Survey and Review

Abstract

The two articles in this issue's Surveys and Review section could not be more different. One looks backward, and the other, forward. The paper “John von Neumann's Analysis of Gaussian Elimination and the Origins of Modern Numerical Analysis,” by Joseph Grcar, provides a historical account of von Neumann's and Goldstine's analysis of Gaussian elimination and argues that one can trace the roots of numerical analysis, as practiced today, to this important paper. There is a wealth of biographical material on both von Neumann and Goldstine, which is a pleasure to read. The article then dives into the contributions of these figures on error analysis of Gaussian elimination. The presentation is clear, using modern notation with which readers will be familiar, and provides convincing evidence of the importance of their contributions. The goal of molecular dynamic simulation is to employ computational techniques to predict the properties of materials from knowledge of their molecular structure. The idea is simple enough— given a model for the forces between the atoms and a model for how they interact with their surroundings, one can use Newton's Second Law to derive the equations of motion of each atom and compute away. To actually do the calculation is not so simple. There are plenty of computational challenges to overcome in order to be able to complete the calculation in a reasonable amount of time. For average protein molecules, the number of pairwise interactions can make a direct calculation of the interactions computationally prohibitive. Therefore, approximation schemes are necessary to make the calculation tractable on today's machines. The game is to come up with approximation schemes that trade off accuracy for computational efficiency. The second paper, “Fast Analytical Methods for Macroscopic Electrostatic Models in Biomolecular Simulations,” by Xu and Cai, is a review of approaches to developing efficient methods in molecular dynamics simulation. The focus of the article is on solvation, where the molecule is immersed in an environment with which it reacts. Solvation presents specific challenges. The paper goes over ways in which approximations lead to fast algorithms. The article provides a view into the world of biomolecular simulations and exposes where analysis has made a difference and where further developments are needed.