# Harbir Antil - Research

## PDE Constrained Optimization (Abstract Formulation)

Optimization problem with constraints given by partial differential
equations (PDE) can be written as

**inf** *J(y,u)*

**over** *(y,u)* * ∈ K*_{1} x *K*_{2}

**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 *K*_{1} ⊂ *Y, K*_{2} ⊂ *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
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 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.