On Adaptive Haar Approximations of Random Flows

The adaptive approximations for some characteristic of random functions defined on arbitrary irregular grids are discussed in this paper. The mentioned functions can be examined as flows of random real values associated with an irregular grid. This paper considers the question of choosing an adaptive enlargement of the initial grid. The mentioned enlargement essentially depends on the formulation of the criterion in relation to which adaptability is considered. Several criteria are considered here, among which there are several criteria applicable to the processing of random flows. In particular, the criteria corresponding to the mathematical expectation, dispersion, as well as autocorrelation and cross-correlation of two random flows are considered. It is possible to consider criteria corresponding to various combinations of the mentioned characteristics. The number of knots of the initial (generally speaking, irregular) grid can be arbitrary, and the main grid can be any subset of the initial one. Decomposition algorithms are proposed, taking into account the nature of the changes in the initial flow. The number of arithmetic operations in the proposed algorithms is proportional to the length of the initial flow. Sequential processing of the initial flow is possible in real time.

A State-of-the-Art Review on Empirical Data Collection for External Governed Pedestrians Complex Movement

Complex movement patterns of pedestrian traffic, ranging from unidirectional to multidirectional flows, are frequently observed in major public infrastructure such as transport hubs. These multidirectional movements can result in increased number of conflicts, thereby influencing the mobility and safety of pedestrian facilities. Therefore, empirical data collection on pedestrians’ complex movement has been on the rise in the past two decades. Although there are several reviews of mathematical simulation models for pedestrian traffic in the existing literature, a detailed review examining the challenges and opportunities on empirical studies on the pedestrians complex movements is limited in the literature. The overall aim of this study is to present a systematic review on the empirical data collection for uni- and multidirectional crowd complex movements. We first categorized the complex movements of pedestrian crowd into two general categories, namely, external governed movements and internal driven movements based on the interactions with the infrastructure and among pedestrians, respectively. Further, considering the hierarchy of movement complexity, we decomposed the externally governed movements of pedestrian traffic into several unique movement patterns including straight line, turning, egress and ingress, opposing, weaving, merging, diverging, and random flows. Analysis of the literature showed that empirical data were highly rich in straight line and egress flow while medium rich in turning, merging, weaving, and opposing flows, but poor in ingress, diverging, and random flows. We put emphasis on the need for the future global collaborative efforts on data sharing for the complex crowd movements.

Topological data analysis and diagnostics of compressible magnetohydrodynamic turbulence

The predictions of mean-field electrodynamics can now be probed using direct numerical simulations of random flows and magnetic fields. When modelling astrophysical magnetohydrodynamics, it is important to verify that such simulations are in agreement with observations. One of the main challenges in this area is to identify robust quantitative measures to compare structures found in simulations with those inferred from astrophysical observations. A similar challenge is to compare quantitatively results from different simulations. Topological data analysis offers a range of techniques, including the Betti numbers and persistence diagrams, that can be used to facilitate such a comparison. After describing these tools, we first apply them to synthetic random fields and demonstrate that, when the data are standardized in a straightforward manner, some topological measures are insensitive to either large-scale trends or the resolution of the data. Focusing upon one particular astrophysical example, we apply topological data analysis to H iobservations of the turbulent interstellar medium (ISM) in the Milky Way and to recent magnetohydrodynamic simulations of the random, strongly compressible ISM. We stress that these topological techniques are generic and could be applied to any complex, multi-dimensional random field.