Research

My research currently concentrates around two main themes: Numerical analysis of multiscale methods, in particular highly oscillatory ordinary differential equations (ODEs), and mathematical biology with emphasis on collective dynamics and swarming.

Multiscale computation is one of the fastest growing areas of numerical analysis and computational mathematics with applications in engineering, biology, physics and more. It has long been understood that nature may appear different across different spatial and temporal scales – from the sub-atomic to the entire universe. Traditional computational methods typically fall short at meeting the demands of multiscale systems due to their inherited complexities. Over the last decade, a multi-physics, multi-level approach has inspired new computational strategies that aim at taking advantage of scale separation.

The field of multiscale modeling and simulations typically involve two stages. First the identification and analysis of coarse-grained, reduced or slow models (or coordinates) that characterize the macroscopic behavior of the multiscale system. This is typically obtained by applying the theory of averaging, homogenization, asymptotic expansions or other multiscale methods. Example of applications which I have developed, or am currently working on include highly oscillatory ODEs, stochastic differential equations, models in statistical physics, population dynamics, and more.

The second stage is the development and rigorous numerical analysis of new computational methods that gain efficiency by taking advantage of scale separation, for example, high frequency wave propagation and highly oscillatory dynamical systems. My current research in this area involves generalizations and applications to a wide class of problems motivated by physics. These include almost-periodic fast oscillations over tori, three-scale systems involving new types of slow variables without a bounded derivative, systems in near-resonance, detection of slow manifolds, generalizations to stochastic differential equations and more.

Collective motion and animal swarms. Collective motion is one of the most prevailing phenomena in nature. The emergence of large-scale synchronization among moving individuals can observed on a wide variety of scales - from single cell micro-organisms to mammals. It have given rise to a new field of interdisciplinary scientific research termed collective motion which spans biologists (typically interested on properties of individual species), physicist and mathematicians (interested in the emergence of order in many-particle systems of self-propelled particles), computer scientists (swarm optimization methods) and engineers (robotic swarms).

Within this field, I am mostly interested in understanding the multiscale structure of the swarming phenomenon. In particular, the identification of coarse grained coordinates that characterize the macroscopic swarm dynamics and the coupling between the microscopic scale of the individual animals and that of the crowd.

My main work is on models involving stochastic effect and can be divided into two main approaches. The first is a numerical and theoretical analysis of simplified models showing collective motion. The second involves analysis and coarse-graining of experiments. Collaborating with experimental physicists (Dr. Averaham Be'er from the Ben Gurion University) and biologists (Prof. Amir Ayali from the Tel-Aviv University and Colin Ingham form Microdish BV, Utrecht, The Netherlands), I am working on both a bottom-up approach for analyzing, modeling, simulating and formalizing real-life examples of animal group behavior. The goal is to identify the principle mechanisms leading to collective behavior and formalizing a mathematical theory of large scale synchronization in complex systems of moving animals and particles. By bottom-up approach, I mean using actual experiments as a basis for mathematical modeling and abstraction. This includes developing video-analysis algorithms and statistical methods for analyzing experiments, mathematical modeling and simulations to describe the multiscale nature of the dynamics and developing abstract models which are amenable to analytical study.

The principal species I am working on are bacteria (in collaboration with Avraham Be'er and Colin Ingham) and locusts (in collaboration with Amir Ayali).

Estimation of entropy. CADEE: CopulA Decomposition Entropy Estimate.

In Collaboration with Yoram Louzoun.

Matlab code available here.