[petsc-users] Integrating TAO into SNES and TS

[Justin Chang]

I guess I will have to write my own code then :)

I am not all that familiar with Variational Inequalities at the moment, but
if my Jacobian is symmetric and positive definite and I only have lower and
upper bounds, doesn't the problem simply reduce to that of a convex
optimization? That is, with SNES act as if it were Tao?

On Fri, Apr 3, 2015 at 6:35 PM, Barry Smith <bsmith at mcs.anl.gov> wrote:

>
>   Justin,
>
>    We haven't done anything with TS to handle variational inequalities. So
> you can either write your own backward Euler (outside of TS) that solves
> each time-step problem either as 1) an optimization problem using Tao or 2)
> as a variational inequality using SNES.
>
>    More adventurously you could look at the TSTHETA code in TS (which is a
> general form that includes Euler, Backward Euler and Crank-Nicolson and see
> if you can add the constraints to the SNES problem that is solved; in
> theory this is straightforward but it would require understanding the
> current code (which Jed, of course, overwrote :-). I think you should do
> this.
>
>   Barry
>
>
> > On Apr 3, 2015, at 12:31 PM, Justin Chang <jchang27 at uh.edu> wrote:
> >
> > I am solving the following anisotropic transient diffusion equation
> subject to 0 bounds:
> >
> > du/dt = div[D*grad[u]] + f
> >
> > Where the dispersion tensor D(x) is symmetric and positive definite.
> This formulation violates the discrete maximum principles so one of the
> ways to ensure nonnegative concentrations is to employ convex optimization.
> I am following the procedures in Nakshatrala and Valocchi (2009) JCP and
> Nagarajan and Nakshatrala (2011) IJNMF.
> >
> > The Variational Inequality method works gives what I want for my
> transient case, but what if I want to implement the Tao methodology in TS?
> That is, what TS functions do I need to set up steps a) through e) for each
> time step (also the Jacobian remains the same for all time steps so I would
> only call this once). Normally I would just call TSSolve() and let the
> libraries and functions do everything, but I would like to incorporate
> TaoSolve into every time step.
> >
> > Thanks,
> >
> > --
> > Justin Chang
> > PhD Candidate, Civil Engineering - Computational Sciences
> > University of Houston, Department of Civil and Environmental Engineering
> > Houston, TX 77004
> > (512) 963-3262
> >
> > On Thu, Apr 2, 2015 at 6:53 PM, Barry Smith <bsmith at mcs.anl.gov> wrote:
> >
> >   An alternative approach is for you to solve it as a (non)linear
> variational inequality. See src/snes/examples/tutorials/ex9.c
> >
> >   How you should proceed depends on your long term goal. What problem do
> you really want to solve? Is it really a linear time dependent problem with
> 0 bounds on U? Can the problem always be represented as an optimization
> problem easily? What are  and what will be the properties of K? For example
> if K is positive definite then likely the bounds will remain try without
> explicitly providing the constraints.
> >
> >   Barry
> >
> > > On Apr 2, 2015, at 6:39 PM, Justin Chang <jchang27 at uh.edu> wrote:
> > >
> > > Hi everyone,
> > >
> > > I have a two part question regarding the integration of the following
> optimization problem
> > >
> > > min 1/2 u^T*K*u + u^T*f
> > > S.T. u >= 0
> > >
> > > into SNES and TS
> > >
> > > 1) For SNES, assuming I am working with a linear FE equation, I have
> the following algorithm/steps for solving my problem
> > >
> > > a) Set an initial guess x
> > > b) Obtain residual r and jacobian A through functions
> SNESComputeFunction() and SNESComputeJacobian() respectively
> > > c) Form vector b = r - A*x
> > > d) Set Hessian equal to A, gradient to A*x, objective function value
> to 1/2*x^T*A*x + x^T*b, and variable (lower) bounds to a zero vector
> > > e) Call TaoSolve
> > >
> > > This works well at the moment, but my question is there a more
> "efficient" way of doing this? Because with my current setup, I am making a
> rather bold assumption that my problem would converge in one SNES iteration
> without the bounded constraints and does not have any unexpected
> nonlinearities.
> > >
> > > 2) How would I go about doing the above for time-stepping problems? At
> each time step, I want to solve a convex optimization subject to the lower
> bounds constraint. I plan on using backward euler and my resulting jacobian
> should still be compatible with the above optimization problem.
> > >
> > > Thanks,
> > >
> > > --
> > > Justin Chang
> > > PhD Candidate, Civil Engineering - Computational Sciences
> > > University of Houston, Department of Civil and Environmental
> Engineering
> > > Houston, TX 77004
> > > (512) 963-3262
> >
> >
> >
> >
> > --
> > Justin Chang
> > PhD Candidate, Civil Engineering - Computational Sciences
> > University of Houston, Department of Civil and Environmental Engineering
> > Houston, TX 77004
> > (512) 963-3262
>
>


-- 
Justin Chang
PhD Candidate, Civil Engineering - Computational Sciences
University of Houston, Department of Civil and Environmental Engineering
Houston, TX 77004
(512) 963-3262
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