Discrete Calculus: Applied Analysis on Graphs for Computational Science

The field of discrete calculus, also known as "discrete exterior calculus", focuses on finding a proper set of definitions and differential operators that make it possible to operate the machinery of multivariate calculus on a finite, discrete space. In contrast to traditional goals of finding an accurate discretization of conventional multivariate calculus, discrete calculus establishes a separate, equivalent calculus that operates purely in the discrete space without any reference to an underlying continuous process.

This unique text brings together into a single framework current research in the three areas of discrete calculus, complex networks, and algorithmic content extraction. Although there have been a few intersections in the literature between these disciplines, they have developed largely independently of one another, yet researchers working in any one of these three areas can strongly benefit from the tools and techniques being used in the others. Many example applications from several fields of computational science are provided to demonstrate the usefulness of this framework to a broad range of problems. Readers are assumed to be familiar with the basics of vector calculus, graph theory, and linear algebra.

Graduate students and researchers interested in discrete calculus, complex networks, image processing and computer graphics will find this text/reference a clear introduction to the foundations of discrete calculus as well as a useful guide to have readily available for their work.

Dr. Leo J. Grady is a Senior Research Scientist with Siemens Corporate Research in Princeton, New Jersey, USA. Dr. Jonathan R. Polimeni is a Research Fellow at the Massachusetts General Hospital in Boston, Massachusetts, USA, and Instructor in Radiology at Harvard Medical School, Boston, Massachusetts, USA.