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) ∈ 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 K1Y, K2U 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: