Research and Education in Computational Science and Engineering

Over the past two decades the field of computational science and engineering (CSE) has penetrated both basic and applied research in academia, industry, and laboratories to advance discovery, optimize systems, support decision-makers, and educate the scientific and engineering workforce. Informed by centuries of theory and experiment, CSE performs computational experiments to answer questions that neither theory nor experiment alone is equipped to answer. CSE provides scientists and engineers of all persuasions with algorithmic inventions and software systems that transcend disciplines and scales. Carried on a wave of digital technology, CSE brings the power of parallelism to bear on troves of data. Mathematics-based advanced computing has become a prevalent means of discovery and innovation in essentially all areas of science, engineering, technology, and society; and the CSE community is at the core of this transformation. However, a combination of disruptive developments---including the architectural complexity of extreme-scale computing, the data revolution that engulfs the planet, and the specialization required to follow the applications to new frontiers---is redefining the scope and reach of the CSE endeavor. This report describes the rapid expansion of CSE and the challenges to sustaining its bold advances. The report also presents strategies and directions for CSE research and education for the next decade.Comment: Major revision, to appear in SIAM Revie

Modeling and simulating chemical reactions

Many students are familiar with the idea of modeling chemical reactions in terms of ordinary differential equations. However, these deterministic reaction rate equations are really a certain large-scale limit of a sequence of finer-scale probabilistic models. In studying this hierarchy of models, students can be exposed to a range of modern ideas in applied and computational mathematics. This article introduces some of the basic concepts in an accessible manner and points to some challenges that currently occupy researchers in this area. Short, downloadable MATLAB codes are listed and described

Teaching of Ordinary Differential Equations Using the Assumptions of the PBL Method

Mathematics is of fundamental importance to any natural sciences program (because it provides analytical and approximate results that can be simulated and modeled) and perhaps to other areas of human knowledge. Ordinary differential equations (ODEs) are especially of fundamental importance to engineering programs because the modeling of all phenomena of interest for these programs involves ODE solutions; at the same time, students experience difficulties when learning ODEs and about their applications to real physical scenarios. In this study, the problem-based learning method was used to study a group of mechanical engineering students at theSENAICIMATECUniversityCenter. The study analyzes the effectiveness of ODE instruction given through this program using the specified methodology. To consolidate the study, Conceptual Field Theory is used together with an evaluation of the results to observe student behaviors and attitudes in relation to actions arising from the method. At the end of the study, a questionnaire was completed by the students, with which they evaluated the effectiveness of the teaching methodology employed

Evolution of the Modern ODE Course

The rapid development of technology in the latter part of the twentieth century has revolutionized the teaching of differential equations. In this paper we will try to trace the evolution of this important change. We tried to include the most important efforts in this regard, but we apologize in advance if some efforts have slipped our attention

Epidemiology and the SIR Model: Historical Context to Modern Applications

We suggest the use of historical documents and primary sources, as well as data and articles from recent events, to teach students about mathematical epidemiology. We propose a project suitable -- in different versions -- as part of a class syllabus, as an undergraduate research project, and as an extra credit assignment. Throughout this project, students explore mathematical, historical, and sociological aspects of the SIR model and approach data analysis and interpretation. Based on their work, students form opinions on public health decisions and related consequences. Feedback from students has been encouraging. We begin our project by having students read excerpts of documents from the early 1900s discussing the Indian plague epidemic. We then guide students through the derivation of the SIR model by analyzing the seminal 1927 Kermack and McKendrick paper, which is based on data from the Indian epidemiological event they have studied. After understanding the historical importance of the SIR model, we consider its modern applications focusing on the Ebola outbreak of 2014-2016 in West Africa. Students fit SIR models to available compiled data sets. The subtleties in the data provide opportunities for students to consider the data and SIR model assumptions critically. Additionally, social attitudes of the outbreak are explored; in particular, local attitudes towards government health recommendations