Mathematics of Gravitational Lensing: Multiple Imaging and Magnification

The mathematical theory of gravitational lensing has revealed many generic and global properties. Beginning with multiple imaging, we review Morse-theoretic image counting formulas and lower bound results, and complex-algebraic upper bounds in the case of single and multiple lens planes. We discuss recent advances in the mathematics of stochastic lensing, discussing a general formula for the global expected number of minimum lensed images as well as asymptotic formulas for the probability densities of the microlensing random time delay functions, random lensing maps, and random shear, and an asymptotic expression for the global expected number of micro-minima. Multiple imaging in optical geometry and a spacetime setting are treated. We review global magnification relation results for model-dependent scenarios and cover recent developments on universal local magnification relations for higher order caustics.Comment: 25 pages, 4 figures. Invited review submitted for special issue of General Relativity and Gravitatio

Optimization using surrogate models and partially converged computational fluid dynamics simulations

Efficient methods for global aerodynamic optimization using computational fluid dynamics simulations should aim to reduce both the time taken to evaluate design concepts and the number of evaluations needed for optimization. This paper investigates methods for improving such efficiency through the use of partially converged computational fluid dynamics results. These allow surrogate models to be built in a fraction of the time required for models based on converged results. The proposed optimization methodologies increase the speed of convergence to a global optimum while the computer resources expended in areas of poor designs are reduced. A strategy which combines a global approximation built using partially converged simulations with expected improvement updates of converged simulations is shown to outperform a traditional surrogate-based optimization

Shape optimization of the carotid artery bifurcation

A parametric CAD model of the human carotid artery bifurcation is employed in an initial exploration of the response of shear stress to the variation of the angle of the internal carotid artery and the width of the sinus bulb. Design of experiment and response surface technologies are harnessed for the first time in such an application with the aim of developing a better understanding of the relationship between geometry (anatomy) and sites of arterial disease

Temporal Multivariate Networks

Networks that evolve over time, or dynamic graphs, have been of interest to the areas of information visualization and graph drawing for many years. Typically, its the structure of the dynamic graph that evolves as vertices and edges are added or removed from the graph. In a multivariate scenario, however, attributes play an important role and can also evolve over time. In this chapter, we characterize and survey methods for visualizing temporal multivariate networks. We also explore future applications and directions for this emerging area in the fields of information visualization and graph drawing