ITR - (ASE) - (sim): Discrete Differential Calculus (DDC)

几个世纪以来，科学家、工程师和数学家一直在使用微分学建立预测模型，这是一种非常宝贵的工具，可以使用可测量量之间简洁、连续的关系来描述复杂性。然而，自然规律的历史微分模型与当前计算机的纯粹数字性质相冲突：如果不特别注意，将连续方程转换为离散数字通常会导致有限的预测能力，因为离散计算可能不忠实于我们试图模拟的潜在数学和物理原理。在这个项目中，研究者和他的同事们开发并适应了基于离散几何的方法，用于简洁的离散化，以及离散操作，这些方法忠实于它们所代表的连续世界。我们的目标是创建一种离散微分学，它不再需要对连续模型进行离散化，而是从一开始就“内置”的计算。这种方法有望为直接适用于数字计算的建模提供理论和计算基础，并保证更高的保真度。这项研究是经典建模和新型计算工具的协同混合，有望成为基础和计算重要方法的丰富来源，以及跨学科的相互作用。研究者期望在物理科学、数学、工程和信息技术之间产生一种互惠互利的思想流动，从这种离散计算的统一方法中出现。这种新颖的方法源于对建模的仔细重新制定，预计将具体地提高预测能力和一系列算法，这些算法将在无数应用中发挥实际作用——从天气预报、地震工程、建筑和汽车安全，到生物医学模拟和医学成像诊断。

For centuries, scientists, engineers, and mathematicians have been building predictive models of the world around them using Differential Calculus, an invaluable tool in describing complexity using concise, continuous relations between measurable quantities. However, the historical differential modeling of the laws of nature clashes with the purely digital nature of current computers: if no special care is used, turning continuous equations into discrete numbers often results in a limited predictive power, as discrete computations can be unfaithful to the underlying mathematics and physical principles we are trying to simulate. In this project, the investigator and his colleagues develop and adapt discrete-geometry based methodologies for concise discretizations, as well as discrete operations, which are faithful to the continuous world they are meant to represent. The goal is to create a Discrete Differential Calculus, for which computation is no longer an afterthought requiring a discretization of continuous models, but is ``built-in'' from the start. This approach promises to provide theoretical and computational foundations for modeling that are directly amenable to digital computations, with guaranteed higher fidelity.This research, a synergetic mixture of classical modeling and novel computational tools, promises to be a rich source of fundamental and computationally important methods, and interdisciplinary interactions. The investigator anticipates a mutually beneficial flow of ideas among physical sciences, mathematics, engineering and Information Technology to emerge from this unifying approach to discrete computations. The novel methodology resulting from this careful reformulation of modeling is expected to concretely result in an improved predictive power and a host of algorithms which will be practically useful in countless applications---from weather forecasts, earthquake engineering, building and car safety, to biomedical simulation and diagnostics from medical imaging.