PCFIELDSPLIT#

Preconditioner created by combining separate preconditioners for individual collections of variables (that may overlap) called splits. See the users manual section on “Solving Block Matrices” for more details.

Options Database Keys#

-pc_fieldsplit_%d_fields <a,b,..> - indicates the fields to be used in the %d’th split
-pc_fieldsplit_default - automatically add any fields to additional splits that have not been supplied explicitly by -pc_fieldsplit_%d_fields
-pc_fieldsplit_block_size - size of block that defines fields (i.e. there are bs fields) when the matrix is not of MatType MATNEST
-pc_fieldsplit_type <additive,multiplicative,symmetric_multiplicative,schur,gkb> - type of relaxation or factorization splitting
-pc_fieldsplit_schur_precondition <self,selfp,user,a11,full> - default is a11; see PCFieldSplitSetSchurPre()
-pc_fieldsplit_schur_fact_type <diag,lower,upper,full> - set factorization type when using -pc_fieldsplit_type schur; see PCFieldSplitSetSchurFactType()
-pc_fieldsplit_dm_splits <true,false> (default is true) - Whether to use DMCreateFieldDecomposition() for splits
-pc_fieldsplit_detect_saddle_point - automatically finds rows with zero diagonal and uses Schur complement with no preconditioner as the solver

Options prefixes for inner solvers when using the Schur complement preconditioner are -fieldsplit_0_ and -fieldsplit_1_ . The options prefix for the inner solver when using the Golub-Kahan biadiagonalization preconditioner is -fieldsplit_0_ For all other solvers they are -fieldsplit_%d_ for the %d’th field; use -fieldsplit_ for all fields.

To set options on the solvers for each block append -fieldsplit_ to all the PC options database keys. For example, -fieldsplit_pc_type ilu -fieldsplit_pc_factor_levels 1

To set the options on the solvers separate for each block call PCFieldSplitGetSubKSP() and set the options directly on the resulting KSP object

Notes#

Use PCFieldSplitSetFields() to set splits defined by “strided” entries or with a MATNEST and PCFieldSplitSetIS() to define a split by an arbitrary collection of entries.

If no splits are set, the default is used. If a DM is associated with the PC and it supports DMCreateFieldDecomposition(), then that is used for the default. Otherwise if the matrix is not MATNEST, the splits are defined by entries strided by bs, beginning at 0 then 1, etc to bs-1. The block size can be set with PCFieldSplitSetBlockSize(), if this is not called the block size defaults to the blocksize of the second matrix passed to KSPSetOperators()/PCSetOperators().

For the Schur complement preconditioner if

\[J = \left[\begin{array}{cc} A_{00} & A_{01} \\ A_{10} & A_{11} \end{array}\right]\]

the preconditioner using full factorization is logically

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

where the action of \(\text{inv}(A_{00})\) is applied using the KSP solver with prefix -fieldsplit_0_. \(S\) is the Schur complement

\[S = A_{11} - A_{10} \text{ksp}(A_{00}) A_{01}\]

which is usually dense and not stored explicitly. The action of \(\text{ksp}(S)\) is computed using the KSP solver with prefix -fieldsplit_splitname_ (where splitname was given in providing the SECOND split or 1 if not given). For PCFieldSplitGetSubKSP() when field number is 0, it returns the KSP associated with -fieldsplit_0_ while field number 1 gives -fieldsplit_1_ KSP. By default \(A_{11}\) is used to construct a preconditioner for \(S\), use PCFieldSplitSetSchurPre() for all the possible ways to construct the preconditioner for \(S\).

The factorization type is set using -pc_fieldsplit_schur_fact_type <diag, lower, upper, full>. full is shown above, diag gives

\[\left[\begin{array}{cc} \text{inv}(A_{00}) & 0 \\ 0 & -\text{ksp}(S) \end{array}\right]\]

Note that, slightly counter intuitively, there is a negative in front of the \(\text{ksp}(S)\) so that the preconditioner is positive definite. For SPD matrices \(J\), the sign flip can be turned off with PCFieldSplitSetSchurScale() or by command line -pc_fieldsplit_schur_scale 1.0. The lower factorization is the inverse of

\[\left[\begin{array}{cc} A_{00} & 0 \\ A_{10} & S \end{array}\right]\]

where the inverses of A_{00} and S are applied using KSPs. The upper factorization is the inverse of

\[\left[\begin{array}{cc} A_{00} & A_{01} \\ 0 & S \end{array}\right]\]

where again the inverses of \(A_{00}\) and \(S\) are applied using KSPs.

If only one set of indices (one IS) is provided with PCFieldSplitSetIS() then the complement of that IS is used automatically for a second submatrix.

The fieldsplit preconditioner cannot currently be used with the MATBAIJ or MATSBAIJ data formats if the blocksize is larger than 1. Generally it should be used with the MATAIJ or MATNEST MatType

The forms of these preconditioners are closely related, if not identical, to forms derived as “Distributive Iterations”, see, for example, page 294 in “Principles of Computational Fluid Dynamics” by Pieter Wesseling [Wes09]. One can also use PCFIELDSPLIT inside a smoother resulting in “Distributive Smoothers”.

See “A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations” [EHS+08].

The Constrained Pressure Preconditioner (CPR) can be implemented using PCCOMPOSITE with PCGALERKIN. CPR first solves an \(R A P\) subsystem, updates the residual on all variables (PCCompositeSetType(pc,PC_COMPOSITE_MULTIPLICATIVE)), and then applies a simple ILU like preconditioner on all the variables.

The generalized Golub-Kahan bidiagonalization preconditioner (GKB) can be applied to symmetric \(2 \times 2\) block matrices of the shape

\[\left[\begin{array}{cc} A_{00} & A_{01} \\ A_{01}' & 0 \end{array}\right]\]

with \(A_{00}\) positive semi-definite. The implementation follows [Ari13]. Therein, we choose \(N := 1/\nu * I\) and the \((1,1)\)-block of the matrix is modified to \(H = _{A00} + \nu*A_{01}*A_{01}'\). A linear system \(Hx = b\) has to be solved in each iteration of the GKB algorithm. This solver is chosen with the option prefix -fieldsplit_0_.

Developer Note#

The Schur complement functionality of PCFIELDSPLIT should likely be factored into its own PC thus simplifying the implementation of the preconditioners and their user API.

References#

[Ari13]

Mario Arioli. Generalized Golub–Kahan bidiagonalization and stopping criteria. SIAM Journal on Matrix Analysis and Applications, 34(2):571–592, 2013.

[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/.

[Wes09]

Pieter Wesseling. Principles of computational fluid dynamics. Volume 29. Springer Science & Business Media, 2009.