PC (Preconditioners) - Low-level Interface

The PC (Preconditioner) component provides methods for preconditioning linear systems to accelerate the convergence of iterative solvers. Preconditioners are essential for efficiently solving large sparse linear systems with KSP.

Overview

Preconditioners transform the linear system $Ax = b$ into an equivalent system that is easier to solve iteratively. PETSc provides a wide variety of preconditioners:

Basic methods: Jacobi, block Jacobi, SOR, ILU, ICC
Multigrid: Algebraic multigrid (GAMG, HYPRE BoomerAMG), geometric multigrid (MG)
Domain decomposition: Additive Schwarz (ASM), block Jacobi with local solves
Direct solvers: LU, Cholesky (using external packages like MUMPS, SuperLU, UMFPACK)
Sparse approximations: ILU(k), ICC(k), approximate inverses
Physics-based: Field-split, composite preconditioners for coupled systems
Matrix-free: Shell preconditioners for matrix-free implementations

The PC object can be used standalone or automatically by KSP during the solution process.

Basic Usage

using PETSc, MPI

# Initialize MPI and PETSc
MPI.Init()
petsclib = PETSc.getlib()
PETSc.initialize(petsclib)

# Create a PC object
pc = LibPETSc.PCCreate(petsclib, LibPETSc.PETSC_COMM_SELF)

# Set the preconditioner type (a String or one of the LibPETSc.PC* constants)
LibPETSc.PCSetType(petsclib, pc, LibPETSc.PCILU)

# Set the operator matrix
# LibPETSc.PCSetOperators(petsclib, pc, A, A)

# Configure preconditioner-specific options
# For ILU: set fill level
# LibPETSc.PCFactorSetLevels(petsclib, pc, 2)

# Set options from command line/options database
LibPETSc.PCSetFromOptions(petsclib, pc)

# Set up the preconditioner
LibPETSc.PCSetUp(petsclib, pc)

# Apply the preconditioner: y = P^{-1} x
# LibPETSc.PCApply(petsclib, pc, x_vec, y_vec)

# Cleanup
LibPETSc.PCDestroy(petsclib, pc)

# Finalize PETSc and MPI
PETSc.finalize(petsclib)
MPI.Finalize()

Integration with KSP

Preconditioners are typically used with KSP solvers:

# Create KSP and get its PC (owned by the KSP)
ksp = LibPETSc.KSPCreate(petsclib, LibPETSc.PETSC_COMM_SELF)
pc = LibPETSc.KSPGetPC(petsclib, ksp)

# Configure the preconditioner
LibPETSc.PCSetType(petsclib, pc, "gamg")

# For GAMG, set additional options
LibPETSc.PCGAMGSetType(petsclib, pc, LibPETSc.PCGAMGAGG)
LibPETSc.PCGAMGSetNlevels(petsclib, pc, 10)

Common Preconditioner Types

Available through PCSetType:

Direct Solvers

PCLU: LU factorization
PCCHOLESKY: Cholesky factorization (symmetric positive definite)
PCQR: QR factorization

Incomplete Factorizations

PCILU: Incomplete LU factorization
PCICC: Incomplete Cholesky factorization
Configure fill with PCFactorSetLevels

Iterative/Relaxation Methods

PCJACOBI: Jacobi (diagonal scaling)
PCSOR: Successive over-relaxation
PCPBJACOBI: Point-block Jacobi
PCBJACOBI: Block Jacobi with local solves

Multigrid Methods

PCMG: Geometric multigrid
PCGAMG: Algebraic multigrid (PETSc's built-in)
PCHYPRE: HYPRE preconditioners (BoomerAMG, ParaSails, etc.)
PCML: ML algebraic multigrid (Trilinos)

Domain Decomposition

PCASM: Additive Schwarz method
PCGASM: Generalized additive Schwarz
PCBDDC: Balancing domain decomposition by constraints

Physics-Based

PCFIELDSPLIT: Field-split for coupled systems
PCLSC: Least-squares commutators (for saddle-point problems)
PCCOMPOSITE: Combine multiple preconditioners

Special Purpose

PCSHELL: User-defined shell preconditioner
PCNONE: No preconditioning (identity)
PCKSP: Use another KSP as preconditioner
PCREDISTRIBUTE: Redistribute matrix for better load balancing

Multigrid Configuration

For geometric multigrid (PCMG):

LibPETSc.PCSetType(petsclib, pc, LibPETSc.PCMG)
LibPETSc.PCMGSetLevels(petsclib, pc, nlevels, C_NULL)  # C_NULL: one communicator for all levels
# Set up grid hierarchy, smoothers, coarse solver

For algebraic multigrid (PCGAMG):

LibPETSc.PCSetType(petsclib, pc, LibPETSc.PCGAMG)
LibPETSc.PCGAMGSetType(petsclib, pc, LibPETSc.PCGAMGAGG)  # Aggregation
LibPETSc.PCGAMGSetNSmooths(petsclib, pc, 1)
LibPETSc.PCGAMGSetThreshold(petsclib, pc, [0.0], 1)

Field Split for Coupled Systems

For systems with multiple fields (e.g., velocity-pressure):

LibPETSc.PCSetType(petsclib, pc, LibPETSc.PCFIELDSPLIT)
LibPETSc.PCFieldSplitSetType(petsclib, pc, LibPETSc.PC_COMPOSITE_SCHUR)

# Define fields using index sets
# LibPETSc.PCFieldSplitSetIS(petsclib, pc, "velocity", velocity_is)
# LibPETSc.PCFieldSplitSetIS(petsclib, pc, "pressure", pressure_is)

External Solver Packages

PETSc can use external direct solver packages:

MUMPS: Parallel direct solver
SuperLU: Sparse direct solver
UMFPACK: Unsymmetric multifrontal solver
Pardiso: Intel MKL parallel direct solver
STRUMPACK: Structured sparse solver

Set via PCFactorSetMatSolverType after choosing PCLU or PCCHOLESKY.

Performance Considerations

Jacobi/Block Jacobi: Fast to apply, limited effectiveness
ILU/ICC: Good for moderately difficult problems, configure fill with PCFactorSetLevels
AMG: Excellent for elliptic PDEs, automatic coarse grid construction
Direct solvers: Robust but memory-intensive for large 3D problems
Field split: Essential for coupled multi-physics problems

Function Reference

PETSc.LibPETSc.PCASMCreateSubdomains — Method

outis::Vector{IS} = PCASMCreateSubdomains(petsclib::PetscLibType, A::AbstractPetscMat, n::PetscInt)

Creates the index sets for the overlapping Schwarz preconditioner, PCASM, for any problem on a general grid.

Collective

Input Parameters:

A - The global matrix operator
n - the number of local blocks

Output Parameter:

outis - the array of index sets defining the subdomains

Level: advanced

See also: PCASM, PCASMSetLocalSubdomains(), PCASMDestroySubdomains()

External Links