Method set

This method sets the parameters defining the preconditioner p. More precisely, the parameter identified by what is assigned the value contained in val.

`what`	`character(len=*)`.
	The parameter to be set. It can be specified through its name; the string is case-insensitive. See Tables 2-8.
`val`	`integer` or `character(len=*)` or `real(psb_spk_)` or `real(psb_dpk_)`, `intent(in)`.
	The value of the parameter to be set. The list of allowed values and the corresponding data types is given in Tables 2-8. When the value is of type `character(len=*)`, it is also treated as case insensitive.
`info`	`integer, intent(out)`.
	Error code. If no error, 0 is returned. See Section 7 for details.
`ilev`	`integer, optional, intent(in)`.
	For the multilevel preconditioner, the level at which the preconditioner parameter has to be set. The levels are numbered in increasing order starting from the finest one, i.e., level 1 is the finest level. If `ilev` is not present, the parameter identified by `what` is set at all levels that are appropriate (see Tables 2-8).
`ilmax`	`integer, optional, intent(in)`.
	For the multilevel preconditioner, when both `ilev` and `ilmax` are present, the settings are applied at all levels `ilev:ilmax`. When `ilev` is present but `ilmax` is not, then the default is `ilmax=ilev`. The levels are numbered in increasing order starting from the finest one, i.e., level 1 is the finest level.
`pos`	`character(len=*), optional, intent(in)`.
	Whether the other arguments apply only to the pre-smoother (`’PRE’`) or to the post-smoother (`’POST’`). If `pos` is not present, the other arguments are applied to both smoothers. If the preconditioner is one-level or the parameter identified by `what` does not concern the smoothers, `pos` is ignored.
`idx`	`integer, optional, intent(in)`.
	An auxiliary input argument that can be passed to the underlying objects.

A variety of preconditioners can be obtained by setting the appropriate preconditioner parameters. These parameters can be logically divided into four groups, i.e., parameters defining

A list of the parameters that can be set, along with their allowed and default values, is given in Tables 2-8.

Remark 2. A smoother is usually obtained by combining two objects: a smoother (’SMOOTHER_TYPE’) and a local solver (’SUB_SOLVE’), as specified in Tables 7-9. For example, the block-Jacobi smoother using ILU(0) on the blocks is obtained by combining the block-Jacobi smoother object with the ILU(0) solver object. Similarly, the hybrid Gauss-Seidel smoother (see Note in Table 7) is obtained by combining the block-Jacobi smoother object with a single sweep of the Gauss-Seidel solver object, while the point-Jacobi smoother is the result of combining the block-Jacobi smoother object with a single sweep of the point-Jacobi solver object. In the same way are obtained the ℓ₁-versions of the smoothers. However, for simplicity, shortcuts are provided to set all versions of point-Jacobi, hybrid (forward) Gauss-Seidel, and hybrid backward Gauss-Seidel, i.e., the previous smoothers can be defined just by setting ’SMOOTHER_TYPE’ to certain specific values (see Tables 7), without the need to set ’SUB_SOLVE’ as well.

The smoother and solver objects are arranged in a hierarchical manner. When specifying a smoother object, its parameters, including the local solver, are set to their default values, and when a solver object is specified, its defaults are also set, overriding in both cases any previous settings even if explicitly specified. Therefore if the user sets a smoother, and wishes to use a solver different from the default one, the call to set the solver must come after the call to set the smoother.

Similar considerations apply to the point-Jacobi, Gauss-Seidel and block-Jacobi coarsest-level solvers, and shortcuts are available in this case too (see Table 5).

Remark 3. The polynomial-accelerated smoother described in Tables 7 and 9 redefines a sweep or iteration as corresponding to the degree of the polynomial used. Consequently, the ’SMOOTHER_SWEEPS’ option is overridden by the ’POLY_DEGREE’ option. This smoother is paired with a base smoother object, whose iterations are accelerated using the specified polynomial smoothing technique. By default, the ℓ₁-Jacobi smoother serves as the base smoother, offering theoretical guarantees on the resulting convergence factor [15, 27]. Alternative combinations are experimental and lack established guarantees.

Remark 4. Many of the coarsest-level solvers apply to a specific coarsest-matrix layout; therefore, setting the solver after the layout may change the layout to either distributed or replicated. Similarly, setting the layout after the solver may change the solver.

More precisely, UMFPACK and SuperLU require the coarsest-level matrix to be replicated, while SuperLU_Dist and KRM require it to be distributed. In these cases, setting the coarsest-level solver implies that the layout is redefined according to the solver, ovverriding any previous settings. MUMPS, point-Jacobi, hybrid Gauss-Seidel and block-Jacobi can be applied to replicated and distributed matrices, thus their choice does not modify any previously specified layout. It is worth noting that, when the matrix is replicated, the point-Jacobi, hybrid Gauss-Seidel and block-Jacobi solvers and their ℓ₁- versions reduce to the corresponding local solver objects (see Remark 2). For the point-Jacobi and Gauss-Seidel solvers, these objects correspond to a single point-Jacobi sweep and a single Gauss-Seidel sweep, respectively, which are very poor solvers.

On the other hand, the distributed layout can be used with any solver but UMFPACK and SuperLU; therefore, if any of these two solvers has already been selected, the coarsest-level solver is changed to block-Jacobi, with the previously chosen solver applied to the local blocks. Likewise, the replicated layout can be used with any solver but SuperLu_Dist and KRM; therefore, if SuperLu_Dist or KRM have been previously set, the coarsest-level solver is changed to the default sequential solver.

In a parallel setting with many cores, we suggest to the users to change the default coarsest solver for using the KRM choice, i.e. a parallel distributed iterative solution of the coarsest system based on Krylov methods.

Remark 4. The argument idx can be used to allow finer control for those solvers; for instance, by specifying the keyword ’MUMPS_IPAR_ENTRY’ and an appropriate value for idx, it is possible to set any entry in the MUMPS integer control array. See also Sec. 6.


`what`	data type	`val`	default	comments

`’ML_CYCLE’`	`character(len=*)`	`’VCYCLE’` `’WCYCLE’` `’KCYCLE’` `’ADD’`	`’VCYCLE’`	Multilevel cycle: V-cycle, W-cycle, K-cycle, and additive composition.

`’CYCLE_SWEEPS’`	`integer`	Any integer number ≥ 1	1	Number of multilevel cycles.

Table 2: Parameters defining the multilevel cycle and the number of cycles to be applied.

Note. The aggregation algorithm stops when at least one of the following criteria is met: the coarse size threshold,

the minimum coarsening ratio, or the maximum number of levels is reached.

Therefore, the actual number of levels may be smaller than the specified maximum number of levels.


`what`	data type	`val`	default	comments

`’MIN_COARSE_SIZE_PER_PROCESS’`	`integer`	Any number > 0	200	Coarse size threshold per process. The aggregation stops if the global number of variables of the computed coarsest matrix is lower than or equal to this threshold multiplied by the number of processes (see Note).

`’MIN_COARSE_SIZE’`	`integer`	Any number > 0	-1	Coarse size threshold. The aggregation stops if the global number of variables of the computed coarsest matrix is lower than or equal to this threshold (see Note). If negative, it is ignored in favour of the default for `’MIN_COARSE_SIZE_PER_PROCESS’`.

`’MIN_CR_RATIO’`	`real`	Any number > 1	1.5	Minimum coarsening ratio. The aggregation stops if the ratio between the global matrix dimensions at two consecutive levels is lower than or equal to this threshold (see Note).

`’MAX_LEVS’`	`integer`	Any integer number > 1	20	Maximum number of levels. The aggregation stops if the number of levels reaches this value (see Note).

`’PAR_AGGR_ALG’`	`character(len=*)`	’DEC’, ’SYMDEC’, ’COUPLED’	’DEC’	Parallel aggregation algorithm. the `SYMDEC` option applies decoupled aggregation to the sparsity pattern of A + A^T.

`’AGGR_TYPE’`	`character(len=*)`	`’SOC1’`, `’SOC2’`, `’MATCHBOXP’`	`’SOC1’`	Type of aggregation algorithm: currently, for the decoupled aggregation we implement two measures of strength of connection, the one by Vaněk, Mandel and Brezina [35], and the one by Gratton et al [24]. The coupled aggregation is based on a parallel version of the half-approximate matching implemented in the MatchBox-P software package [9].

`’AGGR_SIZE’`	`integer`	Any integer power of 2, with aggr_size ≥ 2	4	Maximum size of aggregates when the coupled aggregation based on matching is applied. For aggressive coarsening with size of aggregate larger than 8 we recommend the use of smoothed prolongators. Used only with ’COUPLED’ and ’MATCHBOXP’

`’AGGR_PROL’`	`character(len=*)`	`’SMOOTHED’`, `’UNSMOOTHED’`	`’SMOOTHED’`	Prolongator used by the aggregation algorithm: smoothed or unsmoothed (i.e., tentative prolongator).

Table 3: Parameters defining the aggregation algorithm.

Note. Different thresholds at different levels, such as those used in [35, Section 5.1], can be easily set by invoking the rou-

tine set with the parameter ilev.


`what`	data type	`val`	default	comments

`’AGGR_ORD’`	`character(len=*)`	’NATURAL’ ’DEGREE’	’NATURAL’	Initial ordering of indices for the decoupled aggregation algorithm: either natural ordering or sorted by descending degrees of the nodes in the matrix graph.

`’AGGR_THRESH’`	`real(kind_parameter)`	Any real number ∈ [0,1]	0.01	The threshold θ in the strength of connection algorithm. See also the note at the bottom of this table.

`’AGGR_FILTER’`	`character(len=*)`	’FILTER’ ’NOFILTER’	’NOFILTER’	Matrix used in computing the smoothed prolongator: filtered or unfiltered.

Table 4: Parameters defining the aggregation algorithm (continued).

Note. Defaults for COARSE_SOLVE and COARSE_SUBSOLVE are chosen in the following order:

single precision version – MUMPS if installed, then SLU if installed, ILU otherwise;

double precision version – UMF if installed, then MUMPS if installed, then SLU if installed, ILU otherwise.

Note. Further options for coarse solvers are contained in Table 6.

For a first use it is suggested to use the default options obtained by simply selecting the solver type.


`what`	data type	`val`	default	comments

`’COARSE_MAT’`	`character(len=*)`	`’DIST’` `’REPL’`	`’REPL’`	Coarsest matrix layout: distributed among the processes or replicated on each of them.

`’COARSE_SOLVE’`	`character(len=*)`	`’MUMPS’` `’UMF’` `’SLU’` `’SLUDIST’` `’ILU’` `’JACOBI’` `’GS’` `’BJAC’` `’KRM’` `’L1-JACOBI’` `’L1-BJAC’` `’L1-FBGS’`	See Note.	Solver used at the coarsest level: sequential LU from MUMPS, UMFPACK, or SuperLU (plus triangular solve); distributed LU from MUMPS or SuperLU_Dist (plus triangular solve); point-Jacobi, hybrid Gauss-Seidel or block-Jacobi and related ℓ₁-versions; Krylov Method (flexible Conjugate Gradient) coupled with the block-Jacobi preconditioner with ILU(0) on the blocks. Note that UMF and SLU require the coarsest matrix to be replicated, SLUDIST, JACOBI, GS, BJAC and KRM require it to be distributed, MUMPS can be used with either a replicated or a distributed matrix. When any of the previous solvers is specified, the matrix layout is set to a default value which allows the use of the solver (see Remark 4, p. 21). Note also that UMFPACK and SuperLU_Dist are available only in double precision.

`’COARSE_SUBSOLVE’`	`character(len=*)`	`’ILU’` `’ILUT’` `’MILU’` `’MUMPS’` `’SLU’` `’UMF’` `’INVT’` `’INVK’` `’AINV’`	See Note.	Solver for the diagonal blocks of the coarsest matrix, in case the block Jacobi solver is chosen as coarsest-level solver: ILU(p), ILU(p,t), MILU(p), LU from MUMPS, SuperLU or UMFPACK (plus triangular solve), Approximate Inverses INVK(p,q), INVT(p₁,p2,t₁,t₂) and AINV(t); note that approximate inverses are specifically suited for GPUs since they do not employ triangular system solve kernels, see [3]. Note that UMFPACK and SuperLU_Dist are available only in double precision.





`what`	data type	`val`	default	comments

`’COARSE_SWEEPS’`	`integer`	Any integer number > 0	10	Number of sweeps when `JACOBI`, `GS` or `BJAC` is chosen as coarsest-level solver.

`’COARSE_FILLIN’`	`integer`	Any integer number ≥ 0	0	Fill-in level p of the ILU factorizations and first fill-in for the approximate inverses.

`’COARSE_ILUTHRS’`	`real(kind_parameter)`	Any real number ≥ 0	0	Drop tolerance t in the ILU(p,t) factorization and first drop-tolerance for the approximate inverses.

Table 5: Parameters defining the solver at the coarsest level (continued).


`what`	data type	`val`	default	comments

`’BJAC_STOP’`	`character(len=*)`	`’FALSE’` `’TRUE’`	`’FALSE’`	Select whether to use a stopping criterion for the Block-Jacobi method used as a coarse solver.

`’BJAC_TRACE’`	`character(len=*)`	`’FALSE’` `’TRUE’`	`’FALSE’`	Select whether to print a trace for the calculated residual for the Block-Jacobi method used as a coarse solver.

`’BJAC_ITRACE’`	`integer`	Any integer > 0	-1	Number of iterations after which a trace is to be printed.

`’BJAC_RESCHECK’`	`integer`	Any integer > 0	-1	Number of iterations after which a residual is to be calculated.

`’BJAC_STOPTOL’`	`real(kind_parameter)`	Any real < 1	0	Tolerance for the stopping criterion on the residual.

`’KRM_METHOD’`	`character(len=*)`	`’CG’` `’FCG’` `’CGS’` `’CGR’` `’BICG’` `’BICGSTAB’` `’BICGSTABL’` `’RGMRES’`	`’FCG’`	A string that defines the iterative method to be used when employing a Krylov method `’KRM’` as a coarse solver. CG the Conjugate Gradient method; FCG the Flexible Conjugate Gradient method; CGS the Conjugate Gradient Stabilized method; GCR the Generalized Conjugate Residual method; FCG the Flexible Conjugate Gradient method; BICG the Bi-Conjugate Gradient method; BICGSTAB the Bi-Conjugate Gradient Stabilized method; BICGSTABL the Bi-Conjugate Gradient Stabilized method with restarting; RGMRES the Generalized Minimal Residual method with restarting. Refer to the PSBLAS guide [21] for further information.

`’KRM_KPREC’`	`character(len=*)`	Table 1	`’BJAC’`	The one-level preconditioners from the Table 1 can be used for the coarse Krylov solver.

`’KRM_SUB_SOLVE’`	`character(len=*)`	Table 5	`’ILU’`	Solver for the diagonal blocks of the coarsest matrix preconditioner, in case the block Jacobi solver is chosen as `’KRM_KPREC’`: ILU(p), ILU(p,t), MILU(p), LU from MUMPS, SuperLU or UMFPACK (plus triangular solve), Approximate Inverses INVK(p,q), INVT(p₁,p2,t₁,t₂) and AINV(t); The same caveat from Table 5 applies here.

`’KRM_GLOBAL’`	`character(len=*)`	`’TRUE’`, `’FALSE’`	`’FALSE’`	Choose between a global Krylov solver, all unknowns on a single node, or a distributed one. The default choice is the distributed solver.

`’KRM_EPS’`	`real(kind_parameter)`	Real < 1	10^-6	The stopping tolerance.

`’KRM_IRST’`	`integer`	Integer ≥ 1	30	An integer specifying the restart parameter. This is employed for the BiCGSTABL or RGMRES methods, otherwise it is ignored.

`’KRM_ISTOPC’`	`integer`	Integers 1,2,3	2	If 1 then the method uses the normwise backward error in the infinity norm; if 2, the it uses the relative residual in the 2-norm; if 3 the relative residual reduction in the 2-norm is used instead; refer to the PSBLAS [21] guide for the details.

`’KRM_ITMAX’`	`integer`	Integer ≥ 1	40	The maximum number of iterations to perform.

`’KRM_ITRACE’`	`integer`	Integer ≥ 0	-1	If > 0 print out an informational message about convergence every `’KRM_ITRACE’` iterations. If = 0 print a message in case of convergence failure.

`’KRM_FILLIN’`	`integer`	Integer ≥ 0	0	Fill-in level p of the ILU factorizations and first fill-in for the approximate inverses.

Table 6: Additional parameters defining the solver at the coarsest level.


`what`	data type	`val`	default	comments

`’SMOOTHER_TYPE’`	`character(len=*)`	`’JACOBI’` `’GS’` `’BGS’` `’BJAC’` `’AS’` `’L1-JACOBI’` `’L1-BJAC’` `’L1-FBGS’` `’POLY’`	`’FBGS’`	Type of smoother used in the multilevel preconditioner: point-Jacobi, hybrid (forward) Gauss-Seidel, hybrid backward Gauss-Seidel, block-Jacobi, ℓ₁-Jacobi, ℓ₁–hybrid (forward) Gauss-Seidel, ℓ₁-point-Jacobi and Additive Schwarz, polynomial accelerators; see [15] and Remark 3 (p. 21). It is ignored by one-level preconditioners.

`’SUB_SOLVE’`	`character(len=*)`	`’JACOBI’` `’GS’` `’BGS’` `’ILU’` `’ILUT’` `’MILU’` `’MUMPS’` `’SLU’` `’UMF’` `’INVT’` `’INVK’` `’AINV’`	GS and BGS for pre- and post-smoothers of multilevel preconditioners, respectively ILU for block-Jacobi and Additive Schwarz one-level preconditioners	The local solver to be used with the smoother or one-level preconditioner (see Remark 2, page 24): point-Jacobi, hybrid (forward) Gauss-Seidel, hybrid backward Gauss-Seidel, ILU(p), ILU(p,t), MILU(p), LU from MUMPS, SuperLU or UMFPACK (plus triangular solve), Approximate Inverses INVK(p,q), INVT(p₁,p2,t₁,t₂) and AINV(t); note that approximate inverses are specifically suited for GPUs since they do not employ triangular system solve kernels, see [3]. See Note for details on hybrid Gauss-Seidel.

`’SMOOTHER_SWEEPS’`	`integer`	Any integer number ≥ 0	1	Number of sweeps of the smoother or one-level preconditioner. In the multilevel case, no pre-smother or post-smoother is used if this parameter is set to 0 together with `pos=’PRE’` or `pos=’POST’`, respectively. Is ignored if the smoother is `’POLY’`

`’POLY_DEGREE’`	`integer`	Any integer number ≥ 1 and ≤ 30	1	Degree of the polynomial accelerator, is equal to the number of matrix-vector products performed by the smoother. Is ignored if the smoother is not `’POLY’`

Table 7: Parameters defining the smoother or the details of the one-level preconditioner.


`what`	data type	`val`	default	comments

`’SUB_OVR’`	`integer`	Any integer number ≥ 0	1	Number of overlap layers, for Additive Schwarz only.
`’SUB_RESTR’`	`character(len=*)`	`’HALO’` `’NONE’`	`’HALO’`	Type of restriction operator, for Additive Schwarz only: HALO for taking into account the overlap, `’NONE’` for neglecting it. Note that HALO must be chosen for the classical Addditive Schwarz smoother and its RAS variant.

`’SUB_PROL’`	`character(len=*)`	`’SUM’` `’NONE’`	`’NONE’`	Type of prolongation operator, for Additive Schwarz only: `’SUM’` for adding the contributions from the overlap, `’NONE’` for neglecting them. Note that `’SUM’` must be chosen for the classical Additive Schwarz smoother, and `’NONE’` for its RAS variant.

`’SUB_FILLIN’`	`integer`	Any integer number ≥ 0	0	Fill-in level p of the incomplete LU factorizations.

`’SUB_ILUTHRS’`	`real(kind_parameter)`	Any real number ≥ 0	0	Drop tolerance t in the ILU(p,t) factorization.

`’MUMPS_LOC_GLOB’`	`character(len=*)`	`’LOCAL_SOLVER’` `’GLOBAL_SOLVER’`	`’GLOBAL_SOLVER’`	Whether MUMPS should be used as a distributed solver, or as a serial solver acting only on the part of the matrix local to each process.

`’MUMPS_IPAR_ENTRY’`	`integer`	Any integer number	0	Set an entry in the MUMPS integer control array, as chosen via the `idx` optional argument.

`’MUMPS_RPAR_ENTRY’`	`real`	Any real number	0	Set an entry in the MUMPS real control array, as chosen via the `idx` optional argument.

Table 8: Parameters defining the smoother or the details of the one-level preconditioner (continued).


`what`	data type	`val`	default	comments

`’POLY_VARIANT’`	`character(len=*)`	`’CHEB_4’` `’CHEB_4_OPT’` `’CHEB_1_OPT’`	`’CHEB_4’`	Select the type of polynomial accelerator. The `’CHEB_4’` and `’CHEB_4_OPT’` types are those based on the Chebyshev polynomials of the 4^th-kind described in [27]. The `’CHEB_1_OPT’` version is the one described in [15] and based on the Chebyshev polynomials of the 1^st-kind.

`’POLY_RHO_ESTIMATE’`	`character(len=*)`	`’POLY_RHO_EST_POWER’`	`’POLY_RHO_EST_POWER’`	Algorithm for estimating the spectral radius of the smoother to which the polynomial acceleration is applied. The only implemented algorithm is the power method; see also the two following options.

`’POLY_RHO_ESTIMATE_ITERATIONS’`	`integer`	Any integer number ≥ 1	20	Number of iterations for the spectral radius estimate.

`’POLY_RHO_BA’`	`real(kind_parameter)`	Any real number ∈ (0,1]	1	Sets an estimate of the spectral radius of the base smoother to which the polynomial accelerator is applied.

Table 9: Parameters defining the smoother or the details of the one-level preconditioner (continued).

5.2 Method set