Discrete processes and their continuous limits
Year: 2020
Type: article
Abstract: The possibility that a discrete process can be fruitfully approximated by a continuous one, with the latter involving a differential system, is fascinating. Important theoretical insights, as well as significant computational efficiency gains may lie in store. A great success story in this regard are the Navier-Stokes equations, which model many phenomena in fluid flow rather well. Recent years saw many attempts to formulate more such continuous limits, and thus harvest theoretical and practical advantages, in diverse areas including mathematical biology, economics, finance, computational optimization, image processing, game theory, and machine learning.Caution must be applied as well, however. In fact, it is often the case that the given discrete process is richer in possibilities than its continuous differential system limit, and that a further study of the discrete process is practically rewarding. Furthermore, there are situations where the continuous limit process may provide important qualitative, but not quantitative, information about the actual discrete process. This paper considers several case studies of such continuous limits and demonstrates success as well as cause for caution. Consequences are discussed.
Source: Journal of Dynamics and Games
DOI: 10.3934/jdg.2020008
Author: Uri M. Ascher
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W3019897459