Research Spotlights

Abstract

This issue contains two Research Spotlights articles in disparate fields of applied math. The first paper, “Mortality Implications of Mortality Plateaus” by Trifon I. Missov and James W. Vaupel, focuses on probabilistic models for hazard analysis in biological systems. It has long been observed in many different biological populations that the force of mortality (also known as the hazard rate) becomes constant for very long-lived individuals, a concept known as a mortality plateau. Such a mortality plateau is observed, for example, in human populations among those older than 110 years. By assuming the existence of a mortality plateau in a mathematical model for mortality in a heterogeneous population, the authors are able to give a number of implications of restrictions on the model. For example, one standard mortality model is a Gompertz model, in which for each individual the mortality rate increases exponentially, which implies that the population hazard function has an asymptotic mortality plateau; however, the article shows a set of other possible shapes of mortality functions that would also give rise to such a mortality plateau. This paper will appeal to SIAM Review readers with an interest in actuarial science, demography, epidemiology, evolutionary biology, survival analysis, or probabilistic modeling techniques. The second Research Spotlights paper, “Natural Preconditioning and Iterative Methods for Saddle Point Systems” by Jennifer Pestana and Andrew J. Wathen, provides a mini-survey of iterative solution methods for locally quadratic extremal problems with linear constraints. Saddle point problems arise in computational optimization and partial differential equations. The article gives an application-independent overview of the types of iterative numerical methods that can be used to solve such saddle point problems, with a focus on Krylov subspace methods. The methods are illustrated for three examples: Stokes fluid flow, interpolation, and circuits in a resistive network. The application-independent approach proves to be useful. By viewing saddle point problems in a unified way, the authors are able to specify efficient preconditioning methods. This article would be of interest to readers in many different fields such as fluid dynamics, structural mechanics, economics, data reconstruction, and optimization.