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
