[petsc-users] Integrating TAO into SNES and TS

Tue Sep 1 17:14:38 CDT 2015

So in VI, the i-th component/cell/point of x is "corrected" if it is below
lb_i or above ub_i. All other x_i that satisfy their respective bounds
remain untouched. Hence only the corrected components will violated the
original equation

In optimization, every single x_i will be "shifted" such that every single
x_i meets lb_i and ub_i. Hence it's possible all components will violate
the original equation

Is that the general idea here?

On Tue, Sep 1, 2015 at 12:48 PM, Barry Smith <bsmith at mcs.anl.gov> wrote:

>
>    I think we are talking past each other.
>
>    With bound constraint VI's there is no optimization, there is an
> equation F(x) = 0 (where F may be linear or nonlinear) and constraints  a
> <= x <= c. With VIs the equation F_i(x) is simply not satisfied if x_i is
> on a bound (that is x_i = a_i or x_i = b_i),
>
>    With optimization if you have an equality constraint and inequality
> constraints; if to satisfy an inequality constraint FORCES an equality
> constraint to not be satisfied then the constraints are not compatible and
> the problem isn't properly posed.
>
>    Barry
>
>
> > On Sep 1, 2015, at 4:15 AM, Justin Chang <jychang48 at gmail.com> wrote:
> >
> > But if I add those linear equality constraint equations to my original
> problem, would they not be satisfied anyway? Say I add this to my weak form:
> >
> > Ax = b
> >
> > But once i subject x to some bounded constraints, Ax != b. Unless I add
> some sort of penalty where extra weighting is added to this property...
> >
> > On Tue, Sep 1, 2015 at 3:02 AM, Matthew Knepley <knepley at gmail.com>
> wrote:
> > On Tue, Sep 1, 2015 at 3:46 AM, Justin Chang <jychang48 at gmail.com>
> wrote:
> > I would like to simultaneously enforce both discrete maximum principle
> and local mass/species balance. Because even if a locally conservative
> scheme like RT0 is used, as soon as these bounded constraints are applied,
> i lose the mass balance.
> >
> > What I am saying is, can't you just add "linear equality constraints" as
> more equations?
> >
> >   Thanks,
> >
> >     Matt
> >
> > On Tue, Sep 1, 2015 at 2:33 AM, Matthew Knepley <knepley at gmail.com>
> wrote:
> > On Tue, Sep 1, 2015 at 3:11 AM, Justin Chang <jychang48 at gmail.com>
> wrote:
> > Barry,
> >
> > That's good to know thanks.
> >
> > On a related note, is it possible for VI to one day include linear
> equality constraints?
> >
> > How are these different from just using more equations?
> >
> >   Thanks,
> >
> >     Matt
> >
> > Thanks,
> > Justin
> >
> > On Mon, Aug 31, 2015 at 7:13 PM, Barry Smith <bsmith at mcs.anl.gov> wrote:
> >
> > > On Aug 31, 2015, at 7:36 PM, Justin Chang <jychang48 at gmail.com> wrote:
> > >
> > > Coming back to this,
> > >
> > > Say I now want to ensure the DMP for advection-diffusion equations.
> The linear operator is now asymmetric and non-self-adjoint (assuming I do
> something like SUPG or finite volume), meaning I cannot simply solve this
> problem without any manipulation (e.g. normalizing the equations) using
> TAO's optimization solvers. Does this statement also hold true for SNESVI?
> >
> >   SNESVI doesn't care about symmetry etc
> >
> > >
> > > Thanks,
> > > Justin
> > >
> > > On Fri, Apr 3, 2015 at 7:38 PM, Barry Smith <bsmith at mcs.anl.gov>
> wrote:
> > >
> > > > On Apr 3, 2015, at 7:35 PM, Justin Chang <jchang27 at uh.edu> wrote:
> > > >
> > > > 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?
> > >
> > >   Yes, I think that is essentially correctly.
> > >
> > >   Barry
> > >
> > > >
> > > > 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
> > >
> > >
> >
> >
> >
> >
> >
> > --
> > What most experimenters take for granted before they begin their
> experiments is infinitely more interesting than any results to which their
> experiments lead.
> > -- Norbert Wiener
> >
> >
> >
> >
> > --
> > What most experimenters take for granted before they begin their
> experiments is infinitely more interesting than any results to which their
> experiments lead.
> > -- Norbert Wiener
> >
>
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.mcs.anl.gov/pipermail/petsc-users/attachments/20150901/7799619f/attachment-0001.html>