# Events for 03/28/2023 from all calendars

## Nonlinear Partial Differential Equations

**Time: ** 3:00PM - 4:00PM

**Location: ** BLOC 302

**Speaker: **Boualem Khouider, University of Vicotria

**Title: ***Optimal transport for particle image velocimetry*

**Abstract: **Particle image velocimetry (PIV) is a non intrusive method used to measure the velocity field in laboratory experiments. Small particles, immersed in the fluid, are illuminated by a pulsating laser gun and a suspended camera records the particles' positions from the reflected laser light. Cross correlation algorithms are traditionally used to retrieve the flow velocity from the successive images by measuring the average displacement of look-a-like clusters belonging to two successive images. We propose a new method for PIV,
based on the L2 optimal mass transportation (OT) problem. The suspended particles are modelled by a network of Gaussian-like distributions and the flow fluid is approximated by the optimal transport map of distribution networks associated with successive images. We derive rigorous bounds on the approximation error in terms of the model parameters, namely the size of the Gaussians and the noise level. To obtain the numerical solution of the OT problem, we solve the associated Monge-Ampere equation using a PDE based Newton-like method combined with an efficient spectral method for the underlying linearized PDE. Numerical experiments based on two synthetic flow fields, consisting of a plane shear and an array of vortices are used to validate the methodology. We also consider the case of particles with different masses that are randomly seeded and compare the OT-PIV method with a typical cross correlation algorithm for the case of real data. Using a combination of theory and numerical experiments, we demonstrate that, in the
presence of particles with different weights/brightness, the OT method is more accurate for the largest/brightest particles and it is more faithful when the particles are far enough from each other, making it more suitable for the so-called particle tracking regime of PIV, i.e, when the seeding density is low. In deed, using numerical experiments, we demonstrate that for low seeding densities, the OT method performs better than the traditional cross correlation algorith