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
where \(A\) denotes the matrix representation of a linear operator,
\(b\) is the right-hand-side vector, and \(x\) 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. [FGN92] [vdV03], or [Saa03]) provide an
overview of the theory of such methods.
The KSP package, discussed in
Krylov Methods, provides many popular Krylov subspace
iterative methods; the PC module, described in
Preconditioners, includes a variety of preconditioners.
Using KSP#
To solve a linear system with KSP, one must first create a solver
context with the command
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 Matrix-Free Matrices
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 right hand size and solution vectors using the
command
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.
Convergence Tests 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 Block Jacobi and Overlapping Additive Schwarz Preconditioners) or
to log certain performance data using the PETSc profiling facilities (as
discussed in Profiling). In this case, the user can
optionally explicitly call
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,
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 Using External Linear Solvers) one may use
the options -ksp_type preonly (or the equivalent -ksp_type none)
-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.
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 the equivalent KSPNONE), or others; see
KSP Objects 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 the equivalent none), or others (see
KSP Objects 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
scale=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 Krylov vectors in GMRES is the unmodified (classical) Gram-Schmidt method, which can be set with
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
or via the options database with
-ksp_gmres_cgs_refinement_type <refine_never,refine_ifneeded,refine_always>
The values for KSPGMRESCGSRefinementType() are
KSP_GMRES_CGS_REFINE_NEVER, KSP_GMRES_CGS_REFINE_IFNEEDED
and KSP_GMRES_CGS_REFINE_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_CG_SYMMETRIC);
Note that this option is not valid for all matrices.
Some KSP types do not support preconditioning. For instance,
the CGLS algorithm does not involve a preconditioner; any preconditioner
set to work with the KSP object is ignored if KSPCGLS 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);
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 (1) by
where \(M_L\) and \(M_R\) indicate preconditioning matrices (or, matrices from which the preconditioner is to be constructed). If \(M_L = I\) in (2), right preconditioning results, and the residual of (1),
is preserved. In contrast, the residual is altered for left (\(M_R = I\)) and symmetric preconditioning, as given by
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,PC_RIGHT);
Attempting to use right preconditioning for a method that does not currently support it results in an error message of the form
KSPSetUp_Richardson:No right preconditioning for KSPRICHARDSON
We summarize the defaults for the residuals used in KSP convergence
monitoring within KSP Objects. 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
Method |
KSPType |
Options Database Name |
|---|---|---|
Richardson |
|
|
Chebyshev |
|
|
Conjugate Gradient [HS52] |
|
|
Pipelined Conjugate Gradients [GV14] |
|
|
Pipelined Conjugate Gradients (Gropp) |
|
|
Pipelined Conjugate Gradients with Residual Replacement |
|
|
Conjugate Gradients for the Normal Equations |
|
|
Flexible Conjugate Gradients [Not00] |
|
|
Pipelined, Flexible Conjugate Gradients [SSM16] |
|
|
Conjugate Gradients for Least Squares |
|
|
Conjugate Gradients with Constraint (1) |
|
|
Conjugate Gradients with Constraint (2) |
|
|
Conjugate Gradients with Constraint (3) |
|
|
Conjugate Gradients with Constraint (4) |
|
|
BiConjugate Gradient |
|
|
BiCGSTAB [vandVorst92] |
|
|
Improved BiCGSTAB |
|
|
QMRCGSTAB [CGS+94] |
|
|
Flexible BiCGSTAB |
|
|
Flexible BiCGSTAB (variant) |
|
|
Enhanced BiCGSTAB(L) |
|
|
Minimal Residual Method [PS75] |
|
|
Generalized Minimal Residual [SS86] |
|
|
Flexible Generalized Minimal Residual [Saa93] |
|
|
Deflated Generalized Minimal Residual |
|
|
Pipelined Generalized Minimal Residual [GAMV13] |
|
|
Pipelined, Flexible Generalized Minimal Residual [SSM16] |
|
|
Generalized Minimal Residual with Accelerated Restart |
|
|
Conjugate Residual [EES83] |
|
|
Generalized Conjugate Residual |
|
|
Pipelined Conjugate Residual |
|
|
Pipelined, Flexible Conjugate Residual [SSM16] |
|
|
FETI-DP |
|
|
Conjugate Gradient Squared [Son89] |
|
|
Transpose-Free Quasi-Minimal Residual (1) [Fre93] |
|
|
Transpose-Free Quasi-Minimal Residual (2) |
|
|
Least Squares Method |
|
|
Symmetric LQ Method [PS75] |
|
|
TSIRM |
|
|
Python Shell |
|
|
Shell for no |
|
|
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 constrained 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.
Convergence Tests#
The default convergence test, KSPConvergedDefault(), is based on the
\(l_2\)-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 \(k\) if
where \(r_k = b - A x_k\). Divergence is detected if
These parameters, as well as the maximum number of allowable iterations, can be set with the routine
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
\(l_2\) 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 is given in the KSPConvergedReason manual page.
You can use KSPGetConvergedReason() after
KSPSolve() to see why convergence/divergence was detected.
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
\(l_2\) 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 Writing PETSc Programs 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 \(l_2\)-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 KSPMonitorTrueResidual() prints the true
residual at each iteration by actually computing the residual using the
formula \(r = b - Ax\), 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 \(10^{-10}\), 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
Then after the KSP solve one calls
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_explicit draw. Or they can be dumped to the
screen in ASCII text via -ksp_view_eigenvalues and
-ksp_view_eigenvalues_explicit.
Flexible Krylov Methods#
Standard Krylov methods require that the preconditioner be a linear operator, thus, for example, a standard KSP method
cannot use a KSP in its preconditioner, as is common in the Block-Jacobi method PCBJACOBI, for example.
Flexible Krylov methods are a subset of methods that allow (with modest additional requirements
on memory) the preconditioner to be nonlinear. For example, they can be used with the PCKSP preconditioner.
The flexible KSP methods have the label “Flexible” in KSP Objects.
One can use KSPMonitorDynamicTolerance() to control the tolerances used by inner KSP solvers in PCKSP, PCBJACOBI, and PCDEFLATION.
In addition to supporting PCKSP, the flexible methods support KSP*SetModifyPC(), for example, KSPFGMRESSetModifyPC(), these functions
allow the user to provide a callback function that changes the preconditioner at each Krylov iteration. Its calling sequence is as follows.
PetscErrorCode f(KSP ksp,PetscInt total_its,PetscInt its_since_restart,PetscReal res_norm,void *ctx);
Pipelined Krylov Methods#
Standard Krylov methods have one or more global reductions resulting from the computations of inner products or norms in each iteration.
These reductions need to block until all MPI ranks have received the results. For a large number of MPI ranks (this number is machine dependent
but can be above 10,000 ranks) this synchronization is very time consuming and can significantly slow the computation. Pipelined Krylov
methods overlap the reduction operations with local computations (generally the application of the matrix-vector products and precondtiioners)
thus effectively “hiding” the time of the reductions. In addition, they may reduce the number of global synchronizations by rearranging the
computations in a way that some of them can be collapsed, e.g., two or more calls to MPI_Allreduce() may be combined into one call.
The pipeline KSP methods have the label “Pipeline” in KSP Objects.
Special configuration of MPI may be necessary for reductions to make asynchronous progress, which is important for performance of pipelined methods. See What steps are necessary to make the pipelined solvers execute efficiently? for details.
Other KSP Options#
To obtain the solution vector and right hand side from a KSP
context, one uses
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 even require 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.
Preconditioners#
As discussed in Preconditioning within KSP, 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
In PETSc Preconditioners (partial list) 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.
Method |
PCType |
Options Database Name |
|---|---|---|
Jacobi |
|
|
Block Jacobi |
|
|
SOR (and SSOR) |
|
|
SOR with Eisenstat trick |
|
|
Incomplete Cholesky |
|
|
Incomplete LU |
|
|
Additive Schwarz |
|
|
Generalized Additive Schwarz |
|
|
Algebraic Multigrid |
|
|
Balancing Domain Decomposition by Constraints |
|
|
Linear solver |
|
|
Combination of preconditioners |
|
|
LU |
|
|
Cholesky |
|
|
No preconditioning |
|
|
Shell for user-defined |
|
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.
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 Memory Allocation for Sparse Matrix Factorization 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 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 [Eis81] 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.
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- NaturalMATORDERINGND- Nested DissectionMATORDERING1WD- One-way DissectionMATORDERINGRCM- Reverse Cuthill-McKeeMATORDERINGQMD- 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 Matrix Factorization.
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 an 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, Memory Allocation for Sparse Matrix Factorization 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.
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 (or equivalently KSPType=KSPNONE); 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 \(b_g-1\), where \(b_g\) 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
or
KSP Tutorial ex8.
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
[CS97]. 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
consecutive 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.
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().
Algebraic Multigrid (AMG) Preconditioners#
PETSc has a native algebraic multigrid preconditioner PCGAMG –
gamg – and interfaces to three external AMG packages: hypre, ML
and AMGx (CUDA platforms only), that can be downloaded in the
configuration phase (eg, --download-hypre ) and used by
specifiying that command line parameter (eg, -pc_type hypre).
Hypre is relatively monolithic in that a PETSc matrix is converted 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
[ISG15]. 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,
used 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.,
\(x_1,y_1,z_1,x_2,y_2,...\)).
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 \(P_i < P\) processors are desired on a level
\(i\) then the first \(P_i\) 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 (\(q\)) is selected
and the equations on the first \(q\) processes are moved 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 PCGAMG with -help to get full listing of GAMG
parameters with short parameter descriptions. The rate of coarsening is
critical in AMG performance – too slow 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 <-1>
and -pc_gamg_aggressive_coarsening <N> 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 \(0.01 - 0.05\). There
are additional parameters for changing the weights on coarse grids.
The parallel MIS algorithms requires symmetric weights/matrix. Thus PCGAMG
will automatically make the graph symmetric if it is not symmetric. Since this
has additional cost users should indicate the symmetry of the matrices they
provide by calling
MatSetOption``(mat,``MAT_SYMMETRIC,``PETSC_TRUE`` (or PETSC_FALSE))
or
MatSetOption``(mat,``MAT_STRUCTURALLY_SYMMETRIC,``PETSC_TRUE`` (or PETSC_FALSE))
. If they know that the matrix will always have symmetry, despite future changes
to the matrix (with, for example, MatSetValues()) then they should also call
MatSetOption``(mat,``MAT_SYMMETRY_ETERNAL,``PETSC_TRUE`` (or PETSC_FALSE))
or
MatSetOption``(mat,``MAT_STRUCTURAL_SYMMETRY_ETERNAL,``PETSC_TRUE`` (or PETSC_FALSE)).
Using this information allows the algorithm to skip the unnecessary computations.
Trouble shooting algebraic multigrid methods: If GAMG, ML, AMGx 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>
or PCGAMGSetThreshold() to regulate coarsening rates, higher values decrease
coarsening rate. Squaring the graph is the second mechanism for
increasing coarsening rate. Use -pc_gamg_aggressive_coarsening <N>, or
PCGAMGSetAggressiveLevels(pc,N), to aggressive ly coarsen (MIS-2) the graph on the finest N
levels. A high threshold (e.g., \(x=0.08\)) will result in an
expensive but potentially powerful preconditioner, and a low threshold
(e.g., \(x=0.0\)) 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 ratio 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 \(1.3\) 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 \(-1.0\) 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 one 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 using aggressive coarsening
(-pc_gamg_aggressive_coarsening <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.
Adaptive Interpolation#
Interpolation transfers a function from the coarse space to the fine space. We would like this process to be accurate for the functions resolved by the coarse grid, in particular the approximate solution computed there. By default, we create these matrices using local interpolation of the fine grid dual basis functions in the coarse basis. However, an adaptive procedure can optimize the coefficients of the interpolator to reproduce pairs of coarse/fine functions which should approximate the lowest modes of the generalized eigenproblem
where \(A\) is the system matrix and \(M\) is the smoother. Note that for defect-correction MG, the interpolated solution from the coarse space need not be as accurate as the fine solution, for the same reason that updates in iterative refinement can be less accurate. However, in FAS or in the final interpolation step for each level of Full Multigrid, we must have interpolation as accurate as the fine solution since we are moving the entire solution itself.
Injection should accurately transfer the fine solution to the coarse grid. Accuracy here means that the action of a coarse dual function on either should produce approximately the same result. In the structured grid case, this means that we just use the same values on coarse points. This can result in aliasing.
Restriction is intended to transfer the fine residual to the coarse space. Here we use averaging (often the transpose of the interpolation operation) to damp out the fine space contributions. Thus, it is less accurate than injection, but avoids aliasing of the high modes.
For a multigrid cycle, the interpolator \(P\) is intended to accurately reproduce “smooth” functions from the coarse space in the fine space, keeping the energy of the interpolant about the same. For the Laplacian on a structured mesh, it is easy to determine what these low-frequency functions are. They are the Fourier modes. However an arbitrary operator \(A\) will have different coarse modes that we want to resolve accurately on the fine grid, so that our coarse solve produces a good guess for the fine problem. How do we make sure that our interpolator \(P\) can do this?
We first must decide what we mean by accurate interpolation of some functions. Suppose we know the continuum function \(f\) that we care about, and we are only interested in a finite element description of discrete functions. Then the coarse function representing \(f\) is given by
and similarly the fine grid form is
Now we would like the interpolant of the coarse representer to the fine grid to be as close as possible to the fine representer in a least squares sense, meaning we want to solve the minimization problem
Now we can express \(P\) as a matrix by looking at the matrix elements \(P_{ij} = \phi^F_i P \phi^C_j\). Then we have
so that our discrete optimization problem is
and we will treat each row of the interpolator as a separate optimization problem. We could allow an arbitrary sparsity pattern, or try to determine adaptively, as is done in sparse approximate inverse preconditioning. However, we know the supports of the basis functions in finite elements, and thus the naive sparsity pattern from local interpolation can be used.
We note here that the BAMG framework of Brannick et al. [BBKL11] does not use fine and coarse functions spaces, but rather a fine point/coarse point division which we will not employ here. Our general PETSc routine should work for both since the input would be the checking set (fine basis coefficients or fine space points) and the approximation set (coarse basis coefficients in the support or coarse points in the sparsity pattern).
We can easily solve the above problem using QR factorization. However, there are many smooth functions from the coarse space that we want interpolated accurately, and a single \(f\) would not constrain the values \(P_{ij}\) well. Therefore, we will use several functions \(\{f_k\}\) in our minimization,
where
or alternatively
We thus have a standard least-squares problem
where
which can be solved using LAPACK.
We will typically perform this optimization on a multigrid level \(l\) when the change in eigenvalue from level \(l+1\) is relatively large, meaning
This indicates that the generalized eigenvector associated with that eigenvalue was not adequately represented by \(P^l_{l+1}\), and the interpolator should be recomputed.
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 Dirichlet 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:
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
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().
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 Multigrid Preconditioners.
Let \(B_1\) and \(B_2\) represent the application of two
preconditioners of type type1 and type2. The preconditioner
\(B = B_1 + B_2\) 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
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: \(B_1\) and \(B_2\); \(y = B x\) is obtained from
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 \(A\) 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
\(B_1\), \(B_2\), 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.
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.
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 encapsulated 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
A large number of parameters affect the multigrid behavior. The command
PCMGSetType(PC pc,PCMGType mode);
indicates which form of multigrid to apply [SBjorstadG96].
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
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
Matrix-Free Matrices); 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 \(b - Ax\), 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
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.
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
where each \(A_{ij}\) 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 \(A_{0*}\), \(A_{1*}\) etc. On a process, the blocks may be stored one block followed by another
or interlaced, for example with two blocks
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 Block Matrices 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
On occasion the user may provide another matrix that is used to construct parts of the preconditioner
For notational simplicity define \(\text{ksp}(A,Ap)\) to mean
approximately solving a linear system using KSP with operator
\(A\) and preconditioner built from matrix \(Ap\).
For matrices defined with any number of blocks there are three “block” algorithms available: block Jacobi,
block Gauss-Seidel,
which is implemented 3 as
and symmetric block Gauss-Seidel
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 \(A_{00}, A_{01}\) 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., \(A\) itself). As discussed above in
Combining Preconditioners, 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
\(A_{00}, A_{11}\) 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
\(A_{01},A_{10}\) 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
The preconditioner is accessed with -pc_fieldsplit_type schur and is
implemented as
Where \(\hat{S} = A_{11} - A_{10} \text{ksp}(A_{00},Ap_{00}) A_{01}\) 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
This is done to ensure the preconditioner is positive definite for a common class of problems, saddle points with a positive definite \(A_{00}\): for these the Schur complement is negative definite.
The effectiveness of the Schur complement preconditioner depends on the
availability of a good preconditioner \(\hat Sp\) for the Schur
complement matrix. In general, you are responsible for supplying
\(\hat Sp\) via
PCFieldSplitSchurPrecondition(pc,PC_FIELDSPLIT_SCHUR_PRE_USER,Sp).
In the absence of a good problem-specific \(\hat Sp\), 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)
\(\hat Sp = A_{11}\) is used to build a preconditioner for
\(\hat S\).
Otherwise, -pc_fieldsplit_schur_precondition self will set
\(\hat Sp = \hat S\) and use the Schur complement matrix itself to
build the preconditioner.
The problem with the last approach is that \(\hat S\) 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 \(\hat S\) by inverting \(A_{00}\), but only
approximately, so as to ensure the sparsity of \(\hat Sp\) as much
as possible. Specifically, using
-pc_fieldsplit_schur_precondition selfp will assemble
\(\hat Sp = A_{11} - A_{10} \text{inv}(A_{00}) A_{01}\).
By default \(\text{inv}(A_{00})\) is the inverse of the diagonal of
\(A_{00}\), but using
-fieldsplit_1_mat_schur_complement_ainv_type lump will lump
\(A_{00}\) first. Using
-fieldsplit_1_mat_schur_complement_ainv_type blockdiag will use the
inverse of the block diagonal of \(A_{00}\). 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 \(\hat{S}\) the operator
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
\(A_{10} A_{01}\) the user can provide their own matrix. This is
done by attaching the matrix/matrices to the \(Sp\) matrix they
provide with
PetscObjectCompose((PetscObject)Sp,"LSC_L",(PetscObject)L);
PetscObjectCompose((PetscObject)Sp,"LSC_Lp",(PetscObject)Lp);
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. This approach is only guaranteed to work for left preconditioning (see KSPSetPCSide()); for example it
may not work in some situations with KSPFGMRES.
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);
The Amat should be the first matrix argument used with
KSPSetOperators(), SNESSetJacobian(), or TSSetIJacobian().
The PETSc solvers will now
handle the null space during the solution process.
If the right hand side of linear system is not in the range of Amat, that is it is not
orthogonal to the null space of Amat transpose, then the residual
norm of the Krylov iteration will not converge to zero; it will converge to a non-zero value while the
solution is converging to the least squares solution of the linear system. One can, if one desires,
apply MatNullSpaceRemove() with the null space of Amat transpose to the right hand side before calling
KSPSolve(). Then the residual norm will converge to zero.
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.
If the matrix is non-symmetric and you wish to solve the transposed linear system
you must provide the null space of the transposed matrix with MatSetTransposeNullSpace().
Using External Linear Solvers#
PETSc interfaces to several external linear solvers (also see Acknowledgments). To use these solvers, one may:
Run
configurewith the additional options--download-packagenamee.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(or equivalently-ksp_type none)-pc_type <pctype>-pc_factor_mat_solver_type <packagename>. For eg:-ksp_type preonly-pc_type lu-pc_factor_mat_solver_type superlu_dist.
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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 (or equivalently -ksp_type none) -pc_type lu
-pc_factor_mat_solver_type mumps -mat_mumps_icntl_7 3:
KSPSetType(ksp,KSPPREONLY); (or equivalently KSPSetType(ksp,KSPNONE))
KSPGetPC(ksp,&pc);
PCSetType(pc,PCLU);
PCFactorSetMatSolverType(pc,MATSOLVERMUMPS);
PCFactorSetUpMatSolverType(pc);
PCFactorGetMatrix(pc,&F);
icntl=7; ival = 3;
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 Options for External Solvers. 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 Options for External Solvers, 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.
Using a MPI parallel linear solver from a non-MPI program#
Using PETSc’s MPI linear solver server it is possible to use multiple MPI processes to solve a
a linear system when the application code, including the matrix generation, is run on a single
MPI rank (with or without OpenMP). The application code must be built with MPI and must call
PetscIntialize() at the very beginning of the program and end with PetscFinalize(). The
application code may utilize OpenMP.
The code may create multiple matrices and KSP objects and call KSPSolve(), similarly the
code may utilize the SNES nonlinear solvers, the TS ODE integrators, and the TAO optimization algorithms
which use KSP.
Amdahl’s law makes clear that parallelizing only a portion of a numerical code can only provide a limited improvement in the computation time; thus it is crucial to understand what phases of a computation must be parallelized (via MPI, OpenMP, or some other model) to ensure a useful increase in performance. One of the crucial phases is likely the generation of the matrix entries; the use of MatSetPreallocationCOO() and MatSetValuesCOO() in an OpenMP code allows parallelizing the generation of the matrix.
The program must then be launched using the standard approaches for launching MPI programs with the option -mpi_linear_solver_server and options to utilize the PCMPI preconditioners; for example, -ksp_type preonly and pc_type mpi. Any standard solver options may be passed to the parallel solvers using the options prefix -mpi_; for example, -mpi_ksp_type cg. The option -mpi_linear_solver_server_view will print a summary of all the systems solved by the MPI linear solver server.
Footnotes
- 2
See Algebraic Multigrid (AMG) Preconditioners for information on using algebraic multigrid.
- 3
This may seem an odd way to implement since it involves the “extra” multiply by \(-A_{11}\). The reason is this is implemented this way is that this approach works for any number of blocks that may overlap.
References
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X.-C. Cai and M. Sarkis. A restricted additive Schwarz preconditioner for general sparse linear systems. Technical Report CU-CS 843-97, Computer Science Department, University of Colorado-Boulder, 1997. (accepted by SIAM J. of Scientific Computing).
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Tony F Chan, Efstratios Gallopoulos, Valeria Simoncini, Tedd Szeto, and Charles H Tong. A quasi-minimal residual variant of the Bi-CGSTAB algorithm for nonsymmetric systems. SIAM Journal on Scientific Computing, 15(2):338–347, 1994.
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S. Eisenstat. Efficient implementation of a class of CG methods. SIAM J. Sci. Stat. Comput., 2:1–4, 1981.
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S.C. Eisenstat, H.C. Elman, and M.H. Schultz. Variational iterative methods for nonsymmetric systems of linear equations. SIAM Journal on Numerical Analysis, 20(2):345–357, 1983.
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Yousef Saad. Iterative Methods for Sparse Linear Systems. SIAM, 2nd edition, 2003. doi:10.1016/S1570-579X(01)80025-2.
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Barry F. Smith, Petter Bjørstad, and William D. Gropp. Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, 1996. URL: http://www.mcs.anl.gov/~bsmith/ddbook.html.
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Peter Sonneveld. CGS, a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 10:36–52, 1989.
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H. van der Vorst. Iterative Krylov Methods for Large Linear Systems. Cambridge University Press, 2003. ISBN 9780521818285.
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H. A. van der Vorst. BiCGSTAB: a fast and smoothly converging variant of BiCG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 13:631–644, 1992.