Approximation of Divergence-free Vector Fields on Regular Lattices

随着计算能力的提高和存储成本的降低，“大数据”——一个用来指大量获取或模拟数据的术语——正变得越来越普遍。如今，大量的研究工作都集中在如何有效地管理和分析大数据上。另一方面，这项研究计划旨在通过设计更多信息采集和处理工具来减少数据量。特别关注的是在计算机图形学、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，在这些学科中，流体流动通常由一个离散的非定常三维速度场来描述：一组有限的时变速度矢量在整个模拟领域中有策略地分布。流动可视化的一个基本任务是用有限的速度向量逼近连续流场。社区中通常的做法是独立地插值每个速度分量。这种直接的方法是首选的，因为它给从业者访问各种优化的插值例程。然而，它也有缺点，因为它没有考虑到实践中遇到的许多流场是不可压缩的，即流体密度在流体流过该域时不会改变。

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.