Let me back up a bit. Say I am solving -div[D(x)grad[c]] = 0 on a unit cube with a hole. Interior boundary has DirichletBC c = 1 and exterior boundary has DirichletBC c = 0. If I solve the above in the strong sense (e.g., finite volume, finite difference, etc) -div[D(x)grad[c]] = 0 at every grid cell (hence local mass conservation).
According to discrete maximum principle and the prescribed BCs, my bounds *should* be between 0 <= c <= 1. However, if D(x) is aniosotropic, the KSP/SNES solver will return c's less than 0 and/or greater than 1. Local mass balance still exists for all cells regardless. Now, if TAO is used and Vec XL and XU of TaoSetVariable bounds are set to 0 and 1 respectively, I now ensure that 0 <= c <= 1 even for anisotropy. However, our experiments have shown that the "corrected" regions i.e., the regions where it would have violated DMP otherwise, no longer satisfy local mass balance i.e., div[D(x)grad[c]] != at the element level. The unaffected regions, i.e. the regions where DMP was satisfied regardless, are still locally conservative. We have shown through MATLAB's *quadprog* that we can achieve both DMP and local mass balance if we applied the mixed form (using least-squares finite element method) of the diffusion and subject the optimization solve to satisfy both upper/lower bounds as well as equality constraints (which is divergence of flux = 0 in Omega). If no bounds/constraints were set, the objective function is equal to zero. If either only bounds or constraints, the objective function is equal to some non-zero value. If both bounds and constraints, the objective function is a slightly larger non-zero value. Basically, the more constraints I apply, the further and further away my overall solution may drift from the original discretization. I am curious to see if 1) something like this is possible with TAO and 2) if I can achieve the SNESVI equivalent in the sense I have described. Hope this makes a little more sense. Thanks, Justin On Tue, Sep 1, 2015 at 4:29 PM, Barry Smith <[email protected]> wrote: > > I really cannot understand what you are saying. > > > On Sep 1, 2015, at 5:14 PM, Justin Chang <[email protected]> wrote: > > > > 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 > > I think what you say above is backwards. First of all x_i would never > be < lb_i or > ub_i since it is constrained to be between them. It is > just that with a VI IF x_i is lb_i or ub_i then F_i(x) is allowed to be > nonzero (that is that equation is allowed to violated). > > > > > 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 > > No. With constraints in optimization, they must all be satisfied for one > to say that one has a solution. > > With VIs the F(x) is NOT a constraint and should not be thought of as a > constraint; only the bounds are constraints. Or think of F(x) as a special > kind of constraint (that doesn't exist with optimization) that is allowed > to be violated in a a very special way (when its x_i variable is on a > constraint). > > Barry > > > > > > Is that the general idea here? > > > > On Tue, Sep 1, 2015 at 12:48 PM, Barry Smith <[email protected]> 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 <[email protected]> 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 <[email protected]> > wrote: > > > On Tue, Sep 1, 2015 at 3:46 AM, Justin Chang <[email protected]> > 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 <[email protected]> > wrote: > > > On Tue, Sep 1, 2015 at 3:11 AM, Justin Chang <[email protected]> > 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 <[email protected]> > wrote: > > > > > > > On Aug 31, 2015, at 7:36 PM, Justin Chang <[email protected]> > 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 <[email protected]> > wrote: > > > > > > > > > On Apr 3, 2015, at 7:35 PM, Justin Chang <[email protected]> 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 <[email protected]> > 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 <[email protected]> > 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 <[email protected]> > 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 <[email protected]> > 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 > > > > > > > > >
