[petsc-users] petsc-users Digest, Vol 90, Issue 34

Ed Bueler elbueler at alaska.edu
Tue Jun 14 19:31:54 CDT 2016

Justin --

2) If so, then what exactly is the difference between these TAO
> complementarity solvers and the SNESVINEWTONSSLS solver? From the online
> manuals and documentation, both claim to be variational inequality solvers.

There are problems like lubrication flows, including shallow ice sheets on
non-trivial bedrock topography, which are "inequality-constrained PDEs".
Some of these problems are known to have well-posed (or sometimes
likely-to-be well-posed) variational inequality formulations.  Equivalently
they can be posed as nonlinear complementarity problems (i.e. in finite
dimensions one can show equivalence between variational inequalities and
nonlinear complementarity problems).

However, because one can also show that the Jacobian is not symmetric in
some of the above cases, e.g. ice sheets, it follows that these are *not*
optimization problems.  That is, because the Jacobian is not symmetric, it
cannot be the Hessian of some (possibly-unknown) smooth objective function.

For these problems one can use PETSc's SNESVINEWTON{R,S}SSLS.  I.e. it
works when I have tried it.

It is not clear to me if TAO is set up to not have an objective function at
all.  (That is a question for the TAO developers?)  In the SNESVINEWTONXSLS
solvers an L^2 merit function (i.e. square of the residual) replaces the
not-even-present objective function when doing line search, just as happens
in solving nonlinear equations.


Ed Bueler
Dept of Math and Stat and Geophysical Institute
University of Alaska Fairbanks
Fairbanks, AK 99775-6660
301C Chapman and 410D Elvey
907 474-7693 and 907 474-7199  (fax 907 474-5394)
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