cashocs in a nutshell

As newcomer to cashocs, we recommend the paper Blauth - cashocs: A Computational, Adjoint-Based Shape Optimization and Optimal Control Software, which gives an overview over cashocs and its capabilities. Moreover, for a comprehensive description of cashocs, we refer to its tutorial.

In the following, we briefly showcase how cashocs can be used to solve a distributed optimal control problem with a Poisson equation as PDE constaint.

Since cashocs is based on FEniCS, most of the user input consists of definining the objects (such as the state system and cost functional) via UFL forms. If one has a functioning code for the forward problem and the evaluation of the cost functional, the necessary modifications to optimize the problem in cashocs are minimal. Consider, e.g., the following optimization problem

\[\begin{split}&\min\; J(y,u) = \frac{1}{2} \int_{\Omega} \left( y - y_d \right)^2 \text{ d}x + \frac{\alpha}{2} \int_{\Omega} u^2 \text{ d}x \\ &\text{ subject to } \quad \left\lbrace \quad \begin{alignedat}{2} -\Delta y &= u \quad &&\text{ in } \Omega,\\ y &= 0 \quad &&\text{ on } \Gamma. \end{alignedat} \right.\end{split}\]

Note, that the problem is treated in detail in the corresponding cashocs tutorial.

For our purposes, we assume that a mesh for this problem is defined and that a suitable function space is chosen. This can, e.g., be achieved via

from fenics import *
import cashocs

config = cashocs.load_config('path_to_config')
mesh, subdomains, boundaries, dx, ds, dS = cashocs.regular_mesh(25)
V = FunctionSpace(mesh, 'CG', 1)

The config object, which is created from a .ini file, is used to determine the parameters for the optimization algorithms. This is where the user can finely tune the behavior of the algorithms.

To define the state problem, we then define a state variable y, an adjoint variable p and a control variable u, and write the PDE as a weak form

y = Function(V)
p = Function(V)
u = Function(V)
e = inner(grad(y), grad(p)) - u*p*dx
bcs = cashocs.create_dirichlet_bcs(V, Constant(0), boundaries, [1,2,3,4])

Finally, we define the cost functional and the optimization problem

y_d = Expression('sin(2*pi * x[0] * sin(2*pi*x[1]))', degree=1)
alpha = 1e-6
J = 1/2*(y - y_d) * (y - y_d) * dx + alpha/2*u*u*dx
opt_problem = cashocs.OptimalControlProblem(e, bcs, J, y, u, p, config)
opt_problem.solve()

The only major difference between cashocs and FEniCS code is that one has to use fenics.Function objects for states and adjoints, and that fenics.TrialFunction and fenics.TestFunction are not needed to define the state equation. Other than that, the syntax would also be valid with FEniCS, at least for this problem.

For a detailed discussion of the features of cashocs and its usage we refer to the cashocs tutorial.