Approximation of Divergence-free Vector Fields on Regular Lattices

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As computational power increases and storage becomes cheaper, 'big-data' - a term used to refer to huge quantities of acquired or simulated data - is becoming more and more pervasive. A lot of research effort nowadays is focused towards efficiently managing and analyzing big-data. This research program, on the other hand, aims at cutting down the amount of data by devising more informative acquisition and processing tools. The particular focus is the visualization and simulation of flow data that arises in disciplines such as computer graphics, climatology, automotive design and petroleum engineering, where the fluid flow is usually described by a discrete unsteady three-dimensional velocity field: a finite set of time-varying velocity vectors strategically distributed throughout the simulation domain. A fundamental task in flow visualization is the approximation of a continuous flow field from the finite set of velocity vectors. The usual practice in the community is to interpolate each velocity component independently. This straightforward approach is preferred since it gives the practitioner access to various optimized interpolation routines. However, it has its drawbacks since it does not take into account the fact that many flow fields encountered in practice are incompressible, i.e. the fluid density does not change as the fluid flows through the domain.The long-term aim of this research program is to improve the quality of flow simulations and visualizations by seeking continuous flow representations that are more efficient and at the same time, preserve the incompressible nature of the flow. Owing to their incompressibility, these representations will be more accurate thus allowing domain experts to simulate flows at coarser resolutions without sacrificing quality. Due to their efficiency, they also have the potential of capturing more details in simulations and visualizations without increasing the computational budget. Additionally, by addressing some of the theoretical challenges, we also hope to gain a better fundamental understanding of the complex dynamics of fluids.