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,  unusual weather patterns, spikes in neural networks and intermittent energy  dissipation in turbulent fluid flows.

 Prediction of rogue waves:
 Farazmand and Sapsis, Submitted 2016

 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



 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 quantifying and computing  these structures.

 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