Numerical solution of large-scale nonlinear optimization problems, with focus on optimization problems in which the constraints involve the solution of (systems of) partial differential equations. These problems include optimal control, optimal design, parameter estimation, and inverse problems.
The goal is to identify the location and the intensity of the source of a hazardous material from measurements of the concentration at sensor locations. The problem geometry is related to a two-story structure, which is modeled in two dimensions. Our approach amounts to solving an inverse (regularized) PDE-constrained optimization problem (optimal control problem) with distributed controls.
The constraints are given by the advection-diffusion equations. The advection term simulates the movement of the air induced by the building's HVAC system, for example:
A typical sensor distribution is as follows:
Below you will find figures of simulated sources along with our solution, using a modest number of sensors.
The goal is to control the temperature on the inflow boundary of a fluid flow that is adjecent to a solid, in order to achieve a given temperature (in our case 90º F) inside the solid (see figure below). This problem amounts to solving an optimal control problem with boundary controls. The governing PDE is of the advection-difusion type, with discontinuous diffusion coefficients, and an advective component only in the fluid domain.
The contour plots below illustrate the uncontrolled temperature distribution, as well as our results, i.e. the controlled temperature.
