenvalue
estimate is not accurate (too low) then the solvers can fail with and
indefinite preconditioner message. One can run with ``-info`` and grep
for “GAMG” to get these estimates or use ``-ksp_view``. These highest
eigenvalues are generally between 1.5-3.0. For symmetric positive
definite systems CG is a better eigenvalue estimator
``-mg_levels_esteig_ksp_type cg``. Indefinite matrix messages are often
caused by bad Eigen estimates. Explicitly damped Jacobi or Krylov
smoothers can provide an alternative to Chebyshev and *hypre* has
alternative smoothers.

**Now am I solving alright, can I expect better?** If you find that you
are getting nearly on digit in reduction of the residual per iteration
and are using a modest number of point smoothing steps (e.g., 1-4
iterations of SOR), then you may be fairly close to textbook multigrid
efficiency. Although you also need to check the setup costs. This can be
determined by running with ``-log_view`` and check that the time for the
Galerkin coarse grid construction (``MatPtAP``) is not (much) more than
the time spent in each solve (``KSPSolve``). If the ``MatPtAP`` time is
too large then one can increase the coarsening rate by decreasing the
threshold and squaring the coarsening graph
(``-pc_gamg_square_graph <N>``, squares the graph on the finest N
levels). Likewise if your ``MatPtAP`` time is small and your convergence
rate is not ideal then you could decrease the coarsening rate.

PETSc’s AMG solver is constructed as a framework for developers to
easily add AMG capabilities, like a new AMG methods or an AMG component
like a matrix triple product. Contact us directly if you are interested
in contributing.

Balancing Domain Decomposition by Constraints
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

PETSc provides the Balancing Domain Decomposition by Constraints (BDDC)
method for preconditioning parallel finite element problems stored in
unassembled format (see ``MATIS``). BDDC is a 2-level non-overlapping
domain decomposition method which can be easily adapted to different
problems and discretizations by means of few user customizations. The
application of the preconditioner to a vector consists in the static
condensation of the residual at the interior of the subdomains by means
of local Dirichet solves, followed by an additive combination of Neumann
local corrections and the solution of a global coupled coarse problem.
Command line options for the underlying ``KSP`` objects are prefixed by
``-pc_bddc_dirichlet``, ``-pc_bddc_neumann``, and ``-pc_bddc_coarse``
respectively.

The current implementation supports any kind of linear system, and
assumes a one-to-one mapping between subdomains and MPI processes.
Complex numbers are supported as well. For non-symmetric problems, use
the runtime option ``-pc_bddc_symmetric 0``.

Unlike conventional non-overlapping methods that iterates just on the
degrees of freedom at the interface between subdomain, ``PCBDDC``
iterates on the whole set of degrees of freedom, allowing the use of
approximate subdomain solvers. When using approximate solvers, the
command line switches ``-pc_bddc_dirichlet_approximate`` and/or
``-pc_bddc_neumann_approximate`` should be used to inform ``PCBDDC``. If
any of the local problems is singular, the nullspace of the local
operator should be attached to the local matrix via
``MatSetNullSpace()``.

At the basis of the method there’s the analysis of the connected
components of the interface for the detection of vertices, edges and
faces equivalence classes. Additional information on the degrees of
freedom can be supplied to ``PCBDDC`` by using the following functions:

-  ``PCBDDCSetDofsSplitting()``

-  ``PCBDDCSetLocalAdjacencyGraph()``

-  ``PCBDDCSetPrimalVerticesLocalIS()``

-  ``PCBDDCSetNeumannBoundaries()``

-  ``PCBDDCSetDirichletBoundaries()``

-  ``PCBDDCSetNeumannBoundariesLocal()``

-  ``PCBDDCSetDirichletBoundariesLocal()``

Crucial for the convergence of the iterative process is the
specification of the primal constraints to be imposed at the interface
between subdomains. ``PCBDDC`` uses by default vertex continuities and
edge arithmetic averages, which are enough for the three-dimensional
Poisson problem with constant coefficients. The user can switch on and
off the usage of vertices, edges or face constraints by using the
command line switches ``-pc_bddc_use_vertices``, ``-pc_bddc_use_edges``,
``-pc_bddc_use_faces``. A customization of the constraints is available
by attaching a ``MatNullSpace`` object to the preconditioning matrix via
``MatSetNearNullSpace()``. The vectors of the ``MatNullSpace`` object
should represent the constraints in the form of quadrature rules;
quadrature rules for different classes of the interface can be listed in
the same vector. The number of vectors of the ``MatNullSpace`` object
corresponds to the maximum number of constraints that can be imposed for
each class. Once all the quadrature rules for a given interface class
have been extracted, an SVD operation is performed to retain the
non-singular modes. As an example, the rigid body modes represent an
effective choice for elasticity, even in the almost incompressible case.
For particular problems, e.g. edge-based discretization with Nedelec
elements, a user defined change of basis of the degrees of freedom can
be beneficial for ``PCBDDC``; use ``PCBDDCSetChangeOfBasisMat()`` to
customize the change of basis.

The BDDC method is usually robust with respect to jumps in the material
parameters aligned with the interface; for PDEs with more than one
material parameter you may also consider to use the so-called deluxe
scaling, available via the command line switch
``-pc_bddc_use_deluxe_scaling``. Other scalings are available, see
``PCISSetSubdomainScalingFactor()``,
``PCISSetSubdomainDiagonalScaling()`` or
``PCISSetUseStiffnessScaling()``. However, the convergence properties of
the BDDC method degrades in presence of large jumps in the material
coefficients not aligned with the interface; for such cases, PETSc has
the capability of adaptively computing the primal constraints. Adaptive
selection of constraints could be requested by specifying a threshold
value at command line by using ``-pc_bddc_adaptive_threshold x``. Valid
values for the threshold ``x`` ranges from 1 to infinity, with smaller
values corresponding to more robust preconditioners. For SPD problems in
2D, or in 3D with only face degrees of freedom (like in the case of
Raviart-Thomas or Brezzi-Douglas-Marini elements), such a threshold is a
very accurate estimator of the condition number of the resulting
preconditioned operator. Since the adaptive selection of constraints for
BDDC methods is still an active topic of research, its implementation is
currently limited to SPD problems; moreover, because the technique
requires the explicit knowledge of the local Schur complements, it needs
the external package MUMPS.

When solving problems decomposed in thousands of subdomains or more, the
solution of the BDDC coarse problem could become a bottleneck; in order
to overcome this issue, the user could either consider to solve the
parallel coarse problem on a subset of the communicator associated with
``PCBDDC`` by using the command line switch
``-pc_bddc_coarse_redistribute``, or instead use a multilevel approach.
The latter can be requested by specifying the number of requested level
at command line (``-pc_bddc_levels``) or by using ``PCBDDCSetLevels()``.
An additional parameter (see ``PCBDDCSetCoarseningRatio()``) controls
the number of subdomains that will be generated at the next level; the
larger the coarsening ratio, the lower the number of coarser subdomains.

For further details, see the example
`KSP Tutorial ex59 <https://www.mcs.anl.gov/petsc/petsc-current/src/ksp/ksp/tutorials/ex59.c>`__
and the online documentation for ``PCBDDC``.

Shell Preconditioners
^^^^^^^^^^^^^^^^^^^^^

The shell preconditioner simply uses an application-provided routine to
implement the preconditioner. To set this routine, one uses the command

::

   PCShellSetApply(PC pc,PetscErrorCode (*apply)(PC,Vec,Vec));

Often a preconditioner needs access to an application-provided data
structured. For this, one should use

::

   PCShellSetContext(PC pc,void *ctx);

to set this data structure and

::

   PCShellGetContext(PC pc,void **ctx);

to retrieve it in ``apply``. The three routine arguments of ``apply()``
are the ``PC``, the input vector, and the output vector, respectively.

For a preconditioner that requires some sort of “setup” before being
used, that requires a new setup every time the operator is changed, one
can provide a routine that is called every time the operator is changed
(usually via ``KSPSetOperators()``).

::

   PCShellSetSetUp(PC pc,PetscErrorCode (*setup)(PC));

The argument to the ``setup`` routine is the same ``PC`` object which
can be used to obtain the operators with ``PCGetOperators()`` and the
application-provided data structure that was set with
``PCShellSetContext()``.

.. _sec_combining-pcs:

Combining Preconditioners
^^^^^^^^^^^^^^^^^^^^^^^^^

The ``PC`` type ``PCCOMPOSITE`` allows one to form new preconditioners
by combining already-defined preconditioners and solvers. Combining
preconditioners usually requires some experimentation to find a
combination of preconditioners that works better than any single method.
It is a tricky business and is not recommended until your application
code is complete and running and you are trying to improve performance.
In many cases using a single preconditioner is better than a
combination; an exception is the multigrid/multilevel preconditioners
(solvers) that are always combinations of some sort, see :any:`sec_mg`.

Let :math:`B_1` and :math:`B_2` represent the application of two
preconditioners of type ``type1`` and ``type2``. The preconditioner
:math:`B = B_1 + B_2` can be obtained with

::

   PCSetType(pc,PCCOMPOSITE);
   PCCompositeAddPCType(pc,type1);
   PCCompositeAddPCType(pc,type2);

Any number of preconditioners may added in this way.

This way of combining preconditioners is called additive, since the
actions of the preconditioners are added together. This is the default
behavior. An alternative can be set with the option

::

   PCCompositeSetType(PC pc,PCCompositeType PC_COMPOSITE_MULTIPLICATIVE);

In this form the new residual is updated after the application of each
preconditioner and the next preconditioner applied to the next residual.
For example, with two composed preconditioners: :math:`B_1` and
:math:`B_2`; :math:`y = B x` is obtained from

.. math::

   \begin{aligned}
   y    = B_1 x \\
   w_1  = x - A y \\
   y    = y + B_2 w_1\end{aligned}

Loosely, this corresponds to a Gauss-Seidel iteration, while additive
corresponds to a Jacobi iteration.

Under most circumstances, the multiplicative form requires one-half the
number of iterations as the additive form; however, the multiplicative
form does require the application of :math:`A` inside the
preconditioner.

In the multiplicative version, the calculation of the residual inside
the preconditioner can be done in two ways: using the original linear
system matrix or using the matrix used to build the preconditioners
:math:`B_1`, :math:`B_2`, etc. By default it uses the “preconditioner
matrix”, to use the ``Amat`` matrix use the option

::

   PCSetUseAmat(PC pc);

The individual preconditioners can be accessed (in order to set options)
via

::

   PCCompositeGetPC(PC pc,PetscInt count,PC *subpc);

For example, to set the first sub preconditioners to use ILU(1)

::

   PC subpc;
   PCCompositeGetPC(pc,0,&subpc);
   PCFactorSetFill(subpc,1);

One can also change the operator that is used to construct a particular
PC in the composite PC call PCSetOperators() on the obtained PC.

These various options can also be set via the options database. For
example, ``-pc_type`` ``composite`` ``-pc_composite_pcs`` ``jacobi,ilu``
causes the composite preconditioner to be used with two preconditioners:
Jacobi and ILU. The option ``-pc_composite_type`` ``multiplicative``
initiates the multiplicative version of the algorithm, while
``-pc_composite_type`` ``additive`` the additive version. Using the
``Amat`` matrix is obtained with the option ``-pc_use_amat``. One sets
options for the sub-preconditioners with the extra prefix ``-sub_N_``
where ``N`` is the number of the sub-preconditioner. For example,
``-sub_0_pc_ifactor_fill`` ``0``.

PETSc also allows a preconditioner to be a complete linear solver. This
is achieved with the ``PCKSP`` type.

::

   PCSetType(PC pc,PCKSP PCKSP);
   PCKSPGetKSP(pc,&ksp);
    /* set any KSP/PC options */

From the command line one can use 5 iterations of biCG-stab with ILU(0)
preconditioning as the preconditioner with
``-pc_type ksp -ksp_pc_type ilu -ksp_ksp_max_it 5 -ksp_ksp_type bcgs``.

By default the inner ``KSP`` solver uses the outer preconditioner
matrix, ``Pmat``, as the matrix to be solved in the linear system; to
use the matrix that defines the linear system, ``Amat`` use the option

::

   PCSetUseAmat(PC pc);

or at the command line with ``-pc_use_amat``.

Naturally, one can use a ``PCKSP`` preconditioner inside a composite
preconditioner. For example,
``-pc_type composite -pc_composite_pcs ilu,ksp -sub_1_pc_type jacobi -sub_1_ksp_max_it 10``
uses two preconditioners: ILU(0) and 10 iterations of GMRES with Jacobi
preconditioning. However, it is not clear whether one would ever wish to
do such a thing.

.. _sec_mg:

Multigrid Preconditioners
^^^^^^^^^^^^^^^^^^^^^^^^^

A large suite of routines is available for using geometric multigrid as
a preconditioner [2]_. In the ``PC`` framework, the user is required to
provide the coarse grid solver, smoothers, restriction and interpolation
operators, and code to calculate residuals. The ``PC`` package allows
these components to be encapuslated within a PETSc-compliant
preconditioner. We fully support both matrix-free and matrix-based
multigrid solvers.

A multigrid preconditioner is created with the four commands

::

   KSPCreate(MPI_Comm comm,KSP *ksp);
   KSPGetPC(KSP ksp,PC *pc);
   PCSetType(PC pc,PCMG);
   PCMGSetLevels(pc,PetscInt levels,MPI_Comm *comms);

A large number of parameters affect the multigrid behavior. The command

::

   PCMGSetType(PC pc,PCMGType mode);

indicates which form of multigrid to apply :cite:`1sbg`.

For standard V or W-cycle multigrids, one sets the ``mode`` to be
``PC_MG_MULTIPLICATIVE``; for the additive form (which in certain cases
reduces to the BPX method, or additive multilevel Schwarz, or multilevel
diagonal scaling), one uses ``PC_MG_ADDITIVE`` as the ``mode``. For a
variant of full multigrid, one can use ``PC_MG_FULL``, and for the
Kaskade algorithm ``PC_MG_KASKADE``. For the multiplicative and full
multigrid options, one can use a W-cycle by calling

::

   PCMGSetCycleType(PC pc,PCMGCycleType ctype);

with a value of ``PC_MG_CYCLE_W`` for ``ctype``. The commands above can
also be set from the options database. The option names are
``-pc_mg_type [multiplicative, additive, full, kaskade]``, and
``-pc_mg_cycle_type`` ``<ctype>``.

The user can control the amount of smoothing by configuring the solvers
on the levels. By default, the up and down smoothers are identical. If
separate configuration of up and down smooths is required, it can be
requested with the option ``-pc_mg_distinct_smoothup`` or the routine

::

   PCMGSetDistinctSmoothUp(PC pc);

The multigrid routines, which determine the solvers and
interpolation/restriction operators that are used, are mandatory. To set
the coarse grid solver, one must call

::

   PCMGGetCoarseSolve(PC pc,KSP *ksp);

and set the appropriate options in ``ksp``. Similarly, the smoothers are
controlled by first calling

::

   PCMGGetSmoother(PC pc,PetscInt level,KSP *ksp);

and then setting the various options in the ``ksp.`` For example,

::

   PCMGGetSmoother(pc,1,&ksp);
   KSPSetOperators(ksp,A1,A1);

sets the matrix that defines the smoother on level 1 of the multigrid.
While

::

   PCMGGetSmoother(pc,1,&ksp);
   KSPGetPC(ksp,&pc);
   PCSetType(pc,PCSOR);

sets SOR as the smoother to use on level 1.

To use a different pre- or postsmoother, one should call the following
routines instead.

::

   PCMGGetSmootherUp(PC pc,PetscInt level,KSP *upksp);
   PCMGGetSmootherDown(PC pc,PetscInt level,KSP *downksp);

Use

::

   PCMGSetInterpolation(PC pc,PetscInt level,Mat P);

and

::

   PCMGSetRestriction(PC pc,PetscInt level,Mat R);

to define the intergrid transfer operations. If only one of these is
set, its transpose will be used for the other.

It is possible for these interpolation operations to be matrix free (see
:any:`sec_matrixfree`); One should then make
sure that these operations are defined for the (matrix-free) matrices
passed in. Note that this system is arranged so that if the
interpolation is the transpose of the restriction, you can pass the same
``mat`` argument to both ``PCMGSetRestriction()`` and
``PCMGSetInterpolation()``.

On each level except the coarsest, one must also set the routine to
compute the residual. The following command suffices:

::

   PCMGSetResidual(PC pc,PetscInt level,PetscErrorCode (*residual)(Mat,Vec,Vec,Vec),Mat mat);

The ``residual()`` function normally does not need to be set if one’s
operator is stored in ``Mat`` format. In certain circumstances, where it
is much cheaper to calculate the residual directly, rather than through
the usual formula :math:`b - Ax`, the user may wish to provide an
alternative.

Finally, the user may provide three work vectors for each level (except
on the finest, where only the residual work vector is required). The
work vectors are set with the commands

::

   PCMGSetRhs(PC pc,PetscInt level,Vec b);
   PCMGSetX(PC pc,PetscInt level,Vec x);
   PCMGSetR(PC pc,PetscInt level,Vec r);

The ``PC`` references these vectors, so you should call ``VecDestroy()``
when you are finished with them. If any of these vectors are not
provided, the preconditioner will allocate them.

One can control the ``KSP`` and ``PC`` options used on the various
levels (as well as the coarse grid) using the prefix ``mg_levels_``
(``mg_coarse_`` for the coarse grid). For example,
``-mg_levels_ksp_type cg`` will cause the CG method to be used as the
Krylov method for each level. Or
``-mg_levels_pc_type ilu -mg_levels_pc_factor_levels 2`` will cause the
ILU preconditioner to be used on each level with two levels of fill in
the incomplete factorization.

.. _sec_block_matrices:

Solving Block Matrices
~~~~~~~~~~~~~~~~~~~~~~

Block matrices represent an important class of problems in numerical
linear algebra and offer the possibility of far more efficient iterative
solvers than just treating the entire matrix as black box. In this
section we use the common linear algebra definition of block matrices
where matrices are divided in a small, problem-size independent (two,
three or so) number of very large blocks. These blocks arise naturally
from the underlying physics or discretization of the problem, for
example, the velocity and pressure. Under a certain numbering of
unknowns the matrix can be written as

.. math::

   \left( \begin{array}{cccc}
   A_{00}   & A_{01} & A_{02} & A_{03} \\
   A_{10}   & A_{11} & A_{12} & A_{13} \\
   A_{20}   & A_{21} & A_{22} & A_{23} \\
   A_{30}   & A_{31} & A_{32} & A_{33} \\
   \end{array} \right),

where each :math:`A_{ij}` is an entire block. On a parallel computer the
matrices are not explicitly stored this way. Instead, each process will
own some of the rows of :math:`A_{0*}`, :math:`A_{1*}` etc. On a
process, the blocks may be stored one block followed by another

.. math::

   \left( \begin{array}{ccccccc}
   A_{{00}_{00}}   & A_{{00}_{01}} & A_{{00}_{02}} & ... & A_{{01}_{00}} & A_{{01}_{02}} & ...  \\
   A_{{00}_{10}}   & A_{{00}_{11}} & A_{{00}_{12}} & ... & A_{{01}_{10}} & A_{{01}_{12}} & ... \\
   A_{{00}_{20}}   & A_{{00}_{21}} & A_{{00}_{22}} & ... & A_{{01}_{20}} & A_{{01}_{22}}  & ...\\
   ... \\
   A_{{10}_{00}}   & A_{{10}_{01}} & A_{{10}_{02}} & ... & A_{{11}_{00}} & A_{{11}_{02}}  & ... \\
   A_{{10}_{10}}   & A_{{10}_{11}} & A_{{10}_{12}} & ... & A_{{11}_{10}} & A_{{11}_{12}}  & ... \\
   ... \\
   \end{array} \right)

or interlaced, for example with two blocks

.. math::

   \left( \begin{array}{ccccc}
   A_{{00}_{00}}   & A_{{01}_{00}} &  A_{{00}_{01}} & A_{{01}_{01}} &  ... \\
   A_{{10}_{00}}   & A_{{11}_{00}} &  A_{{10}_{01}} & A_{{11}_{01}} &  ... \\
   ... \\
   A_{{00}_{10}}   & A_{{01}_{10}} & A_{{00}_{11}} & A_{{01}_{11}} & ...\\
   A_{{10}_{10}}   & A_{{11}_{10}} & A_{{10}_{11}} & A_{{11}_{11}} & ...\\
   ...
   \end{array} \right).

Note that for interlaced storage the number of rows/columns of each
block must be the same size. Matrices obtained with ``DMCreateMatrix()``
where the ``DM`` is a ``DMDA`` are always stored interlaced. Block
matrices can also be stored using the ``MATNEST`` format which holds
separate assembled blocks. Each of these nested matrices is itself
distributed in parallel. It is more efficient to use ``MATNEST`` with
the methods described in this section because there are fewer copies and
better formats (e.g. ``BAIJ`` or ``SBAIJ``) can be used for the
components, but it is not possible to use many other methods with
``MATNEST``. See :any:`sec_matnest` for more on assembling
block matrices without depending on a specific matrix format.

The PETSc ``PCFIELDSPLIT`` preconditioner is used to implement the
“block” solvers in PETSc. There are three ways to provide the
information that defines the blocks. If the matrices are stored as
interlaced then ``PCFieldSplitSetFields()`` can be called repeatedly to
indicate which fields belong to each block. More generally
``PCFieldSplitSetIS()`` can be used to indicate exactly which
rows/columns of the matrix belong to a particular block. You can provide
names for each block with these routines, if you do not provide names
they are numbered from 0. With these two approaches the blocks may
overlap (though generally they will not). If only one block is defined
then the complement of the matrices is used to define the other block.
Finally the option ``-pc_fieldsplit_detect_saddle_point`` causes two
diagonal blocks to be found, one associated with all rows/columns that
have zeros on the diagonals and the rest.

For simplicity in the rest of the section we restrict our matrices to
two by two blocks. So the matrix is

.. math::

   \left( \begin{array}{cc}
   A_{00}   & A_{01} \\
   A_{10}   & A_{11} \\
   \end{array} \right).

On occasion the user may provide another matrix that is used to
construct parts of the preconditioner

.. math::

   \left( \begin{array}{cc}
   Ap_{00}   & Ap_{01} \\
   Ap_{10}   & Ap_{11} \\
   \end{array} \right).

For notational simplicity define :math:`\text{ksp}(A,Ap)` to mean
approximately solving a linear system using ``KSP`` with operator
:math:`A` and preconditioner built from matrix :math:`Ap`.

For matrices defined with any number of blocks there are three “block”
algorithms available: block Jacobi,

.. math::

   \left( \begin{array}{cc}
     \text{ksp}(A_{00},Ap_{00})   & 0 \\
     0   & \text{ksp}(A_{11},Ap_{11}) \\
   \end{array} \right)

block Gauss-Seidel,

.. math::

   \left( \begin{array}{cc}
   I   & 0 \\
   0 & A^{-1}_{11} \\
   \end{array} \right)
   \left( \begin{array}{cc}
   I   & 0 \\
   -A_{10} & I \\
   \end{array} \right)
   \left( \begin{array}{cc}
   A^{-1}_{00}   & 0 \\
   0 & I \\
   \end{array} \right)

which is implemented [3]_ as

.. math::

   \left( \begin{array}{cc}
   I   & 0 \\
     0 & \text{ksp}(A_{11},Ap_{11}) \\
   \end{array} \right)
   \left[
   \left( \begin{array}{cc}
   0   & 0 \\
   0 & I \\
   \end{array} \right)
   +
   \left( \begin{array}{cc}
   I   & 0 \\
   -A_{10} & -A_{11} \\
   \end{array} \right)
   \left( \begin{array}{cc}
   I   & 0 \\
   0 & 0 \\
   \end{array} \right)
   \right]
   \left( \begin{array}{cc}
     \text{ksp}(A_{00},Ap_{00})   & 0 \\
   0 & I \\
   \end{array} \right)

and symmetric block Gauss-Seidel

.. math::

   \left( \begin{array}{cc}
   A_{00}^{-1}   & 0 \\
   0 & I \\
   \end{array} \right)
   \left( \begin{array}{cc}
   I   & -A_{01} \\
   0 & I \\
   \end{array} \right)
   \left( \begin{array}{cc}
   A_{00}   & 0 \\
   0 & A_{11}^{-1} \\
   \end{array} \right)
   \left( \begin{array}{cc}
   I   & 0 \\
   -A_{10} & I \\
   \end{array} \right)
   \left( \begin{array}{cc}
   A_{00}^{-1}   & 0 \\
   0 & I \\
   \end{array} \right).

These can be accessed with
``-pc_fieldsplit_type<additive,multiplicative,``\ ``symmetric_multiplicative>``
or the function ``PCFieldSplitSetType()``. The option prefixes for the
internal KSPs are given by ``-fieldsplit_name_``.

By default blocks :math:`A_{00}, A_{01}` and so on are extracted out of
``Pmat``, the matrix that the ``KSP`` uses to build the preconditioner,
and not out of ``Amat`` (i.e., :math:`A` itself). As discussed above in
:any:`sec_combining-pcs`, however, it is
possible to use ``Amat`` instead of ``Pmat`` by calling
``PCSetUseAmat(pc)`` or using ``-pc_use_amat`` on the command line.
Alternatively, you can have ``PCFieldSplit`` extract the diagonal blocks
:math:`A_{00}, A_{11}` etc. out of ``Amat`` by calling
``PCFieldSplitSetDiagUseAmat(pc,PETSC_TRUE)`` or supplying command-line
argument ``-pc_fieldsplit_diag_use_amat``. Similarly,
``PCFieldSplitSetOffDiagUseAmat(pc,{PETSC_TRUE``) or
``-pc_fieldsplit_off_diag_use_amat`` will cause the off-diagonal blocks
:math:`A_{01},A_{10}` etc. to be extracted out of ``Amat``.

For two by two blocks only there are another family of solvers, based on
Schur complements. The inverse of the Schur complement factorization is

.. math::

   \left[
   \left( \begin{array}{cc}
   I   & 0 \\
   A_{10}A_{00}^{-1} & I \\
   \end{array} \right)
   \left( \begin{array}{cc}
   A_{00}  & 0 \\
   0 & S \\
   \end{array} \right)
   \left( \begin{array}{cc}
   I   & A_{00}^{-1} A_{01} \\
   0 & I \\
   \end{array} \right)
   \right]^{-1}

.. math::

   \left( \begin{array}{cc}
   I   & A_{00}^{-1} A_{01} \\
   0 & I \\
   \end{array} \right)^{-1}
   \left( \begin{array}{cc}
   A_{00}^{-1}  & 0 \\
   0 & S^{-1} \\
   \end{array} \right)
   \left( \begin{array}{cc}
   I   & 0 \\
   A_{10}A_{00}^{-1} & I \\
   \end{array} \right)^{-1}

.. math::

   \left( \begin{array}{cc}
   I   & -A_{00}^{-1} A_{01} \\
   0 & I \\
   \end{array} \right)
   \left( \begin{array}{cc}
   A_{00}^{-1}  & 0 \\
   0 & S^{-1} \\
   \end{array} \right)
   \left( \begin{array}{cc}
   I   & 0 \\
   -A_{10}A_{00}^{-1} & I \\
   \end{array} \right)

.. math::

   \left( \begin{array}{cc}
   A_{00}^{-1}   & 0 \\
   0 & I \\
   \end{array} \right)
   \left( \begin{array}{cc}
   I   & -A_{01} \\
   0 & I \\
   \end{array} \right)
   \left( \begin{array}{cc}
   A_{00}  & 0 \\
   0 & S^{-1} \\
   \end{array} \right)
   \left( \begin{array}{cc}
   I   & 0 \\
   -A_{10} & I \\
   \end{array} \right)
   \left( \begin{array}{cc}
   A_{00}^{-1}   & 0 \\
   0 & I \\
   \end{array} \right).

The preconditioner is accessed with ``-pc_fieldsplit_type schur`` and is
implemented as

.. math::

   \left( \begin{array}{cc}
     \text{ksp}(A_{00},Ap_{00})   & 0 \\
   0 & I \\
   \end{array} \right)
   \left( \begin{array}{cc}
   I   & -A_{01} \\
   0 & I \\
   \end{array} \right)
   \left( \begin{array}{cc}
   I  & 0 \\
     0 & \text{ksp}(\hat{S},\hat{S}p) \\
   \end{array} \right)
   \left( \begin{array}{cc}
   I   & 0 \\
     -A_{10} \text{ksp}(A_{00},Ap_{00}) & I \\
   \end{array} \right).

Where
:math:`\hat{S} = A_{11} - A_{10} \text{ksp}(A_{00},Ap_{00}) A_{01}` is
the approximate Schur complement.

There are several variants of the Schur complement preconditioner
obtained by dropping some of the terms, these can be obtained with
``-pc_fieldsplit_schur_fact_type <diag,lower,upper,full>`` or the
function ``PCFieldSplitSetSchurFactType()``. Note that the ``diag`` form
uses the preconditioner

.. math::

   \left( \begin{array}{cc}
     \text{ksp}(A_{00},Ap_{00})   & 0 \\
     0 & -\text{ksp}(\hat{S},\hat{S}p) \\
   \end{array} \right).

This is done to ensure the preconditioner is positive definite for a
common class of problems, saddle points with a positive definite
:math:`A_{00}`: for these the Schur complement is negative definite.

The effectiveness of the Schur complement preconditioner depends on the
availability of a good preconditioner :math:`\hat Sp` for the Schur
complement matrix. In general, you are responsible for supplying
:math:`\hat Sp` via
``PCFieldSplitSchurPrecondition(pc,PC_FIELDSPLIT_SCHUR_PRE_USER,Sp)``.
In the absence of a good problem-specific :math:`\hat Sp`, you can use
some of the built-in options.

Using ``-pc_fieldsplit_schur_precondition user`` on the command line
activates the matrix supplied programmatically as explained above.

With ``-pc_fieldsplit_schur_precondition a11`` (default)
:math:`\hat Sp = A_{11}` is used to build a preconditioner for
:math:`\hat S`.

Otherwise, ``-pc_fieldsplit_schur_precondition self`` will set
:math:`\hat Sp = \hat S` and use the Schur complement matrix itself to
build the preconditioner.

The problem with the last approach is that :math:`\hat S` is used in
unassembled, matrix-free form, and many preconditioners (e.g., ILU)
cannot be built out of such matrices. Instead, you can *assemble* an
approximation to :math:`\hat S` by inverting :math:`A_{00}`, but only
approximately, so as to ensure the sparsity of :math:`\hat Sp` as much
as possible. Specifically, using
``-pc_fieldsplit_schur_precondition selfp`` will assemble
:math:`\hat Sp = A_{11} - A_{10} \text{inv}(A_{00}) A_{01}`.

By default :math:`\text{inv}(A_{00})` is the inverse of the diagonal of
:math:`A_{00}`, but using
``-fieldsplit_1_mat_schur_complement_ainv_type lump`` will lump
:math:`A_{00}` first. Using
``-fieldsplit_1_mat_schur_complement_ainv_type blockdiag`` will use the
inverse of the block diagonal of :math:`A_{00}`. Option
``-mat_schur_complement_ainv_type`` applies to any matrix of
``MatSchurComplement`` type and here it is used with the prefix
``-fieldsplit_1`` of the linear system in the second split.

Finally, you can use the ``PCLSC`` preconditioner for the Schur
complement with ``-pc_fieldsplit_type schur -fieldsplit_1_pc_type lsc``.
This uses for the preconditioner to :math:`\hat{S}` the operator

.. math:: \text{ksp}(A_{10} A_{01},A_{10} A_{01}) A_{10} A_{00} A_{01} \text{ksp}(A_{10} A_{01},A_{10} A_{01})

which, of course, introduces two additional inner solves for each
application of the Schur complement. The options prefix for this inner
``KSP`` is ``-fieldsplit_1_lsc_``. Instead of constructing the matrix
:math:`A_{10} A_{01}` the user can provide their own matrix. This is
done by attaching the matrix/matrices to the :math:`Sp` matrix they
provide with

::

   PetscObjectCompose((PetscObject)Sp,"LSC_L",(PetscObject)L);
   PetscObjectCompose((PetscObject)Sp,"LSC_Lp",(PetscObject)Lp);

.. _sec_singular:

Solving Singular Systems
~~~~~~~~~~~~~~~~~~~~~~~~

Sometimes one is required to solver singular linear systems. In this
case, the system matrix has a nontrivial null space. For example, the
discretization of the Laplacian operator with Neumann boundary
conditions has a null space of the constant functions. PETSc has tools
to help solve these systems.

First, one must know what the null space is and store it using an
orthonormal basis in an array of PETSc Vecs. The constant functions can
be handled separately, since they are such a common case). Create a
``MatNullSpace`` object with the command

::

   MatNullSpaceCreate(MPI_Comm,PetscBool hasconstants,PetscInt dim,Vec *basis,MatNullSpace *nsp);

Here, ``dim`` is the number of vectors in ``basis`` and ``hasconstants``
indicates if the null space contains the constant functions. If the null
space contains the constant functions you do not need to include it in
the ``basis`` vectors you provide, nor in the count ``dim``.

One then tells the ``KSP`` object you are using what the null space is
with the call

::

   MatSetNullSpace(Mat Amat,MatNullSpace nsp);
   MatSetTransposeNullSpace(Mat Amat,MatNullSpace nsp);

The ``Amat`` should be the *first* matrix argument used with
``KSPSetOperators()``, ``SNESSetJacobian()``, or ``TSSetIJacobian()``.
You can also use ``KSPSetNullspace()``. The PETSc solvers will now
handle the null space during the solution process.

If one chooses a direct solver (or an incomplete factorization) it may
still detect a zero pivot. You can run with the additional options or
``-pc_factor_shift_type NONZERO``
``-pc_factor_shift_amount  <dampingfactor>`` to prevent the zero pivot.
A good choice for the ``dampingfactor`` is 1.e-10.

.. _sec_externalsol:

Using External Linear Solvers
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

PETSc interfaces to several external linear solvers (also see :any:`chapter_acknowledgements`)
at the beginning of this manual). To use these solvers, one may:

#. Run ``configure`` with the additional options
   ``--download-packagename`` e.g. ``--download-superlu_dist``
   ``--download-parmetis`` (SuperLU_DIST needs ParMetis) or
   ``--download-mumps`` ``--download-scalapack`` (MUMPS requires
   ScaLAPACK).

#. Build the PETSc libraries.

#. Use the runtime option: ``-ksp_type preonly`` ``-pc_type <pctype>``
   ``-pc_factor_mat_solver_type <packagename>``. For eg:
   ``-ksp_type preonly`` ``-pc_type lu``
   ``-pc_factor_mat_solver_type superlu_dist``.

.. list-table:: Options for External Solvers
   :name: tab-externaloptions
   :header-rows: 1

   * - MatType
     - PCType
     - MatSolverType
     - Package (``-pc_factor_mat_solver_type``)
   * - ``seqaij``
     - ``lu``
     - ``MATSOLVERESSL``
     - ``essl``
   * - ``seqaij``
     - ``lu``
     - ``MATSOLVERLUSOL``
     -  ``lusol``
   * - ``seqaij``
     - ``lu``
     - ``MATSOLVERMATLAB``
     - ``matlab``
   * - ``aij``
     - ``lu``
     - ``MATSOLVERMUMPS``
     - ``mumps``
   * - ``aij``
     - ``cholesky``
     - -
     - -
   * - ``sbaij``
     - ``cholesky``
     - -
     - -
   * - ``seqaij``
     - ``lu``
     - ``MATSOLVERSUPERLU``
     - ``superlu``
   * - ``aij``
     - ``lu``
     - ``MATSOLVERSUPERLU_DIST``
     - ``superlu_dist``
   * - ``seqaij``
     - ``lu``
     - ``MATSOLVERUMFPACK``
     - ``umfpack``
   * - ``seqaij``
     - ``cholesky``
     - ``MATSOLVERCHOLMOD``
     - ``cholmod``
   * - ``aij``
     - ``lu``
     - ``MATSOLVERCLIQUE``
     -  ``clique``
   * - ``seqaij``
     - ``lu``
     - ``MATSOLVERKLU``
     -  ``klu``
   * - ``dense``
     - ``lu``
     - ``MATSOLVERELEMENTAL``
     -  ``elemental``
   * - ``dense``
     - ``cholesky``
     - -
     - -
   * - ``seqaij``
     - ``lu``
     - ``MATSOLVERMKL_PARDISO``
     - ``mkl_pardiso``
   * - ``aij``
     - ``lu``
     - ``MATSOLVERMKL_CPARDISO``
     - ``mkl_cpardiso``
   * - ``aij``
     - ``lu``
     - ``MATSOLVERPASTIX``
     -  ``pastix``
   * - ``aij``
     - ``cholesky``
     - ``MATSOLVERBAS``
     -  ``bas``
   * - ``aijcusparse``
     - ``lu``
     - ``MATSOLVERCUSPARSE``
     - ``cusparse``
   * - ``aijcusparse``
     - ``cholesky``
     -  -
     -  -
   * - ``aij``
     - ``lu``, ``cholesky``
     - ``MATSOLVERPETSC``
     - ``petsc``
   * - ``baij``
     - -
     - -
     - -
   * - ``aijcrl``
     - -
     - -
     - -
   * - ``aijperm``
     - -
     - -
     - -
   * - ``seqdense``
     - -
     - -
     - -
   * - ``aij``
     - -
     - -
     - -
   * - ``baij``
     - -
     - -
     - -
   * - ``aijcrl``
     - -
     - -
     - -
   * - ``aijperm``
     - -
     - -
     - -
   * - ``seqdense``
     - -
     - -
     - -

The default and available input options for each external software can
be found by specifying ``-help`` at runtime.

As an alternative to using runtime flags to employ these external
packages, procedural calls are provided for some packages. For example,
the following procedural calls are equivalent to runtime options
``-ksp_type preonly`` ``-pc_type lu``
``-pc_factor_mat_solver_type mumps`` ``-mat_mumps_icntl_7 2``:

::

   KSPSetType(ksp,KSPPREONLY);
   KSPGetPC(ksp,&pc);
   PCSetType(pc,PCLU);
   PCFactorSetMatSolverType(pc,MATSOLVERMUMPS);
   PCFactorGetMatrix(pc,&F);
   icntl=7; ival = 2;
   MatMumpsSetIcntl(F,icntl,ival);

One can also create matrices with the appropriate capabilities by
calling ``MatCreate()`` followed by ``MatSetType()`` specifying the
desired matrix type from :any:`tab-externaloptions`. These
matrix types inherit capabilities from their PETSc matrix parents:
``seqaij``, ``mpiaij``, etc. As a result, the preallocation routines
``MatSeqAIJSetPreallocation()``, ``MatMPIAIJSetPreallocation()``, etc.
and any other type specific routines of the base class are supported.
One can also call ``MatConvert()`` inplace to convert the matrix to and
from its base class without performing an expensive data copy.
``MatConvert()`` cannot be called on matrices that have already been
factored.

In :any:`tab-externaloptions`, the base class ``aij`` refers
to the fact that inheritance is based on ``MATSEQAIJ`` when constructed
with a single process communicator, and from ``MATMPIAIJ`` otherwise.
The same holds for ``baij`` and ``sbaij``. For codes that are intended
to be run as both a single process or with multiple processes, depending
on the ``mpiexec`` command, it is recommended that both sets of
preallocation routines are called for these communicator morphing types.
The call for the incorrect type will simply be ignored without any harm
or message.

.. raw:: html

   <hr>

.. [2]
   See :any:`sec_amg` for information on using algebraic multigrid.

.. [3]
   This may seem an odd way to implement since it involves the "extra"
   multiply by :math:`-A_{11}`. The reason is this is implemented this
   way is that this approach works for any number of blocks that may
   overlap.

.. raw:: html

   <hr>

.. bibliography:: ../../tex/petsc.bib
   :filter: docname in docnames

.. bibliography:: ../../tex/petscapp.bib
   :filter: docname in docnames
