Research Highlights

My main research interest is developing mathematical methods and computational tools for the analysis of complex systems arising in nature and engineering. The problems of interest are high-dimensional, multi-scale, nonlinear and chaotic, so that their efficient treatment often requires new techniques and ideas. Some of the problems that I have contributed to are summarized below.



Rare Extreme Events

 Complex irregular behavior is a characteristic of chaotic systems, which is usually  visualized through the time series of an observable. Many natural and engineering systems exhibit a second level of complexity typified by intermittent bursts in the  time series of certain observables. Examples include rogue waves in the ocean,  extreme weather patterns, spikes in neural networks and intermittent energy  dissipation in turbulent fluid flows.



Related Publications:
 
 Variational method for prediction of extreme events:
 Farazmand and Sapsis, Sic. Adv. 2017
 
 Prediction of rogue waves:
 Farazmand and Sapsis, JCP 2017

 Precursor for rare, extreme events in high-dimensional systems:
 Farazmand and Sapsis, PRE 2016

 Intermittent energy dissipation in turbulent flows:
 Farazmand, JFM 2016


Invariant Solutions in High-dimensional Systems

 Valuable information about the global behavior of chaotic systems can be achieved through their simple invariant solutions (i.e. equilibria, traveling waves and periodic orbits). In high-dimensional systems, the unstable invariant solutions are difficult to compute. We developed an adjoint-based method for efficient computation of these solutions. 



 Invariant solutions of Navier-Stokes and intermittency:
 Farazmand, JFM 2016

 Invariant solutions form the backbone of turbulent flows
 
Budanur, Short, Farazmand, Willis, Cvitanović, JFM 2017


Lagrangian Coherent Structures

 Chaotic dynamical systems often exhibit islands of regular behavior (elliptic islands)  interrupting the surrounding chaotic sea that is dominated by hyperbolicity. A fascinating  example of such coherent behavior is vortex tubes in turbulent fluid flow. While elliptic and hyperbolic structures are well understood in steady and time-periodic systems, their treatment  in fully unsteady flows remains a topic of current research. 

 We have developed methods (based on variational principles) for frame-invariant detection and quantification of these structures.


Related Publications:

 Polar rotation-based detection of vortices:
 Farazmand and Haller, Physica D 2016

 Variational theory of shearless LCS:
 Farazmand, Blazevski and Haller, Physica D 2014

 Computation of LCS:
 Farazmand and Haller, Chaos 2013
 Farazmand and Haller, Chaos 2012