Solving Tangent linear System

[nikhilgv91]

Hello,

My goal is to perform PDE-constrained optimization, and I am using MOOSE to perform the simulations. In addition to the simulation result, I would like to compute the solution of the 'Tangent Linear System' using MOOSE. A short explanation for what I would like to do:
My PDE: F (T,v) = 0
T : state variable
v : control variable (real number)
Tangent linear system: [dF(T,v) / dT] * x = - [dF(T,v) / dv] ; linear solve for x

The matrix dF(T,v) / dT is the jacobian of the original PDE. I would like to know if I can modify the executioner (or other appropriate object) so that the tangent linear system can be solved right after the "original" MOOSE solve in a single run of the simulation.

I am aware that MOOSE can perform matrix-free solves, but for my problem I am using a NEWTON solve.

Thanks in advance!

Kind regards,
Nikhil

[YaqiWang]

Yes. You can modify executioners in whatever way. Once you have [dF(T,v) / dT] and [dF(T,v) / dv], you can directly use PETSc to solve for x. If dF(T,v) / dT is fixed in this solve, you do not have to use MOOSE's PJFNK machinery which could be messy with your "original" system. I feel you can evaluate dF(T,v) / dv pretty easily with MOOSE residual evaluation and Control for v. Just my suggestions.

[aeslaughter]

There is a tagging system: https://mooseframework.inl.gov/framework_development/tagging.html

[nikhilgv91]

Thanks for this hint. Are there more detailed tutorials on the use of tagging? I looked at some input files in the module tests that uses tagging, but I don't understand it very well.

[YaqiWang]

I was thinking to use two residual evaluations to evaluate that one with v another with v+epsilon. Then take the difference of the two divided by epsilon.

[nikhilgv91]

The analytical derivative is trivial to compute, so I would prefer to assemble the residual using the analytical expression.

[nikhilgv91]

@aeslaughter and @YaqiWang thanks for your respective suggestions! I was able to solve the Tangent Linear System. I will mark this thread as the solution to my query.

[aeslaughter]

@zachmprince @lynnmunday How does this relate to your optimization work?

[zachmprince]

This is somewhat related. @lynnmunday could speak more to it. Basically we are approximating [dF(T,v) / dv] using an adjoint equation evaluation, at least that is my understanding.

[nikhilgv91]

Is PDE-constrained optimization already already part of the modules?

[lynnmunday]

We have an app that is doing pde constrained optimization but it needs cleaned up with better testing before we can merge it into moose. I think we would have it merged in by the end of March. We are use TAO to do the optimization and we are providing it with a forward and gradient solve. The gradient solve is computed from the adjoint which we get by just solving the system a second time with the adjoint loads and bcs. We do this with the multiapp system.

[nikhilgv91]

@lynnmunday Thanks for the detailed reply!