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

California Institute Of Technology, Pasadena CA

Investigators

Abstract

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.