Harbir Antil - Research
PDE Constrained Optimization (Abstract Formulation)
Optimization problem with constraints given by partial differential
equations (PDE) can be written as
over (y,u) ∈ K1 x K2
subject to: e(y,u) = 0.
Here, J: Y x U → ℜ stands for the objective functional
depending on the state
variables y ∈ Y and the control variables u ∈ U.
The third equation corresponds to the PDE (linear/nonlinear),
and K1 ⊂ Y, K2 ⊂ U refer to the sets
of admissible states and control variables, respectively.
The variable u could be an optimal control (optimal control problems) or a shape parameter (shape optimization
The numerical solution to PDE constrained optimization problems involves a series of theoretical and
- Solving a PDE constrained optimization problem not only require a solution to state equations but
adjoint equations as well.
- The structural interaction between the optimization issue 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 can be achieved using
model reduction techniques.