# Harbir Antil - Research

## PDE Constrained Optimization and Model Reduction with Localized
Nonlinearities

For a description of PDE constrained optimization problems we refer to:
link
For a description of model reduction techniques we refer to:
link

A real time solution to river pollution, let say two factories are discharging
waste in the river and we want to control the discharge and want to attain a
desired concentration of pollutants in the river. We solve the minimization problem using
the model reduction techniques, the reduced problem takes *1 sec* as compared
to *10 min* for full problem. link

## Goal

The goal of this project is to develop optimization algorithm to solve the
time dependent linear and nonlinear PDE constrained optimization problems
with localized nonlinearities, using balanced truncation model reduction
(BTMR) and domain decomposition.
## Collaborators

## Achievements

- We have integrated domain decomposition and model reduction for systems with small localized nonlinearities. In our case, nonlinearities arise from dependence on shape parameters.
- Reduced order minimization of an objective function is carried out subject to the advection-diffusion equations and Stokes equations.
- We have proven error estimated for the error between the solution of the original and the reduced order problem. The error estimate depend on the model reduction error estimate.

## Related Publications

**H. Antil, M. Heinkenschloss, and R.H.W. Hoppe.**

* Domain decomposition and balanced truncation model reduction
for shape optimization of the Stokes system* (2010).

Optimization Methods and Software (2011), 26(4-5):643--669. preprint

Most cited article in this journal. Years 2012-13.

**H. Antil, M. Heinkenschloss, R.H.W. Hoppe, and D.C.
Sorensen.**

* Domain decomposition and model reduction for the numerical
solution of PDE constrained optimization problems with localized
optimization variables * .

Computing and Visualization in Science (2010), 13(6):249--264. link