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.
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 >
