Parallel Algorithms for Summing Floating-Point Numbers

The problem of exactly summing n floating-point numbers is a fundamental problem that has many applications in large-scale simulations and computational geometry. Unfortunately, due to the round-off error in standard floating-point operations, this problem becomes very challenging. Moreover, all existing solutions rely on sequential algorithms which cannot scale to the huge datasets that need to be processed. In this paper, we provide several efficient parallel algorithms for summing n floating point numbers, so as to produce a faithfully rounded floating-point representation of the sum. We present algorithms in PRAM, external-memory, and MapReduce models, and we also provide an experimental analysis of our MapReduce algorithms, due to their simplicity and practical efficiency.Comment: Conference version appears in SPAA 201

Image registration algorithm for molecular tagging velocimetry applied to unsteady flow in Hele-Shaw cell

In order to develop velocimetry methods for confined geometries, we propose to combine image registration and volumetric reconstruction from a monocular video of the draining of a Hele-Shaw cell filled with water. The cell’s thickness is small compared to the other two dimensions (e.g. 1x400 x 800 mm3). We use a technique known as molecular tagging which consists in marking by photobleaching a pattern in the fluid and then tracking its deformations. The evolution of the pattern is filmed with a camera whose principal axis coincides with the cell’s gap. The velocity of the fluid along this direction is not constant. Consequently, tracking the pattern cannot be achieved with classical methods because what is observed is the integral of the marked molecules over the entire cell’s gap. The proposed approach is built on top of direct image registration that we extend to specifically model the volumetric image formation. It allows us to accurately measure the motion and the velocity profiles for the entire volume (including the cell’s gap) which is something usually hard to achieve. The results we obtained are consistent with the theoretical hydrodynamic behaviour for this flow which is known as the Poiseuille flow

Homotopy hyperbolic 3-manifolds are hyperbolic

This paper introduces a rigorous computer-assisted procedure for analyzing hyperbolic 3-manifolds. This procedure is used to complete the proof of several long-standing rigidity conjectures in 3-manifold theory as well as to provide a new lower bound for the volume of a closed orientable hyperbolic 3-manifold