Table of Contents Preface Summary Article Introduction Models and Simulations Computing with Large Data Sets Geometrization of Topology and Physics Noise and Randomness Nonlinearity Beyond Fermat Mathematics for Biology and Medicine Information Technology Individual Contributions List of Contributors with Affiliations

Mathematics -- The Science of Patterns and Algorithms

Models and Simulations

Simulation and computing have become an essential part of modern science and engineering, complementing theory and experimentation. This development, driven by access to ever more powerful computers, poses new challenges to the mathematical sciences.

At the heart of any simulation is a model, a mathematical formulation that captures the structure and form of real-world phenomena. This model is typically surrounded by numerical techniques that produce quantitative information sometimes augmented by estimates of reliability, and by computer visualization that allows results to be manipulated and summarized, leading to qualitative understanding that inspires further analysis. Each of these elements requires sophisticated mathematics.

In modeling complex and poorly understood phenomena, such as weather or the stock market, the modeler encounters data that are uncertain, inaccurate, inconsistent, incomplete, and/or insufficient to determine a solution [MS]. For example, numerous medical conditions produce very similar CAT scan images, yet doctors must decide on the most plausible diagnosis. Typically, the input data, the modeling process and the solution may contain random and systematic errors and thus need statistical and/or probabilistic analysis. On the other hand, solving a problem exactly may be computationally intractable; finding approximate solutions that have sufficient accuracy is then another challenging task, requiring a careful mathematical analysis. The resultant uncertainty assessment provides a mechanism to drive confidence in the validity of the computed solution [GL]. For the simplest cases in which phenomena are nearly linear, powerful mathematical and computational tools have been developed over the last fifty years to estimate and deal with these various forms of uncertainty. Genuinely nonlinear problems, however, are still far beyond reach [MS].

A particular challenge is posed by "model reduction." Technology relies increasingly on models that describe huge systems on fine scales in length and time while data and conclusions may be concentrated at coarser scales only. Unwieldy fine-scale equations need to be replaced with effective coarse-scale descriptions in order to provide more accurate predictions when computation and analysis on the fine scale is totally unfeasible. Stochastic modeling can also be used to provide a huge reduction in complexity. The challenge to the mathematical sciences is to develop systematic approaches to bridge these levels of organization. Important examples include global climate change, environmental remediation, computer-assisted design for manufactured products, neuroscience, and chemical kinetics of drug design [GL]. Similar organization and reduction are needed in settings where the multiple scales do not result from physical scaling in time or length, but are discrete or hierarchical, such as in multilayered descriptions of large networks [MS].

In many of these areas, we have made progress in the past, but we still need many orders of magnitude improvement before we can tackle truly realistic problems. These activities require substantial effort in collaboration between mathematical and other scientists as well as novel mathematical insight to create new techniques that allow qualitative insights needed to make further breakthroughs.