Analysis of advanced discretizations of partial differential equations

利用偏微分方程进行建模和模拟在科学和工程中无处不在，包括天文学、气象学、海洋学、地震学、地球物理学、地质学、经济学、流体力学、固体物理学和量子力学。在过去的几十年里，我们看到可以在计算机上运行的典型模拟的复杂性有了巨大的增长。其中一个原因当然是我们目睹的计算机硬件的惊人技术飞跃。另一方面，也许令人惊讶的是，据估计，快速算法的发展与硬件改进的影响大致相同。事实上，我们期望的更多：现有的快速算法及其理论可以比作冰山一角，大部分宝藏尚未被发现。该计划的主题是对求解偏微分方程的两类重要数值算法的理论理解：自适应方法和几何离散化技术。虽然自适应方法可以被描述为以“最聪明”的方式分配计算资源以最小化努力浪费的算法，但几何离散化技术的目标是保持原始微分方程的基本几何性质。众所周知，后一种技术通常比更传统的方法更好，并且在许多情况下，例如生物化学分子的模拟，它们是唯一合理的选择。当前的项目旨在进一步从数学上理解属于上述两类中的一种或两种的某些数值算法。首先，我们计划设计新的快速算法来求解用于模拟生物膜、大规模海洋流动和电磁现象的各种方程。其次，我们将对所设计的算法进行严格的数学分析，建立一个全面的理论来解释算法在不同情况下的行为。这一点很重要，因为没有算法是完全防傻瓜的，根据手头的特定问题，人们可能想要选择不同的算法。该项目的第三个目标是对一些现有的几何离散化方法进行数学分析，这些方法用于模拟亚核子物质和剧烈的天文事件，如黑洞碰撞。这些方法在实践中相当有效，但并非没有难以解决的问题，目前我们从数学的角度对它们的了解甚少。我们期望严谨的数学处理不仅会给实践者更多的信心，而且还会为解决问题和改进现有算法开辟可能性

Modelling and simulation using partial differential equations are ubiquitous in science and engineering, including astronomy, meteorology, oceanography, seismology, geophysics, geology, economics, fluid mechanics, solid state physics, and quantum mechanics. In the last few decades we have seen an enormous increase in the complexity of a typical simulation that can be run on a computer. One reason is of course the amazing technological leaps in computer hardware we have been witnessing. On the other hand, perhaps surprisingly, it is estimated that the development of fast algorithms has had roughly the same impact as that of the hardware improvements. In fact, we expect much more: The existing fast algorithms and their theory can be compared to the tip of an iceberg, with most of the treasures yet to be discovered. The theme of the proposed project is the theoretical understanding of two important classes of numerical algorithms for solving partial differential equations: Adaptive methods and geometric discretization techniques. While adaptive methods can be described as algorithms that distribute computing resources in a "smartest" way so as to minimize waste of effort, the goal of geometric discretization techniques is to preserve fundamental geometric properties of the original differential equations. The latter techniques are known to be generally preferable to the more conventional methods, and in many cases, such as simulations of bio-chemical molecules, they are the only reasonable choice. The current project aims to further mathematical understanding of certain numerical algorithms belonging to either or both of the aforementioned classes. First, we plan to design new fast algorithms for solving various equations that are used to model bio-membranes, large scale ocean flows, and electromagnetic phenomena. Second, we will perform rigorous mathematical analysis of the designed algorithms, building a comprehensive theory that explains how the algorithms behave in different situations. This is important since no algorithm is completely fool proof, and depending on the particular problem at hand, one might want to choose different algorithms. The third aim of the project is mathematical analysis of some existing geometric discretization methods that are used in simulations of sub-nucleonic matter and of violent astronomical events such as black hole collisions. These methods work reasonably well in practice, but not without difficult unresolved problems, and currently we have a very little understanding of them from mathematical standpoint. We expect that rigorous mathematical treatment will not only give more confidence to the practitioners, but also open up possibilities to resolve the issues and improve upon the existing algorithms.**