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KSP

The KSP (Krylov Subspace Methods) module provides iterative linear solvers for systems of the form Ax = b. PETSc offers a wide variety of Krylov methods and preconditioners.

Overview

KSP provides:

  • Krylov methods: GMRES, CG, BiCGStab, and many more
  • Preconditioners: Jacobi, ILU, multigrid, direct solvers, etc.
  • Runtime configuration: Choose methods via command-line options
  • Convergence monitoring: Built-in residual tracking

Creating a KSP Solver

From a Matrix

# Basic creation with default options
ksp = KSP(A)

# With preconditioner matrix P (for different preconditioning)
ksp = KSP(A, P)

# With options
ksp = KSP(A; 
    ksp_type = "gmres",
    pc_type = "ilu",
    ksp_rtol = 1e-8
)

From a DM

# Create KSP associated with a DM (for multigrid, etc.)
ksp = KSP(dm; 
    ksp_type = "cg",
    pc_type = "mg"
)

From a Sparse Matrix

# Directly from Julia SparseMatrixCSC
using SparseArrays
S = sprand(100, 100, 0.1) + 10I
ksp = KSP(petsclib, MPI.COMM_SELF, S)

Solving

# Solve Ax = b
solve!(x, ksp, b)

# Or allocate solution vector
x = solve(ksp, b)

Common Solver/Preconditioner Options

Krylov Methods (ksp_type)

  • cg - Conjugate Gradient (symmetric positive definite)
  • gmres - Generalized Minimum Residual
  • bicgstab - BiConjugate Gradient Stabilized
  • richardson - Richardson iteration
  • preonly - Apply preconditioner only (for direct solvers)

Preconditioners (pc_type)

  • jacobi - Diagonal scaling
  • ilu - Incomplete LU factorization
  • lu - Direct LU factorization
  • mg - Geometric multigrid
  • gamg - Algebraic multigrid
  • none - No preconditioning

Convergence Options

  • ksp_rtol - Relative tolerance (default: 1e-5)
  • ksp_atol - Absolute tolerance
  • ksp_max_it - Maximum iterations
  • ksp_monitor - Print residual each iteration

Example: Multigrid Solver

ksp = KSP(dm;
    ksp_type = "cg",
    pc_type = "mg",
    pc_mg_levels = 4,
    pc_mg_galerkin = true,
    mg_levels_ksp_type = "richardson",
    mg_levels_pc_type = "jacobi",
    mg_coarse_pc_type = "lu"
)

Functions

PETSc.KSP — Method
KSP(petsclib, comm::MPI.Comm, A::SparseMatrixCSC; options...)

Create a KSP with the sparse matrix A using the petsclib. If petsclib is not given, the default library will be used`.

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PETSc.KSP — Method
KSP(dm::AbstractPetscDM; prefix="", options...)

Create a KSP associated with the dm with optional prefix and options.

The communicator is obtained from dm. The KSP can be used with geometric multigrid when the DM provides grid hierarchy information.

Arguments

  • dm::AbstractPetscDM: The DM object to associate with the KSP
  • prefix::String: Optional prefix for command-line options
  • options...: Additional PETSc options as keyword arguments

External Links

source
PETSc.KSP — Method
KSP(comm::MPI.Comm, A::AbstractPetscMat, P::AbstractPetscMat{PetscLib} = A; prefix="", options...)

Create a KSP using the matrix A and preconditioner construction matrix P with optional prefix and options.

The communicator is obtained from A and if it has size 1 then the garbage collector is set, otherwise the user is responsible for calling destroy.

External Links

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PETSc.get_solution — Method
sol = get_solution(ksp::AbstractPetscKSP)

Returns the soluteion vector associated with the KSP object.

source
PETSc.setcomputeoperators! — Method
setcomputeoperators!(ksp::PetscKSP, ops!::Function)
setcomputeoperators!(ops!::Function, ksp::PetscKSP)

Define ops! to be the compute operators function for the ksp. A call to ops!(A, P, new_ksp) should set the elements of the PETSc matrix linear operator A and preconditioning matrix P based on the new_ksp.

Note

The new_ksp passed to ops! may not be the same as the ksp passed to setcomputeoperators!.

External Links

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PETSc.setcomputerhs! — Method
setcomputerhs!(ksp::AbstractKSP, rhs!::Function)
setcomputerhs!(rhs!::Function, ksp::AbstractKSP)

Define rhs! to be the right-hand side function of the ksp. A call to rhs!(b, new_ksp) should set the elements of the PETSc vector b based on the new_ksp.

Note

The new_ksp passed to rhs! may not be the same as the ksp passed to setcomputerhs!.

External Links

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