Approximation of Divergence-free Vector Fields on Regular Lattices

随着计算能力的增强和存储成本的降低，大数据正变得越来越普遍。大数据是指获取或模拟的海量数据。如今，许多研究工作都集中在高效管理和分析大数据上。另一方面，这个研究项目的目的是通过设计更多的信息获取和处理工具来减少数据量。特别关注的是计算机图形学、气候学、汽车设计和石油工程等学科中出现的流动数据的可视化和模拟，其中流体流动通常由离散的非定常三维速度场描述：在整个模拟域中战略性地分布的一组有限的时变速度矢量。流动可视化的一项基本任务是从有限的速度矢量集中逼近连续的流场。社区中的通常做法是独立地对每个速度分量进行内插。这种直接的方法是首选的，因为它使从业者能够访问各种优化的内插例程。然而，它也有其缺点，因为它没有考虑到实际中遇到的许多流场是不可压缩的，即流体密度不随流体流过区域而变化。

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.