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”（用于指大量获取或模拟的数据）变得越来越普遍。如今，许多研究工作都集中在有效管理和分析大数据上。另一方面，本研究计划旨在通过设计更多信息的采集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稳态三维速度场描述：一组有限的时变速度矢量策略性地分布在整个模拟域中。流动显示的一个基本任务是从有限组速度矢量近似连续流场。通常的做法是对每个速度分量进行独立插值。这种直接的方法是优选的，因为它使从业者能够访问各种优化的插值例程。然而，它有其缺点，因为它没有考虑到在实践中遇到的许多流场是不可压缩的事实，即流体密度不会随着流体流过该域而改变。该研究计划的长期目标是通过寻求更有效的连续流表示来提高流动模拟和可视化的质量，同时，保持流动的不可压缩性质。由于它们的不可压缩性，这些表示将更加准确，从而允许领域专家在不牺牲质量的情况下以更粗糙的分辨率模拟流动。由于它们的效率，它们还具有在模拟和可视化中捕获更多细节的潜力，而不会增加计算预算。此外，通过解决一些理论挑战，我们还希望对流体的复杂动力学有更好的基本了解。

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.