Tutorial

Welcome to the CASHOCS tutorial. In the following, we present several example programs that showcase how CASHOCS can be used to solve optimal control and shape optimization problems.

Note, that we assume that you are (at least somewhat) familiar with PDE constrained optimization and FEniCS. For a introduction to these topics, we can recommend the textbooks

• Optimal Control and general PDE constrained optimization

  • Hinze, Ulbrich, Ulbrich, and Pinnau, Optimization with PDE Constraints

  • Tröltzsch, Optimal Control of Partial Differential Equations

• Shape Optimization

  • Delfour and Zolesio, Shapes and Geometries

  • Sokolowski and Zolesio, Introduction to Shape Optimization

• FEniCS

  • Logg, Mardal, and Wells, Automated Solution of Differential Equations by the Finite Element Method

  • The FEniCS demos

However, we will also provide links to either the underlying theory of PDE constrained optimization or to the relevant documentation of FEniCS in this tutorial.

Note, that an overview over CASHOCS and its capabilities can be found in Blauth, cashocs: A Computational, Adjoint-Based Shape Optimization and Optimal Control Software.

List of all demos:

• Optimal Control Problems
  • Distributed Control of a Poisson Problem
  • Documentation of the Config Files for Optimal Control Problems
  • Control Constraints
  • Neumann Boundary Control
  • Using Multiple Variables and PDEs
  • Coupled Problems - Monolithic Approach
  • Coupled Problems - Picard Iteration
  • Distributed Control of a Stokes Problem
  • Distributed Control for Time Dependent Problems
  • Optimal Control with Nonlinear PDE Constraints
  • Dirichlet Boundary Control
  • Iterative Solvers for State and Adjoint Systems
  • Optimal Control with State Constraints
  • Sparse Control
  • Tracking of the Cost Functional for Optimal Control Problems
• Shape Optimization Problems
  • Shape Optimization with a Poisson Problem
  • Documentation of the Config Files for Shape Optimization Problems
  • Regularization for Shape Optimization Problems
  • Inverse Problem in Electric Impedance Tomography
  • Optimization of an Obstacle in Stokes Flow
  • Remeshing with CASHOCS
  • Custom Scalar Products for Shape Gradient Computation
  • Computing the Shape Stiffness via Distance to the Boundaries
  • Shape Optimization with the p-Laplacian
• Using CASHOCS Exclusively as a Solver
  • CASHOCS as Solver for Optimal Control Problems
  • CASHOCS as Solver for Shape Optimization Problems

Note

We recommend that you start with the introductory demos for optimal control problems, i.e., Distributed Control of a Poisson Problem and Documentation of the Config Files for Optimal Control Problems, as these demonstrate the basic ideas of CASHOCS. Additionally, they are a bit simpler than the introductory tutorials for shape optimization problems, i.e., Shape Optimization with a Poisson Problem and Documentation of the Config Files for Shape Optimization Problems.

Moreover, we note that some of CASHOCS functionality is explained only for optimal control, but not for shape optimization problems. This includes the contents of Coupled Problems - Picard Iteration, Distributed Control for Time Dependent Problems, Iterative Solvers for State and Adjoint Systems, Optimal Control with State Constraints. However, the corresponding functionalities only deal with either the definition of the state system, its (numerical) solution, or the definition of suitable cost functionals. Therefore, they are straightforward to adapt to the case of shape optimization.