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

(1)\[ A x = b,\]

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

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 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

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.

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

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 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

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 <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

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

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),

\[r \equiv b - Ax = b - A M_R^{-1} \, M_R x, \]

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

\[r_L \equiv M_L^{-1} b - M_L^{-1} A x = M_L^{-1} r. \]

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,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

KSPSetNormType(ksp,KSP_NORM_UNPRECONDITIONED);

Table 6 KSP Objects

Method	KSPType	Options Database
Richardson	KSPRICHARDSON	richardson
Chebyshev	KSPCHEBYSHEV	chebyshev
Conjugate Gradient [HS52]	KSPCG	cg
Pipelined Conjugate Gradients [GV14]	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 [Not00]	KSPFCG	fcg
Pipelined, Flexible Conjugate Gradients [SSM16]	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 [vandVorst92]	KSPBCGS	bcgs
Improved BiCGSTAB	KSPIBCGS	ibcgs
QMRCGSTAB [CGS+94]	KSPQMRCGS	qmrcgs
Flexible BiCGSTAB	KSPFBCGS	fbcgs
Flexible BiCGSTAB (variant)	KSPFBCGSR	fbcgsr
Enhanced BiCGSTAB(L)	KSPBCGSL	bcgsl
Minimal Residual Method [PS75]	KSPMINRES	minres
Generalized Minimal Residual [SS86]	KSPGMRES	gmres
Flexible Generalized Minimal Residual [Saa93]	KSPFGMRES	fgmres
Deflated Generalized Minimal Residual	KSPDGMRES	dgmres
Pipelined Generalized Minimal Residual [GAMV13]	KSPPGMRES	pgmres
Pipelined, Flexible Generalized Minimal Residual [SSM16]	KSPPIPEFGMRES	pipefgmres
Generalized Minimal Residual with Accelerated Restart	KSPLGMRES	lgmres
Conjugate Residual [EES83]	KSPCR	cr
Generalized Conjugate Residual	KSPGCR	gcr
Pipelined Conjugate Residual	KSPPIPECR	pipecr
Pipelined, Flexible Conjugate Residual [SSM16]	KSPPIPEGCR	pipegcr
FETI-DP	KSPFETIDP	fetidp
Conjugate Gradient Squared [Son89]	KSPCGS	cgs
Transpose-Free Quasi-Minimal Residual (1) [Fre93]	KSPTFQMR	tfqmr
Transpose-Free Quasi-Minimal Residual (2)	KSPTCQMR	tcqmr
Least Squares Method	KSPLSQR	lsqr
Symmetric LQ Method [PS75]	KSPSYMMLQ	symmlq
TSIRM	KSPTSIRM	tsirm
Python Shell	KSPPYTHON	python
Shell for no KSP method	KSPNONE	none

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

\[\| r_k \|_2 < {\rm max} ( \text{rtol} * \| b \|_2, \text{atol}), \]

where \(r_k = b - A x_k\). Divergence is detected if

\[\| r_k \|_2 > \text{dtol} * \| b \|_2. \]

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 current value of any of these parameters by specifying PETSC_CURRENT as the corresponding tolerance; the defaults are rtol=1e-5, atol=1e-50, dtol=1e5, and maxits=1e4. Using PETSC_DETERMINE will set the parameters back to their initial values when the object’s type was set. 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

KSPSetComputeEigenvalues(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_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 processes have received the results. For a large number of MPI processes (this number is machine dependent but can be above 10,000 processes) 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

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 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

PCSetType(PC pc,PCType method);

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.

Table 7 PETSc Preconditioners (partial list)

Method	PCType	Options Database
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.

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 - 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 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 [CS99]. 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 processes. 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 processes. The simplest way to do this is using a call to PCGASMSetTotalSubdomains(PC pc,PetscInt 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 processes. In general this might not result in a very good subproblem if the single-rank problems corresponding to the coalesced processes 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 process, 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 (e.g., --download-hypre ) and used by specifying that command line parameter (e.g., -pc_type hypre). Hypre is relatively monolithic in that a PETSc matrix is converted into a hypre matrix, and then hypre is called to solve the entire problem. ML is more modular because PETSc only has ML generate the coarse grid spaces (columns of the prolongation operator), which is the core of an AMG method, and then constructs a PCMG with Galerkin coarse grid operator construction. PCGAMG 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 (see Multigrid Preconditioners). PETSc provides a fully supported (smoothed) aggregation AMG, but supports the addition of new methods (-pc_type gamg -pc_gamg_type agg or PCSetType(pc,PCGAMG) and PCGAMGSetType(pc, PCGAMGAGG). Examples of extension are reference implementations of a classical AMG method (-pc_gamg_type classical), a (2D) hybrid geometric AMG method (-pc_gamg_type geo) that are not supported. A 2.5D AMG method DofColumns [ISG15] supports 2D coarsenings extruded in the third dimension. PCGAMG does require the use of MATAIJ matrices. For instance, MATBAIJ matrices are not supported. One can use MATAIJ instead of MATBAIJ without changing any code other than the constructor (or the -mat_type from the command line). For instance, MatSetValuesBlocked works with MATAIJ matrices.

Important parameters for PCGAMGAGG

• Control the generation of the coarse grid

  • -pc_gamg_aggressive_coarsening <n:int:1> Use aggressive coarsening on the finest n levels to construct the coarser mesh. See PCGAMGAGGSetNSmooths(). The larger value produces a faster preconditioner to create and solve, but the convergence may be slower.

  • -pc_gamg_low_memory_threshold_filter <bool:false> Filter small matrix entries before coarsening the mesh. See PCGAMGSetLowMemoryFilter().

  • -pc_gamg_threshold <tol:real:0.0> The threshold of small values to drop when -pc_gamg_low_memory_threshold_filter is used. A negative value means keeping even the locations with 0.0. See PCGAMGSetThreshold()

  • -pc_gamg_threshold_scale <v>:real:1.0> Set a scale factor applied to each coarser level when -pc_gamg_low_memory_threshold_filter is used. See PCGAMGSetThresholdScale().

  • -pc_gamg_mat_coarsen_type <mis|hem|misk:misk> Algorithm used to coarsen the matrix graph. See MatCoarsenSetType().

  • -pc_gamg_mat_coarsen_max_it <it:int:4> Maximum HEM iterations to use. See MatCoarsenSetMaximumIterations().

  • -pc_gamg_aggressive_mis_k <k:int:2> k distance in MIS coarsening (>2 is ‘aggressive’) to use in coarsening. See PCGAMGMISkSetAggressive(). The larger value produces a preconditioner that is faster to create and solve with but the convergence may be slower. This option and the previous option work to determine how aggressively the grids are coarsened.

  • -pc_gamg_mis_k_minimum_degree_ordering <bool:true> Use a minimum degree ordering in the greedy MIS algorithm used to coarsen. See PCGAMGMISkSetMinDegreeOrdering()

• Control the generation of the prolongation for PCGAMGAGG

  • -pc_gamg_agg_nsmooths <n:int:1> Number of smoothing steps to be used in constructing the prolongation. For symmetric problems, generally, one or more is best. For some strongly nonsymmetric problems, 0 may be best. See PCGAMGSetNSmooths().

• Control the amount of parallelism on the levels

  • -pc_gamg_process_eq_limit <n:int:50> Sets the minimum number of equations allowed per process when coarsening (otherwise, fewer MPI processes are used for the coarser mesh). A larger value will cause the coarser problems to be run on fewer MPI processes, resulting in less communication and possibly a faster time to solution. See PCGAMGSetProcEqLim().

  • -pc_gamg_rank_reduction_factors <rn,rn-1,…,r1:int> Set a schedule for MPI rank reduction on coarse grids. See PCGAMGSetRankReductionFactors() This overrides the lessening of processes that would arise from -pc_gamg_process_eq_limit.

  • -pc_gamg_repartition <bool:false> Run a partitioner on each coarser mesh generated rather than using the default partition arising from the finer mesh. See PCGAMGSetRepartition(). This increases the preconditioner setup time but will result in less time per iteration of the solver.

  • -pc_gamg_parallel_coarse_grid_solver <bool:false> Allow the coarse grid solve to run in parallel, depending on the value of -pc_gamg_coarse_eq_limit. See PCGAMGSetParallelCoarseGridSolve(). If the coarse grid problem is large then this can improve the time to solution.

    • -pc_gamg_coarse_eq_limit <n:int:50> Sets the minimum number of equations allowed per process on the coarsest level when coarsening (otherwise fewer MPI processes will be used). A larger value will cause the coarse problems to be run on fewer MPI processes. This only applies if -pc_gamg_parallel_coarse_grid_solver is set to true. See PCGAMGSetCoarseEqLim().

• Control the smoothers

  • -pc_mg_levels <n:int> Set the maximum number of levels to use.

  • -mg_levels_ksp_type <KSPType:chebyshev> If KSPCHEBYSHEV or KSPRICHARDSON is not used, then the Krylov method for the entire multigrid solve has to be a flexible method such as KSPFGMRES. Generally, the stronger the Krylov method the faster the convergence, but with more cost per iteration. See KSPSetType().

  • -mg_levels_ksp_max_it <its:int:2> Sets the number of iterations to run the smoother on each level. Generally, the more iterations , the faster the convergence, but with more cost per multigrid iteration. See PCMGSetNumberSmooth().

  • -mg_levels_ksp_xxx Sets options for the KSP in the smoother on the levels.

  • -mg_levels_pc_type <PCType:jacobi> Sets the smoother to use on each level. See PCSetType(). Generally, the stronger the preconditioner the faster the convergence, but with more cost per iteration.

  • -mg_levels_pc_xxx Sets options for the PC in the smoother on the levels.

  • -mg_coarse_ksp_type <KSPType:none> Sets the solver KSPType to use on the coarsest level.

  • -mg_coarse_pc_type <PCType:lu> Sets the solver PCType to use on the coarsest level.

  • -pc_gamg_asm_use_agg <bool:false> Use PCASM as the smoother on each level with the aggregates defined by the coarsening process are the subdomains. This option automatically switches the smoother on the levels to be PCASM.

  • -mg_levels_pc_asm_overlap <n:int:0> Use non-zero overlap with -pc_gamg_asm_use_agg. See PCASMSetOverlap().

• Control the multigrid algorithm

  • -pc_mg_type <additive|multiplicative|full|kaskade:multiplicative> The type of multigrid to use. Usually, multiplicative is the fastest.

  • -pc_mg_cycle_type <v|w:v> Use V- or W-cycle with -pc_mg_type multiplicative

PCGAMG provides unsmoothed aggregation (-pc_gamg_agg_nsmooths 0) and smoothed aggregation (-pc_gamg_agg_nsmooths 1 or PCGAMGSetNSmooths(pc,1)). Smoothed aggregation (SA), [VanvekMB96], [VanveekBM01], is recommended for symmetric positive definite systems. Unsmoothed aggregation can be useful for asymmetric problems and problems where the highest eigenestimates 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 faster 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 as with 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 require 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 solvers 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 reduce communication costs potentially). However, 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. GAMG will select the number of active processors by fitting the desired number of equations 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 \(P_i\) processes 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 (e.g., LU or SVD) in a relatively short time. GAMG will stop coarsening when the number of the equation on a grid falls below the threshold given 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 using PCGAMG with -help to get a full listing of GAMG parameters with short 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 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, and 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 require symmetric weights/matrices. 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 unnecessary computations.

Troubleshooting algebraic multigrid methods: If PCGAMG, 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 the 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 :pc and grep for PCGAMG to get 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 slowly 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, 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, decreasing the coarsening rate (increasing the threshold) should be tried if convergence is slow.

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), the solvers can fail with an indefinite preconditioner message. One can run with -info and grep for PCGAMG 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. Bad Eigen estimates often cause indefinite matrix messages. 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. However, 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 short and your convergence If the rate is not ideal, you could decrease the coarsening rate.

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

Using algebraic multigrid as a “standalone” solver is possible but not recommended, as it does not accelerate it with a Krylov method. Use a KSPType of KSPRICHARDSON (or equivalently -ksp_type richardson) to achieve this. Using KSPPREONLY will not work since it only applies a single multigrid cycle.

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

\[A x = \lambda M x\]

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

\[f^C = \sum_i f^C_i \phi^C_i,\]

and similarly the fine grid form is

\[f^F = \sum_i f^F_i \phi^F_i.\]

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

\[\min_{P} \| f^F - P f^C \|_2\]

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

\[\begin{aligned} &\phi^F_i f^F - \phi^F_i P f^C \\ = &f^F_i - \sum_j P_{ij} f^C_j \end{aligned}\]

so that our discrete optimization problem is

\[\min_{P_{ij}} \| f^F_i - \sum_j P_{ij} f^C_j \|_2\]

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,

\[\begin{aligned} &\min_{P_{ij}} \sum_k w_k \| f^{F,k}_i - \sum_j P_{ij} f^{C,k}_j \|_2 \\ = &\min_{P_{ij}} \sum_k \| \sqrt{w_k} f^{F,k}_i - \sqrt{w_k} \sum_j P_{ij} f^{C,k}_j \|_2 \\ = &\min_{P_{ij}} \| W^{1/2} \mathbf{f}^{F}_i - W^{1/2} \mathbf{f}^{C} p_i \|_2 \end{aligned}\]

where

\[\begin{aligned} W &= \begin{pmatrix} w_0 & & \\ & \ddots & \\ & & w_K \end{pmatrix} \\ \mathbf{f}^{F}_i &= \begin{pmatrix} f^{F,0}_i \\ \vdots \\ f^{F,K}_i \end{pmatrix} \\ \mathbf{f}^{C} &= \begin{pmatrix} f^{C,0}_0 & \cdots & f^{C,0}_n \\ \vdots & \ddots & \vdots \\ f^{C,K}_0 & \cdots & f^{C,K}_n \end{pmatrix} \\ p_i &= \begin{pmatrix} P_{i0} \\ \vdots \\ P_{in} \end{pmatrix} \end{aligned}\]

or alternatively

\[\begin{aligned} [W]_{kk} &= w_k \\ [f^{F}_i]_k &= f^{F,k}_i \\ [f^{C}]_{kj} &= f^{C,k}_j \\ [p_i]_j &= P_{ij} \end{aligned}\]

We thus have a standard least-squares problem

\[\min_{P_{ij}} \| b - A x \|_2\]

where

\[\begin{aligned} A &= W^{1/2} f^{C} \\ b &= W^{1/2} f^{F}_i \\ x &= p_i \end{aligned}\]

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

\[\frac{|\lambda_l - \lambda_{l+1}|}{|\lambda_l|}.\]

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 (PCBDDC) method for preconditioning parallel finite element problems stored in unassembled format (see MATIS). PCBDDC 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:

• 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 PCBDDC 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 PCBDDC 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 PCBDDC` 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 PCBDDC 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

PCCompositeSetType(pc,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: \(B_1\) and \(B_2\); \(y = B x\) is obtained from

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

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 \(A\) 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.

PCSetType(PC pc,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.

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

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 [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

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 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

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.

Solving Block Matrices with PCFIELDSPLIT

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 a black box. In this section, we use the common linear algebra definition of block matrices, where matrices are divided into 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, such as the velocity and pressure. Under a certain numbering of unknowns, the matrix can be written as

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

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

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

or interlaced, for example, with two blocks

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

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., MATBAIJ or MATSBAIJ) 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 implements the “block” solvers in PETSc, [EHS+08]. 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 (field). You can provide names for each block with these routines; if you do not, they are numbered from 0. With these two approaches, the blocks may overlap (though they generally will not overlap). 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.

Important parameters for PCFIELDSPLIT

• Control the fields used

  • -pc_fieldsplit_detect_saddle_point <bool:false> Generate two fields, the first consists of all rows with a nonzero on the diagonal, and the second will be all rows with zero on the diagonal. See PCFieldSplitSetDetectSaddlePoint().

  • -pc_fieldsplit_dm_splits <bool:true> Use the DM attached to the preconditioner to determine the fields. See PCFieldSplitSetDMSplits() and DMCreateFieldDecomposition().

  • -pc_fieldsplit_%d_fields <f1,f2,…:int> Use f1, f2, .. to define field d. The fn are in the range of 0, …, bs-1 where bs is the block size of the matrix or set with PCFieldSplitSetBlockSize(). See PCFieldSplitSetFields().

    • -pc_fieldsplit_default <bool:true> Automatically add any fields needed that have not been supplied explicitly by -pc_fieldsplit_%d_fields.

  • DMFieldsplitSetIS() Provide the IS that defines a particular field.

• Control the type of the block preconditioner

  • -pc_fieldsplit_type <additive|multiplicative|symmetric_multiplicative|schur|gkb:multiplicative> The order in which the field solves are applied. For symmetric problems where KSPCG is used symmetric_multiplicative must be used instead of multiplicative. additive is the least expensive to apply but provides the worst convergence. schur requires either a good preconditioner for the Schur complement or a naturally well-conditioned Schur complement, but when it works well can be extremely effective. See PCFieldSplitSetType(). gkb is for symmetric saddle-point problems (the lower-right the block is zero).

  • -pc_fieldsplit_diag_use_amat <bool:false> Use the first matrix that is passed to KSPSetJacobian() to construct the block-diagonal sub-matrices used in the algorithms, by default, the second matrix is used.

  • Options for Schur preconditioner: -pc_fieldsplit_type schur

    • -pc_fieldsplit_schur_fact_type <diag|lower|upper|full:diag> See PCFieldSplitSetSchurFactType(). full reduces the iterations but each iteration requires additional field solves.

    • -pc_fieldsplit_schur_precondition <self|selfp|user|a11|full:user> How the Schur complement is preconditioned. See PCFieldSplitSetSchurPre().

      • -fieldsplit_1_mat_schur_complement_ainv_type <diag|lump:diag> Use the lumped diagonal of \(A_{00}\) when -pc_fieldsplit_schur_precondition selfp is used.

    • -pc_fieldsplit_schur_scale <scale:real:-1.0> Controls the sign flip of S for -pc_fieldsplit_schur_fact_type diag. See PCFieldSplitSetSchurScale()

    • fieldsplit_1_xxx controls the solver for the Schur complement system. If a DM provided the fields, use the second field name set in the DM instead of 1.

      • -fieldsplit_1_pc_type lsc -fieldsplit_1_lsc_pc_xxx use the least squares commutators [EHS+06] [SEKW01] preconditioner for the Schur complement with any preconditioner for the least-squares matrix, see PCLSC. If a DM provided the fields, use the second field name set in the DM instead of 1.

    • -fieldsplit_upper_xxx Set options for the solver in the upper solver when -pc_fieldsplit_schur_fact_type upper or full is used. Defaults to using the solver as provided with -fieldsplit_0_xxx.

    • -fieldsplit_1_inner_xxx Set the options for the solver inside the application of the Schur complement; defaults to using the solver as provided with -fieldsplit_0_xxx. If a DM provides the fields use the name of the second field name set in the DM instead of 1.

  • Options for GKB preconditioner: -pc_fieldsplit_type gkb

    • -pc_fieldsplit_gkb_tol <tol:real:1e-5> See PCFieldSplitSetGKBTol().

    • -pc_fieldsplit_gkb_delay <delay:int:5> See PCFieldSplitSetGKBDelay().

    • -pc_fieldsplit_gkb_nu <nu:real:1.0> See PCFieldSplitSetGKBNu().

    • -pc_fieldsplit_gkb_maxit <maxit:int:100> See PCFieldSplitSetGKBMaxit().

    • -pc_fieldsplit_gkb_monitor <bool:false> Monitor the convergence of the inner solver.

• Options for additive and multiplication field solvers:

  • -fieldsplit_%d_xxx Set options for the solver for field number d. For example, -fieldsplit_0_pc_type jacobi. When the fields are obtained from a DM use the field name instead of d.

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

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

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

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

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

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

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

block Gauss-Seidel,

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

which is implemented [3] as

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

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

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

and symmetric block Gauss-Seidel

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

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 \(A_{00}, A_{01}\) 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 is another family of solvers based on Schur complements. The inverse of the Schur complement factorization is

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

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

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

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

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

\[\left( \begin{array}{cc} \text{ksp}(A_{00},Ap_{00}) & 0 \\ 0 & I \\ \end{array} \right) \left( \begin{array}{cc} I & -A_{01} \\ 0 & I \\ \end{array} \right)\]

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

Where \(\hat{S} = A_{11} - A_{10} \text{ksp}(A_{00},Ap_{00}) A_{01}\) 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

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

This is done to ensure the preconditioner is positive definite for a 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 PCFieldSplitSetSchurPre(pc,PC_FIELDSPLIT_SCHUR_PRE_USER,Sp). Without a good problem-specific \(\hat Sp\), you can use some 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 the 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, 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

\[\text{ksp}(A_{10} A_{01},A_{10} A_{01}) A_{10} A_{00} A_{01} \text{ksp}(A_{10} A_{01},A_{10} A_{01}) \]

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 \(A_{10} A_{01}\), users 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 Acknowledgements). To use these solvers, one may:

1. 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).

2. Build the PETSc libraries.

3. 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.

Table 8 Options for External Solvers

MatType	PCType	MatSolverType	Package
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
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 (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: MATSEQAIJ, MATMPIAIJ, 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 PETSc’s MPI parallel linear solvers 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 process (with or without OpenMP). The application code must be built with MPI and must call PetscInitialize() 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.

The program must then be launched using the standard approaches for launching MPI programs with the additional PETSc option -mpi_linear_solver_server. The linear solves are controlled via the options database in the usual manner (using any options prefix you may have provided via KSPSetOptionsPrefix(), for example -ksp_type cg -ksp_monitor -pc_type bjacobi -ksp_view. The solver options cannot be set via the functional interface, for example KSPSetType() etc.

The option -mpi_linear_solver_server_view will print a summary of all the systems solved by the MPI linear solver server when the program completes. By default the linear solver server will only parallelize the linear solve to the extent that it believes is appropriate to obtain speedup for the parallel solve, for example, if the matrix has 1,000 rows and columns the solution will not be parallelized by default. One can use the option -mpi_linear_solver_server_minimum_count_per_rank 5000 to cause the linear solver server to allow as few as 5,000 unknowns per MPI process in the parallel solve.

See PCMPI, PCMPIServerBegin(), and PCMPIServerEnd() for more details on the solvers.

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.

See Application with the MPI linear solver server for a study of the use of PCMPI on a specific PETSc application.

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

[BBKL11]

Achi Brandt, James Brannick, Karsten Kahl, and Irene Livshits. Bootstrap AMG. SIAM Journal on Scientific Computing, 33(2):612–632, 2011.

[CS99]

X.-C. Cai and M. Sarkis. A restricted additive Schwarz preconditioner for general sparse linear systems. SIAM J. Scientific Computing, 21:792–797, 1999.

[CGS+94]

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.

[Eis81]

S. Eisenstat. Efficient implementation of a class of CG methods. SIAM J. Sci. Stat. Comput., 2:1–4, 1981.

[EES83]

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.

[EHS+06]

H.C. Elman, V.E. Howle, J. Shadid, R. Shuttleworth, and R. Tuminaro. Block preconditioners based on approximate commutators. SIAM J. Sci. Comput., 27(5):1651–1668, 2006.

[EHS+08]

H.C. Elman, V.E. Howle, J. Shadid, R. Shuttleworth, and R. Tuminaro. A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations. Journal of Computational Physics, 227(1):1790–1808, 2008. URL: https://www.osti.gov/biblio/920807/.

[FGN92]

R. Freund, G. H. Golub, and N. Nachtigal. Iterative Solution of Linear Systems, pages 57–100. Acta Numerica. Cambridge University Press, 1992.

[Fre93]

Roland W. Freund. A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems. SIAM J. Sci. Stat. Comput., 14:470–482, 1993.

[GAMV13]

P. Ghysels, T.J. Ashby, K. Meerbergen, and W. Vanroose. Hiding global communication latency in the GMRES algorithm on massively parallel machines. SIAM Journal on Scientific Computing, 35(1):C48–C71, 2013.

[GV14]

P. Ghysels and W. Vanroose. Hiding global synchronization latency in the preconditioned conjugate gradient algorithm. Parallel Computing, 40(7):224–238, 2014. 7th Workshop on Parallel Matrix Algorithms and Applications. doi:10.1016/j.parco.2013.06.001.

[HS52]

Magnus R. Hestenes and Eduard Steifel. Methods of conjugate gradients for solving linear systems. J. Research of the National Bureau of Standards, 49:409–436, 1952.

[ISG15]

Tobin Isaac, Georg Stadler, and Omar Ghattas. Solution of nonlinear Stokes equations discretized by high-order finite elements on nonconforming and anisotropic meshes, with application to ice sheet dynamics. SIAM Journal on Scientific Computing, 37(6):804–833, 2015. doi:10.1137/140974407.

[Not00]

Yvan Notay. Flexible conjugate gradients. SIAM Journal on Scientific Computing, 22(4):1444–1460, 2000.

[PS75] (1,2)

C. C. Paige and M. A. Saunders. Solution of sparse indefinite systems of linear equations. SIAM Journal on Numerical Analysis, 12:617–629, 1975.

[SS86]

Y. Saad and M. Schultz. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 44:856–869, 1986.

[Saa93]

Youcef Saad. A flexible inner-outer preconditioned GMRES algorithm. SIAM Journal on Scientific Computing, 14(2):461–469, 1993. doi:10.1137/0914028.

[Saa03]

Yousef Saad. Iterative Methods for Sparse Linear Systems. SIAM, 2nd edition, 2003. doi:10.1016/S1570-579X(01)80025-2.

[SSM16] (1,2,3)

P. Sanan, S. M. Schnepp, and D. A. May. Pipelined, flexible Krylov subspace methods. SIAM Journal on Scientific Computing, 38(5):C441–C470, 2016. doi:10.1137/15M1049130.

[SEKW01]

D. Silvester, H. Elman, D. Kay, and A. Wathen. Efficient preconditioning of the linearized Navier-Stokes equations for incompressible flow. Journal of Computational and Applied Mathematics, 128(1-2):261–279, 2001.

[SBjorstadG96]

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.

[Son89]

Peter Sonneveld. CGS, a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 10:36–52, 1989.

[vdV03]

H. van der Vorst. Iterative Krylov Methods for Large Linear Systems. Cambridge University Press, 2003. ISBN 9780521818285.

[VanveekBM01]

P. Vaněek, M. Brezina, and J. Mandel. Convergence of algebraic multigrid based on smoothed aggregation. Numerische Mathematik, 88(3):559–579, 2001.

[VanvekMB96]

P. Vaněk, J. Mandel, and M. Brezina. Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing, 56(3):179–196, 1996.

[vandVorst92]

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.