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Iterative Solution of Linear and Nonlinear Systems

At a Glance

Questions                 |Objectives                     |Key Points
--------------------------|-------------------------------|-------------------------------------
Does the preconditioner   | See that the preconditioner   | Through a single interface,
affect the convergence    | can be crucial for            | PETSc supports runtime choices 
rate of Krylov solvers?   | convergence.                  | of algorithms and options.
                          |                               |
How can I choose algs.    | Learn the basics of using     | Experimenting with
and options at runtime    | PETSc solvers & understanding | algorithms is essential
when using PETSc?         | output.                       | for good performance.

Before running the examples, you must switch to the bash shell by using

bash

Example 1: Structural Mechanics Beam Deflection:

This code uses MFEM and PETSc/TAO to demonstrate the convergence of Krylov methods.

The source code is included in ex2p.c

Notes: Normally PETSc options can be passed as command line arguments. But because MFEM turns off this capability, PETSc options must be passed either in a file or in the PETSC_OPTIONS environmental variable. See the file rc_ex2p for the PETSc options that are supplied to the application in these examples.

Run 1: Run with Jacobi preconditioner

PETSC_OPTIONS="-pc_type jacobi -ksp_max_it 25" ./ex2p -petscopts rc_ex2p --mesh /projects/ATPESC2017/NumericalPackages/handson/mfem/data/beam-tri.mesh

The first column of the output is the residual norm. The next two are the maximum and minimum estimated eigenvalues of the operator and the final column is the condition number.

Questions

Is the iteration converging?

Read the output at the bottom from -ksp_view ... What Krylov method and preconditioner are being used?

Run 2: Run with the algebraic multigrid preconditioner

./ex2p -petscopts rc_ex2p --mesh /projects/ATPESC2017/NumericalPackages/handson/mfem/data/beam-tri.mesh

Questions

Is the iteration now converging?

Read the output at the bottom from -ksp_view ... What Krylov method and preconditioner are being used?

Run 3: Run with the algebraic multigrid preconditioner but no conjugate gradient method

PETSC_OPTIONS="-ksp_norm_type preconditioned -ksp_type richardson -ksp_max_it 25"  ./ex2p -petscopts rc_ex2p --mesh /projects/ATPESC2017/NumericalPackages/handson/mfem/data/beam-tri.mesh

Questions

Is the iteration now converging?

Run 4: Run with the algebraic multigrid preconditioner but with GMRES and a restart of 10

PETSC_OPTIONS="-ksp_norm_type preconditioned -ksp_type gmres -ksp_gmres_restart 10"  ./ex2p -petscopts rc_ex2p --mesh /projects/ATPESC2017/NumericalPackages/handson/mfem/data/beam-tri.mesh

Now run with a gmres restart of 30

PETSC_OPTIONS="-ksp_norm_type preconditioned -ksp_type gmres -ksp_gmres_restart 30"  ./ex2p -petscopts rc_ex2p --mesh /projects/ATPESC2017/NumericalPackages/handson/mfem/data/beam-tri.mesh

Note the convergence is now very similar to that with CG.

Now attempt to run this in parallel and obtain solver performance data

PETSC_OPTIONS="-log_view -ksp_norm_type preconditioned -ksp_type gmres -ksp_gmres_restart 30" ${MPIEXEC_OMPI} -n 4 ./ex2p -petscopts rc_ex2p --mesh /projects/ATPESC2017/NumericalPackages/handson/mfem/data/beam-tri.mesh

Example 2: Nonlinear Problem:

PETSC_OPTS="-snes_rtol 1.e-10 -snes_view  -pc_type bjacobi -sub_pc_type ilu " ${MPIEXEC_OMPI} -n 4 ./ex10p -m ../../data/beam-quad.mesh --petscopts rc_ex10p -s 3 -rs 2 -dt 3 | more

Note the quadratic convergence; the residual norm exponent doubles until it runs out of digits to double.

Out-Brief

We have used PETSc to demonstrate the use of preconditioned Krylov methods. Many examples are available for various aspects of PETSc functionality, including

 

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