Custom residual vector derivatives for optimization problems

[dschwen]

Reason

Optimization problems (as @lynnmunday is working on as a PI) require the derivatives of the residual vector R of the problem with respect to the input parameters p, *∂R**/∂p***. We would like to utilize automatic differentiation for this.

The compute workflow would be similar to how the Jacobian matrix *∂R**/∂u_i is built, except we would only need ∂**R/∂**p*** at the end of a time step.

Design

• Add an API method to get an "optimizable parameter" const ADReal & getOptimizableParam("name");, where name is a parameter of type Real. It will be implicitly controllable. The method returns a const reference to a value managed by the framework. During the computation of the residual and Jacobian the underlying ADReal value will have no derivative vector entries.
• Add MOOSE syntax to specify which parameters are uses in the optimization process. Those correspond to parameters gotten through the API above. For those selected params a pseudo-DOF ID is reserved and used to seed the derivative vector in the *∂R**/∂p*** calculation stage
• *∂R**/∂p*** calculation stage This will be another loop like the calculation of the Jacobian for ADKernels, except, this will only work for AD Kernels and materials. At the end of a time step, when the solution is converged, we run computeResidual again, but with empty residual vectors for all variables, and populated residual vectors for all optimizable parameters.

Impact

Enable correct gradients for non-linear optimization problems

[YaqiWang]

Can we handle a parameter in a step function z>p ? 1 : 0? I am questioning because I think it could be better to have number of parameters separate residual evaluations to get the derivatives before doing anything further. Assuming everything can be handled locally can be dangerous, otherwise we possibly could call computeResidual multiple times without AD.

[tegrubbs]

Have there been any updates on this issue? My group is interested in this particular use-case described by @dschwen.

[lynnmunday]

@tegrubbs There is an optimization module that was merged that gives you an interface to the petsc optimization solver TAO. We haven't done anything for these automatic derivatives yet, @oanaoana is working on this.