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# Communications in Mathematical Sciences

## Volume 13 (2015)

### Number 1

### Optimal transport for particle image velocimetry

Pages: 269 – 296

DOI: https://dx.doi.org/10.4310/CMS.2015.v13.n1.a13

#### Authors

#### Abstract

We present a new method for particle image velocimetry, a technique using successive laser images of particles immersed in a fluid to measure the velocity field of the fluid flow. The main idea is to recover this velocity field via the solution of the $L^2$-optimal transport problem associated with each pair of successive distributions of tracers. We model the tracers by a network of Gaussian-like distributions and derive rigorous bounds on the approximation error in terms of the model’s parameters. To obtain the numerical solution, we employ Newton’s method, combined with an efficient spectral method, to solve the Monge-Ampère equation associated with the transport problem. We present numerical experiments based on two synthetic flow fields, a plane shear and an array of vortices. Although the theoretical results are derived for the case of a single particle in dimensions one and two, the results are valid in $\mathbb{R}^d , d \geq 1$. Moreover, the numerical experiments demonstrate that these results hold for the case of multiple particles, provided the Monge-Ampère equation is solved on a fine enough grid.

#### Keywords

optimal transport, particle image velocimetry, Monge-Ampère equation, numerical method

#### 2010 Mathematics Subject Classification

35J96, 49M15, 65M06, 76M28, 90C99

Published 16 July 2014