[petsc-users] VI: RS vs SS

[Alexander Lindsay]

I've been working on mechanical contact in MOOSE for a while, and it's led
to me to think about general inequality constraint enforcement. I've been
playing around with both `vinewtonssls` and `vinewtonrsls`. In Benson's and
Munson's Flexible Complementarity Solvers paper, they were able to solve
73.7% of their problems with SS and 65.5% with RS which led them to
conclude that the SS method is generally more robust.  We have had at least
one instance where a MOOSE user reported an order of magnitude reduction
<https://groups.google.com/d/msg/moose-users/Kzimz-HcK2s/y_jnCQGaCAAJ> in
non-linear iterations when switching from SS to RS. Moreover, when running
the problem described in this issue
<https://gitlab.com/petsc/petsc/issues/411#note_229556542>, I get these
results:

num_elements = 100
SS nl iterations = 53
RS nl iterations = 22

num_elements = 1000
SS nl iterations = 123
RS nl iterations = 140

num_elements = 10000
SS: fails to converge within 50 nl iterations during the second time step
whether using a `basic` or `bt` line search
RS: fails to converge within 50 nl iterations during the second time step
whether using a `basic` or `bt` line search (although I believe
`vinewtonrsls` performs a line-search that is guaranteed to keep the
degrees of freedom within their bounds)

So depending on the number of elements, it appears that either SS or RS may
be more performant. I guess since I can get different relative performance
with even the same PDE, it would be silly for me to ask for guidance on
when to use which? In the conclusion of Benson's and Munson's paper, they
mention using mesh sequencing for generating initial guesses on finer
meshes. Does anyone know whether there have been any publications using
PETSc/TAO and mesh sequencing for solving large VI problems?

A related question: what needs to be done to allow SS to run with
`-snes_mf_operator`? RS already appears to support the option.
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