Optimization and Control
Optimization problem with constraints given by partial (ordinary) differential
equations (PDEs / ODEs) can be written as
min J(u,z),
subject to: e(u,z) = 0,
over (u,z) ∈ Uad x Zad.
Here, J: U x Z → ℜ stands for the objective functional
depending on the state
variables u ∈ U and the control variables z ∈ Z.
The equation corresponds to the PDE (linear/nonlinear),
and Uad ⊂ U, Zad ⊂ Z refer to the sets
of admissible states and control variables, respectively.
The variable z could be an optimal control (optimal control problems) or a shape parameter
(shape optimization problems).
The numerical solution to PDE constrained optimization problems involves a series of theoretical and
practical challenges:
- Solving a PDE constrained optimization problem not only require a solution to state equations but
adjoint equations as well.
-
The PDEs may not equalities, but could be complementarity problems or variational inequalities.
Also, these problems may contain uncertainty due to unknown boundary conditions, coefficients, etc.
- The structural interaction between optimization algorithms and the underlying
PDE and the impact of the discretization processes have to be taken
into account.
- The numerical approaches typically lead to large-scale nonlinear programming problems. With
regard to algorithmic complexity, their numerical solution requires the use of efficient
iterative schemes such as multilevel techniques.
- Significant savings both in terms of memory and computational time maybe needed via
model reduction techniques.