Solvers and Preconditioners

There are several solvers available in hypre via different conceptual interfaces:

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System Interfaces

Solvers

Struct

SStruct

FEI

IJ

Jacobi

X

X

SMG

X

X

PFMG

X

X

Split

X

SysPFMG

X

FAC

X

Maxwell

X

BoomerAMG

X

X

X

AMS

X

X

X

ADS

X

X

X

MLI

X

X

X

MGR

X

ParaSails

X

X

X

hypre-ILU

X

Euclid

X

X

X

PILUT

X

X

X

PCG

X

X

X

X

GMRES

X

X

X

X

FlexGMRES

X

X

X

X

LGMRES

X

X

X

BiCGSTAB

X

X

X

X

Hybrid

X

X

X

X

LOBPCG

X

X

X

Note that there are a few additional solvers and preconditioners not mentioned in the table that can be used only through the FEI interface and are described in Paragraph 6.14. The procedure for setup and use of solvers and preconditioners is largely the same. We will refer to them both as solvers in the sequel except when noted. In normal usage, the preconditioner is chosen and constructed before the solver, and then handed to the solver as part of the solver’s setup. In the following, we assume the most common usage pattern in which a single linear system is set up and then solved with a single righthand side. We comment later on considerations for other usage patterns.

Setup:

  1. Pass to the solver the information defining the problem. In the typical user cycle, the user has passed this information into a matrix through one of the conceptual interfaces prior to setting up the solver. In this situation, the problem definition information is then passed to the solver by passing the constructed matrix into the solver. As described before, the matrix and solver must be compatible, in that the matrix must provide the services needed by the solver. Krylov solvers, for example, need only a matrix-vector multiplication. Most preconditioners, on the other hand, have additional requirements such as access to the matrix coefficients.

  2. Create the solver/preconditioner via the Create() routine.

  3. Choose parameters for the preconditioner and/or solver. Parameters are chosen through the Set() calls provided by the solver. Throughout hypre, we have made our best effort to give all parameters reasonable defaults if not chosen. However, for some preconditioners/solvers the best choices for parameters depend on the problem to be solved. We give recommendations in the individual sections on how to choose these parameters. Note that in hypre, convergence criteria can be chosen after the preconditioner/solver has been setup. For a complete set of all available parameters see the Reference Manual.

  4. Pass the preconditioner to the solver. For solvers that are not preconditioned, this step is omitted. The preconditioner is passed through the SetPrecond() call.

  5. Set up the solver. This is just the Setup() routine. At this point the matrix and right hand side is passed into the solver or preconditioner. Note that the actual right hand side is not used until the actual solve is performed.

At this point, the solver/preconditioner is fully constructed and ready for use.

Use:

  1. Set convergence criteria. Convergence can be controlled by the number of iterations, as well as various tolerances such as relative residual, preconditioned residual, etc. Like all parameters, reasonable defaults are used. Users are free to change these, though care must be taken. For example, if an iterative method is used as a preconditioner for a Krylov method, a constant number of iterations is usually required.

  2. Solve the system. This is just the Solve() routine.

Finalize:

  1. Free the solver or preconditioner. This is done using the Destroy() routine.

Synopsis

In general, a solver (let’s call it SOLVER) is set up and run using the following routines, where A is the matrix, b the right hand side and x the solution vector of the linear system to be solved:

/* Create Solver */
int HYPRE_SOLVERCreate(MPI_COMM_WORLD, &solver);

/* Set certain parameters if desired */
HYPRE_SOLVERSetTol(solver, 1.e-8);
...

/* Set up Solver */
HYPRE_SOLVERSetup(solver, A, b, x);

/* Solve the system */
HYPRE_SOLVERSolve(solver, A, b, x);

/* Destroy the solver */
HYPRE_SOLVERDestroy(solver);

In the following sections, we will give brief descriptions of the available hypre solvers with some suggestions on how to choose the parameters as well as references for users who are interested in a more detailed description and analysis of the solvers. A complete list of all routines that are available can be found in the reference manual.