@article{anderson_farazmand_2024, title={Fast and scalable computation of shape-morphing nonlinear solutions with application to evolutional neural networks}, volume={498}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2023.112649}, abstractNote={We develop fast and scalable methods for computing reduced-order nonlinear solutions (RONS). RONS was recently proposed as a framework for reduced-order modeling of time-dependent partial differential equations (PDEs), where the modes depend nonlinearly on a set of time-varying parameters. RONS uses a set of ordinary differential equations (ODEs) for the parameters to optimally evolve the shape of the modes to adapt to the PDE's solution. This method has already proven extremely effective in tackling challenging problems such as advection-dominated flows and high-dimensional PDEs. However, as the number of parameters grow, integrating the RONS equation and even its formation become computationally prohibitive. Here, we develop three separate methods to address these computational bottlenecks: symbolic RONS, collocation RONS and regularized RONS. We demonstrate the efficacy of these methods on two examples: Fokker–Planck equation in high dimensions and the Kuramoto–Sivashinsky equation. In both cases, we observe that the proposed methods lead to several orders of magnitude in speedup and accuracy. Our proposed methods extend the applicability of RONS beyond reduced-order modeling by making it possible to use RONS for accurate numerical solution of linear and nonlinear PDEs. Finally, as a special case of RONS, we discuss its application to problems where the PDE's solution is approximated by a neural network, with the time-dependent parameters being the weights and biases of the network. The RONS equations dictate the optimal evolution of the network's parameters without requiring any training.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Anderson, William and Farazmand, Mohammad}, year={2024}, month={Feb} }
 @article{anderson_farazmand_2024, title={Fisher information and shape-morphing modes for solving the Fokker-Planck equation in higher dimensions}, volume={467}, ISSN={["1873-5649"]}, DOI={10.1016/j.amc.2023.128489}, abstractNote={The Fokker–Planck equation describes the evolution of the probability density associated with a stochastic differential equation. As the dimension of the system grows, solving this partial differential equation (PDE) using conventional numerical methods becomes computationally prohibitive. Here, we introduce a fast, scalable, and interpretable method for solving the Fokker–Planck equation which is applicable in higher dimensions. This method approximates the solution as a linear combination of shape-morphing Gaussians with time-dependent means and covariances. These parameters evolve according to the method of reduced-order nonlinear solutions (RONS) which ensures that the approximate solution stays close to the true solution of the PDE for all times. As such, the proposed method approximates the transient dynamics as well as the equilibrium density, when the latter exists. Our approximate solutions can be viewed as an evolution on a finite-dimensional statistical manifold embedded in the space of probability densities. We show that the metric tensor in RONS coincides with the Fisher information matrix on this manifold. We also discuss the interpretation of our method as a shallow neural network with Gaussian activation functions and time-varying parameters. In contrast to existing deep learning methods, our method is interpretable, requires no training, and automatically ensures that the approximate solution satisfies all properties of a probability density.}, journal={APPLIED MATHEMATICS AND COMPUTATION}, author={Anderson, William and Farazmand, Mohammad}, year={2024}, month={Apr} }
 @article{mamis_farazmand_2023, title={Stochastic compartmental models of the COVID-19 pandemic must have temporally correlated uncertainties}, volume={479}, ISSN={["1471-2946"]}, DOI={10.1098/rspa.2022.0568}, abstractNote={Compartmental models are an important quantitative tool in epidemiology, enabling us to forecast the course of a communicable disease. However, the model parameters, such as the infectivity rate of the disease, are riddled with uncertainties, which has motivated the development and use of stochastic compartmental models. Here, we first show that a common stochastic model, which treats the uncertainties as white noise, is fundamentally flawed since it erroneously implies that greater parameter uncertainties will lead to the eradication of the disease. Then, we present a principled modelling of the uncertainties based on reasonable assumptions on the contacts of each individual. Using the central limit theorem and Doob’s theorem on Gaussian Markov processes, we prove that the correlated Ornstein–Uhlenbeck (OU) process is the appropriate tool for modelling uncertainties in the infectivity rate. We demonstrate our results using a compartmental model of the COVID-19 pandemic and the available US data from the Johns Hopkins University COVID-19 database. In particular, we show that the white noise stochastic model systematically underestimates the severity of the Omicron variant of COVID-19, whereas the OU model correctly forecasts the course of this variant. Moreover, using an SIS model of sexually transmitted disease, we derive an exact closed-form solution for the final distribution of infected individuals. This analytical result shows that the white noise model underestimates the severity of the pandemic because of unrealistic noise-induced transitions. Our results strongly support the need for temporal correlations in modelling of uncertainties in compartmental models of infectious disease.}, number={2269}, journal={PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES}, author={Mamis, Konstantinos and Farazmand, Mohammad}, year={2023}, month={Jan} }
 @article{farazmand_saibaba_2023, title={Tensor-based flow reconstruction from optimally located sensor measurements}, volume={962}, ISSN={["1469-7645"]}, DOI={10.1017/jfm.2023.269}, abstractNote={Reconstructing high-resolution flow fields from sparse measurements is a major challenge in fluid dynamics. Existing methods often vectorize the flow by stacking different spatial directions on top of each other, hence confounding the information encoded in different dimensions. Here, we introduce a tensor-based sensor placement and flow reconstruction method which retains and exploits the inherent multidimensionality of the flow. We derive estimates for the flow reconstruction error, storage requirements and computational cost of our method. We show, with examples, that our tensor-based method is significantly more accurate than similar vectorized methods. Furthermore, the variance of the error is smaller when using our tensor-based method. While the computational cost of our method is comparable to similar vectorized methods, it reduces the storage cost by several orders of magnitude. The reduced storage cost becomes even more pronounced as the dimension of the flow increases. We demonstrate the efficacy of our method on three examples: a chaotic Kolmogorov flow, in situ and satellite measurements of the global sea surface temperature and three-dimensional unsteady simulated flow around a marine research vessel.}, journal={JOURNAL OF FLUID MECHANICS}, author={Farazmand, Mohammad and Saibaba, Arvind K.}, year={2023}, month={May} }
 @article{anderson_farazmand_2022, title={EVOLUTION OF NONLINEAR REDUCED-ORDER SOLUTIONS FOR PDEs WITH CONSERVED QUANTITIES}, volume={44}, ISSN={["1095-7197"]}, url={https://doi.org/10.1137/21M1415972}, DOI={10.1137/21M1415972}, abstractNote={Reduced-order models of time-dependent partial differential equations (PDEs) where the solution is assumed as a linear combination of prescribed modes are rooted in a well-developed theory. However, more general models where the reduced solutions depend nonlinearly on time varying parameters have thus far been derived in an ad hoc manner. Here, we introduce Reduced-order Nonlinear Solutions (RONS): a unified framework for deriving reduced-order models that depend nonlinearly on a set of time-dependent parameters. The set of all possible reduced-order solutions are viewed as a manifold immersed in the function space of the PDE. The parameters are evolved such that the instantaneous discrepancy between reduced dynamics and the full PDE dynamics is minimized. This results in a set of explicit ordinary differential equations on the tangent bundle of the manifold. In the special case of linear parameter dependence, our reduced equations coincide with the standard Galerkin projection. Furthermore, any number of conserved quantities of the PDE can readily be enforced in our framework. Since RONS does not assume an underlying variational formulation for the PDE, it is applicable to a broad class of problems. We demonstrate the efficacy of RONS on three examples: an advection-diffusion equation, the nonlinear Schrödinger equation and Euler’s equation for ideal fluids.}, number={1}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Anderson, William and Farazmand, Mohammad}, year={2022}, pages={A176–A197} }
 @article{asch_j. brady_gallardo_hood_chu_farazmand_2022, title={Model-assisted deep learning of rare extreme events from partial observations}, volume={32}, ISSN={["1089-7682"]}, url={https://doi.org/10.1063/5.0077646}, DOI={10.1063/5.0077646}, abstractNote={To predict rare extreme events using deep neural networks, one encounters the so-called small data problem because even long-term observations often contain few extreme events. Here, we investigate a model-assisted framework where the training data are obtained from numerical simulations, as opposed to observations, with adequate samples from extreme events. However, to ensure the trained networks are applicable in practice, the training is not performed on the full simulation data; instead, we only use a small subset of observable quantities, which can be measured in practice. We investigate the feasibility of this model-assisted framework on three different dynamical systems (Rössler attractor, FitzHugh–Nagumo model, and a turbulent fluid flow) and three different deep neural network architectures (feedforward, long short-term memory, and reservoir computing). In each case, we study the prediction accuracy, robustness to noise, reproducibility under repeated training, and sensitivity to the type of input data. In particular, we find long short-term memory networks to be most robust to noise and to yield relatively accurate predictions, while requiring minimal fine-tuning of the hyperparameters.}, number={4}, journal={CHAOS}, publisher={AIP Publishing}, author={Asch, Anna and J. Brady, Ethan and Gallardo, Hugo and Hood, John and Chu, Bryan and Farazmand, Mohammad}, year={2022}, month={Apr} }
 @article{mendez_farazmand_2022, title={Quantifying rare events in spotting: How far do wildfires spread?}, volume={132}, ISSN={["1873-7226"]}, url={https://doi.org/10.1016/j.firesaf.2022.103630}, DOI={10.1016/j.firesaf.2022.103630}, abstractNote={Spotting refers to the transport of burning pieces of firebrand by wind which, at the time of landing, may ignite new fires beyond the direct ignition zone of the main fire. Spot fires that occur far from the original burn unit are rare but have consequential ramifications since their prediction and control remains challenging. To facilitate their prediction, we examine three methods for quantifying the landing distribution of firebrands: crude Monte Carlo simulations, importance sampling, and large deviation theory (LDT). In particular, we propose an LDT method that accurately and parsimoniously quantifies the low probability events at the tail of the landing distribution. In contrast, Monte Carlo and importance sampling methods are most efficient in quantifying the high probability landing distances near the mode of the distribution. However, they become computationally intractable for quantifying the tail of the distribution due to the large sample size required. We also show that the most probable landing distance grows linearly with the mean characteristic velocity of the wind field. Furthermore, defining the relative landed mass as the proportion of mass landed at a given distance from the main fire, we derive an explicit formula which allows computing this quantity as a function of the landing distribution at a negligible computational cost. We numerically demonstrate our findings on two prescribed wind fields.}, journal={FIRE SAFETY JOURNAL}, author={Mendez, Alexander and Farazmand, Mohammad}, year={2022}, month={Sep} }
 @article{anderson_farazmand_2022, title={Shape-morphing reduced-order models for nonlinear Schrodinger equations}, volume={4}, ISSN={["1573-269X"]}, url={https://doi.org/10.1007/s11071-022-07448-w}, DOI={10.1007/s11071-022-07448-w}, abstractNote={We consider reduced-order modeling of nonlinear dispersive waves described by a class of nonlinear Schrödinger (NLS) equations. We compare two nonlinear reduced-order modeling methods: (i) The reduced Lagrangian approach which relies on the variational formulation of NLS and (ii) the recently developed method of reduced-order nonlinear solutions (RONS). First, we prove the surprising result that, although the two methods are seemingly quite different, they can be obtained from the real and imaginary parts of a single complex-valued master equation. Furthermore, for the NLS equation in a stationary frame, we show that the reduced Lagrangian method fails to predict the correct group velocity of the waves, whereas RONS predicts the correct group velocity. Finally, for the modified NLS equation, where the reduced Lagrangian approach is inapplicable, the RONS reduced-order model accurately approximates the true solutions.}, number={4}, journal={NONLINEAR DYNAMICS}, publisher={Springer Science and Business Media LLC}, author={Anderson, William and Farazmand, Mohammad}, year={2022}, month={Apr} }
 @article{chu_farazmand_2021, title={Data-driven prediction of multistable systems from sparse measurements}, volume={31}, ISSN={["1089-7682"]}, url={https://doi.org/10.1063/5.0046203}, DOI={10.1063/5.0046203}, abstractNote={We develop a data-driven method, based on semi-supervised classification, to predict the asymptotic state of multistable systems when only sparse spatial measurements of the system are feasible. Our method predicts the asymptotic behavior of an observed state by quantifying its proximity to the states in a precomputed library of data. To quantify this proximity, we introduce a sparsity-promoting metric-learning (SPML) optimization, which learns a metric directly from the precomputed data. The optimization problem is designed so that the resulting optimal metric satisfies two important properties: (i) it is compatible with the precomputed library and (ii) it is computable from sparse measurements. We prove that the proposed SPML optimization is convex, its minimizer is non-degenerate, and it is equivariant with respect to the scaling of the constraints. We demonstrate the application of this method on two multistable systems: a reaction–diffusion equation, arising in pattern formation, which has four asymptotically stable steady states, and a FitzHugh–Nagumo model with two asymptotically stable steady states. Classifications of the multistable reaction–diffusion equation based on SPML predict the asymptotic behavior of initial conditions based on two-point measurements with 95% accuracy when a moderate number of labeled data are used. For the FitzHugh–Nagumo, SPML predicts the asymptotic behavior of initial conditions from one-point measurements with 90% accuracy. The learned optimal metric also determines where the measurements need to be made to ensure accurate predictions.}, number={6}, journal={CHAOS}, author={Chu, Bryan and Farazmand, Mohammad}, year={2021}, month={Jun} }
 @article{anderson_farazmand_2021, title={Evolution of nonlinear reduced-order solutions for PDEs with conserved quantities}, url={https://arxiv.org/abs/2104.13515}, journal={SIAM J. on Scientific Computing, In press}, author={Anderson, W and Farazmand, M}, year={2021} }
 @article{mendez_farazmand_2021, title={Investigating climate tipping points under various emission reduction and carbon capture scenarios with a stochastic climate model}, volume={477}, ISSN={["1471-2946"]}, url={https://doi.org/10.1098/rspa.2021.0697}, DOI={10.1098/rspa.2021.0697}, abstractNote={
            We study the mitigation of climate tipping point transitions using an energy balance model. The evolution of the global mean surface temperature is coupled with the
            
              
                
                  CO
                  2
                
              
            
            concentration through the green-house effect. We model the
            
              
                
                  CO
                  2
                
              
            
            concentration with a stochastic delay differential equation (SDDE), accounting for various carbon emission and capture scenarios. The resulting coupled system of SDDEs exhibits a tipping point phenomena: if
            
              
                
                  CO
                  2
                
              
            
            concentration exceeds a critical threshold (around
            
              
                478
                 
                ppm
              
            
            ), the temperature experiences an abrupt increase of about six degrees Celsius. We show that the
            
              
                
                  CO
                  2
                
              
            
            concentration exhibits a transient growth which may cause a climate tipping point, even if the concentration decays asymptotically. We derive a rigorous upper bound for the
            
              
                
                  CO
                  2
                
              
            
            evolution which quantifies its transient and asymptotic growths, and provides sufficient conditions for evading the climate tipping point. Combining this upper bound with Monte Carlo simulations of the stochastic climate model, we investigate the emission reduction and carbon capture scenarios that would avert the tipping point.
          }, number={2256}, journal={PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES}, author={Mendez, Alexander and Farazmand, Mohammad}, year={2021}, month={Dec} }
 @article{mamis_farazmand_2021, title={Mitigation of rare events in multistable systems driven by correlated noise}, volume={104}, ISSN={["2470-0053"]}, url={https://doi.org/10.1103/PhysRevE.104.034201}, DOI={10.1103/PhysRevE.104.034201}, abstractNote={We consider rare transitions induced by colored noise excitation in multistable systems. We show that undesirable transitions can be mitigated by a simple time-delay feedback control if the control parameters are judiciously chosen. We devise a parsimonious method for selecting the optimal control parameters, without requiring any Monte Carlo simulations of the system. This method relies on a new nonlinear Fokker-Planck equation whose stationary response distribution is approximated by a rapidly convergent iterative algorithm. In addition, our framework allows us to accurately predict, and subsequently suppress, the modal drift and tail inflation in the controlled stationary distribution. We demonstrate the efficacy of our method on two examples, including an optical laser model perturbed by multiplicative colored noise.}, number={3}, journal={PHYSICAL REVIEW E}, author={Mamis, Konstantinos and Farazmand, Mohammad}, year={2021}, month={Sep} }
 @article{farazmand_2020, title={MULTISCALE ANALYSIS OF ACCELERATED GRADIENT METHODS}, volume={30}, ISSN={["1095-7189"]}, url={https://doi.org/10.1137/18M1203997}, DOI={10.1137/18M1203997}, abstractNote={Accelerated gradient descent iterations are widely used in optimization. It is known that, in the continuous-time limit, these iterations converge to a second-order differential equation which we refer to as the accelerated gradient flow. Using geometric singular perturbation theory, we show that, under certain conditions, the accelerated gradient flow possesses an attracting invariant slow manifold to which the trajectories of the flow converge asymptotically. We obtain a general explicit expression in the form of functional series expansions that approximates the slow manifold to any arbitrary order of accuracy. To the leading order, the accelerated gradient flow reduced to this slow manifold coincides with the usual gradient descent. We illustrate the implications of our results on three examples.}, number={3}, journal={SIAM JOURNAL ON OPTIMIZATION}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Farazmand, Mohammad}, year={2020}, pages={2337–2354} }
 @article{farazmand_2020, title={Mitigation of tipping point transitions by time-delay feedback control}, volume={30}, ISSN={["1089-7682"]}, url={https://doi.org/10.1063/1.5137825}, DOI={10.1063/1.5137825}, abstractNote={In stochastic multistable systems driven by the gradient of a potential, transitions between equilibria are possible because of noise. We study the ability of linear delay feedback control to mitigate these transitions, ensuring that the system stays near a desirable equilibrium. For small delays, we show that the control term has two effects: (i) a stabilizing effect by deepening the potential well around the desirable equilibrium and (ii) a destabilizing effect by intensifying the noise by a factor of (1−τα)−1/2, where τ and α denote the delay and the control gain, respectively. As a result, successful mitigation depends on the competition between these two factors. We also derive analytical results that elucidate the choice of the appropriate control gain and delay that ensure successful mitigations. These results eliminate the need for any Monte Carlo simulations of the stochastic differential equations and, therefore, significantly reduce the computational cost of determining the suitable control parameters. We demonstrate the application of our results on two examples.}, number={1}, journal={CHAOS}, publisher={AIP Publishing}, author={Farazmand, Mohammad}, year={2020}, month={Jan} }
 @article{katsanoulis_farazmand_serra_haller_2020, title={Vortex boundaries as barriers to diffusive vorticity transport in two-dimensional flows}, volume={5}, ISSN={["2469-990X"]}, url={https://doi.org/10.1103/PhysRevFluids.5.024701}, DOI={10.1103/PhysRevFluids.5.024701}, abstractNote={We put forward the idea of defining vortex boundaries in planar flows as closed material barriers to the diffusive transport of vorticity. Such diffusive vortex boundaries minimize the leakage of vorticity from the fluid mass they enclose when compared to other nearby material curves. Building on recent results on passive diffusion barriers, we develop an algorithm for the automated identification of such structures from general, two-dimensional unsteady flow data. As examples, we identify vortex boundaries as vorticity diffusion barriers in two flows: an explicitly known laminar flow and a numerically generated turbulent Navier--Stokes flow.}, number={2}, journal={PHYSICAL REVIEW FLUIDS}, author={Katsanoulis, Stergios and Farazmand, Mohammad and Serra, Mattia and Haller, George}, year={2020}, month={Feb} }
 @article{are extreme dissipation events predictable in turbulent fluid flows?_2019, url={http://dx.doi.org/10.1103/physrevfluids.4.044606}, DOI={10.1103/physrevfluids.4.044606}, abstractNote={We derive precursors of extreme dissipation events in a turbulent channel flow. Using a recently developed method that combines dynamics and statistics for the underlying attractor, we extract a characteristic state that precedes laminarization events that subsequently lead to extreme dissipation episodes. Our approach utilizes coarse statistical information for the turbulent attractor, in the form of second order statistics, to identify high-likelihood regions in the state space. We then search within this high probability manifold for the state that leads to the most finite-time growth of the flow kinetic energy. This state has both high probability of occurrence and leads to extreme values of dissipation. We use the alignment between a given turbulent state and this critical state as a precursor for extreme events and demonstrate its favorable properties for prediction of extreme dissipation events. Finally, we analyze the physical relevance of the derived precursor and show its robust character for different Reynolds numbers. Overall, we find that our choice of precursor works well at the Reynolds number it is computed at and at higher Reynolds number flows with similar extreme events.}, journal={Physical Review Fluids}, year={2019}, month={Apr} }
 @article{farazmand_sapsis_2019, title={Closed-loop adaptive control of extreme events in a turbulent flow}, volume={100}, ISSN={["2470-0053"]}, url={http://dx.doi.org/10.1103/physreve.100.033110}, DOI={10.1103/PhysRevE.100.033110}, abstractNote={Extreme events that arise spontaneously in chaotic dynamical systems often have an adverse impact on the system or the surrounding environment. As such, their mitigation is highly desirable. Here, we introduce a control strategy for mitigating extreme events in a turbulent shear flow. The controller combines a probabilistic prediction of the extreme events with a deterministic actuator. The predictions are used to actuate the controller only when an extreme event is imminent. When actuated, the controller only acts on the degrees of freedom that are involved in the formation of the extreme events, exerting minimal interference with the flow dynamics. As a result, the attractors of the controlled and uncontrolled systems share the same chaotic core (containing the nonextreme events) and only differ in the tail of their distributions. We propose that such adaptive low-dimensional controllers should be used to mitigate extreme events in general chaotic dynamical systems, beyond the shear flow considered here.}, number={3}, journal={PHYSICAL REVIEW E}, author={Farazmand, Mohammad and Sapsis, Themistoklis P.}, year={2019}, month={Sep} }
 @article{closure to “discussion of ‘extreme events: mechanisms and prediction’” (grigoriu, m. d., and uy, w. i. t., asme appl. mech. rev., 2019, 71(5), p. 055501)_2019, url={http://dx.doi.org/10.1115/1.4043632}, DOI={10.1115/1.4043632}, journal={Applied Mechanics Reviews}, year={2019}, month={Sep} }
 @article{farazmand_sapsis_2019, title={Extreme Events: Mechanisms and Prediction}, volume={71}, DOI={10.1115/1.4042065}, abstractNote={Abstract}, number={5}, journal={Applied Mechanics Reviews}, author={Farazmand, M. and Sapsis, T. P.}, year={2019} }
 @article{farazmand_2019, title={Mitigation of tipping point transitions by time-delay feedback control}, journal={arXiv preprint arXiv:1911.05292}, author={Farazmand, M.}, year={2019} }
 @article{farazmand_sapsis_2019, title={Surface Waves Enhance Particle Dispersion}, url={https://www.mdpi.com/2311-5521/4/1/55}, DOI={10.3390/fluids4010055}, abstractNote={We study the horizontal dispersion of passive tracer particles on the free surface of gravity waves in deep water. For random linear waves with the JONSWAP spectrum, the Lagrangian particle trajectories are computed using an exact nonlinear model known as the John–Sclavounos equation. We show that the single-particle dispersion exhibits an unusual super-diffusive behavior. In particular, for large times t, the variance of the tracer     ⟨ | X  ( t )   | 2  ⟩     increases as a quadratic function of time, i.e.,       ⟨ | X  ( t )  |  2   ⟩ ∼   t 2     . This dispersion is markedly faster than Taylor’s single-particle dispersion theory which predicts that the variance of passive tracers grows linearly with time for large t. Our results imply that the wave motion significantly enhances the dispersion of fluid particles. We show that this super-diffusive behavior is a result of the long-term correlation of the Lagrangian velocities of fluid parcels on the free surface.}, journal={Fluids}, author={Farazmand, Mohammad and Sapsis, Themistoklis}, year={2019}, month={Mar} }
 @article{farazmand_sapsis_2018, title={Physics-based probing and prediction of extreme events}, volume={51}, url={https://www.researchgate.net/profile/Mohammad_Farazmand/publication/323245041_Physics-based_probing_and_prediction_of_extreme_events/links/5a887e4daca272017e5f5bef/Physics-based-probing-and-prediction-of-extreme-events.pdf}, number={1}, journal={SIAM News}, author={Farazmand, M. and Sapsis, T}, year={2018}, month={Jan}, pages={1, 3} }
 @article{a critical comparison of lagrangian methods for coherent structure detection_2017, url={http://dx.doi.org/10.1063/1.4982720}, DOI={10.1063/1.4982720}, abstractNote={We review and test twelve different approaches to the detection of finite-time coherent material structures in two-dimensional, temporally aperiodic flows. We consider both mathematical methods and diagnostic scalar fields, comparing their performance on three benchmark examples: the quasiperiodically forced Bickley jet, a two-dimensional turbulence simulation, and an observational wind velocity field from Jupiter's atmosphere. A close inspection of the results reveals that the various methods often produce very different predictions for coherent structures, once they are evaluated beyond heuristic visual assessment. As we find by passive advection of the coherent set candidates, false positives and negatives can be produced even by some of the mathematically justified methods due to the ineffectiveness of their underlying coherence principles in certain flow configurations. We summarize the inferred strengths and weaknesses of each method, and make general recommendations for minimal self-consistency requirements that any Lagrangian coherence detection technique should satisfy.}, journal={Chaos: An Interdisciplinary Journal of Nonlinear Science}, year={2017}, month={May} }
 @article{farazmand_sapsis_2017, title={A variational approach to probing extreme events in turbulent dynamical systems}, volume={3}, DOI={10.1126/sciadv.1701533}, abstractNote={A variational framework for the analysis and data-driven prediction of extreme events is developed.}, number={9}, journal={Science Advances}, author={Farazmand, M. and Sapsis, T. P.}, year={2017}, pages={e1701533} }
 @article{optimal initial condition of passive tracers for their maximal mixing in finite time_2017, url={http://dx.doi.org/10.1103/physrevfluids.2.054601}, DOI={10.1103/physrevfluids.2.054601}, abstractNote={The efficiency of a fluid mixing device is often limited by fundamental laws and/or design constraints, such that a perfectly homogeneous mixture cannot be obtained in finite time. Here, we address the natural corollary question: Given the best available mixer, what is the optimal initial tracer pattern that leads to the most homogeneous mixture after a prescribed finite time? For ideal passive tracers, we show that this optimal initial condition coincides with the right singular vector (corresponding to the smallest singular value) of a suitably truncated Perron-Frobenius (PF) operator. The truncation of the PF operator is made under the assumption that there is a small length-scale threshold $\ell_\nu$ under which the tracer blobs are considered, for all practical purposes, completely mixed. We demonstrate our results on two examples: a prototypical model known as the sine flow and a direct numerical simulation of two-dimensional turbulence. Evaluating the optimal initial condition through this framework only requires the position of a dense grid of fluid particles at the final instance and their preimages at the initial instance of the prescribed time interval. As such, our framework can be readily applied to flows where such data is available through numerical simulations or experimental measurements.}, journal={Physical Review Fluids}, year={2017}, month={May} }
 @article{reduced-order description of transient instabilities and computation of finite-time lyapunov exponents_2017, url={http://dx.doi.org/10.1063/1.4984627}, DOI={10.1063/1.4984627}, abstractNote={High-dimensional chaotic dynamical systems can exhibit strongly transient features. These are often associated with instabilities that have a finite-time duration. Because of the finite-time character of these transient events, their detection through infinite-time methods, e.g., long term averages, Lyapunov exponents or information about the statistical steady-state, is not possible. Here, we utilize a recently developed framework, the Optimally Time-Dependent (OTD) modes, to extract a time-dependent subspace that spans the modes associated with transient features associated with finite-time instabilities. As the main result, we prove that the OTD modes, under appropriate conditions, converge exponentially fast to the eigendirections of the Cauchy–Green tensor associated with the most intense finite-time instabilities. Based on this observation, we develop a reduced-order method for the computation of finite-time Lyapunov exponents (FTLE) and vectors. In high-dimensional systems, the computational cost of the reduced-order method is orders of magnitude lower than the full FTLE computation. We demonstrate the validity of the theoretical findings on two numerical examples.}, journal={Chaos: An Interdisciplinary Journal of Nonlinear Science}, year={2017}, month={Jun} }
 @article{reduced-order prediction of rogue waves in two-dimensional deep-water waves_2017, url={http://dx.doi.org/10.1016/j.jcp.2017.03.054}, DOI={10.1016/j.jcp.2017.03.054}, abstractNote={We consider the problem of large wave prediction in two-dimensional water waves. Such waves form due to the synergistic effect of dispersive mixing of smaller wave groups and the action of localized nonlinear wave interactions that leads to focusing. Instead of a direct simulation approach, we rely on the decomposition of the wave field into a discrete set of localized wave groups with optimal length scales and amplitudes. Due to the short-term character of the prediction, these wave groups do not interact and therefore their dynamics can be characterized individually. Using direct numerical simulations of the governing envelope equations we precompute the expected maximum elevation for each of those wave groups. The combination of the wave field decomposition algorithm, which provides information about the statistics of the system, and the precomputed map for the expected wave group elevation, which encodes dynamical information, allows (i) for understanding of how the probability of occurrence of rogue waves changes as the spectrum parameters vary, (ii) the computation of a critical length scale characterizing wave groups with high probability of evolving to rogue waves, and (iii) the formulation of a robust and parsimonious reduced-order prediction scheme for large waves. We assess the validity of this scheme in several cases of ocean wave spectra.}, journal={Journal of Computational Physics}, year={2017}, month={Jul} }
 @article{relative periodic orbits form the backbone of turbulent pipe flow_2017, url={http://dx.doi.org/10.1017/jfm.2017.699}, DOI={10.1017/jfm.2017.699}, abstractNote={The chaotic dynamics of low-dimensional systems, such as Lorenz or Rössler flows, is guided by the infinity of periodic orbits embedded in their strange attractors. Whether this is also the case for the infinite-dimensional dynamics of Navier–Stokes equations has long been speculated, and is a topic of ongoing study. Periodic and relative periodic solutions have been shown to be involved in transitions to turbulence. Their relevance to turbulent dynamics – specifically, whether periodic orbits play the same role in high-dimensional nonlinear systems like the Navier–Stokes equations as they do in lower-dimensional systems – is the focus of the present investigation. We perform here a detailed study of pipe flow relative periodic orbits with energies and mean dissipations close to turbulent values. We outline several approaches to reduction of the translational symmetry of the system. We study pipe flow in a minimal computational cell at $Re=2500$, and report a library of invariant solutions found with the aid of the method of slices. Detailed study of the unstable manifolds of a sample of these solutions is consistent with the picture that relative periodic orbits are embedded in the chaotic saddle and that they guide the turbulent dynamics.}, journal={Journal of Fluid Mechanics}, year={2017}, month={Dec} }
 @article{an adjoint-based approach for finding invariant solutions of navier–stokes equations_2016, url={http://dx.doi.org/10.1017/jfm.2016.203}, DOI={10.1017/jfm.2016.203}, abstractNote={We consider the incompressible Navier–Stokes equations with periodic boundary conditions and time-independent forcing. For this type of flow, we derive adjoint equations whose trajectories converge asymptotically to the equilibrium and travelling-wave solutions of the Navier–Stokes equations. Using the adjoint equations, arbitrary initial conditions evolve to the vicinity of a (relative) equilibrium at which point a few Newton-type iterations yield the desired (relative) equilibrium solution. We apply this adjoint-based method to a chaotic two-dimensional Kolmogorov flow. A convergence rate of $100\,\%$ is observed, leading to the discovery of $21$ new steady-state and travelling-wave solutions at Reynolds number $Re=40$. Some of the new invariant solutions have spatially localized structures that were previously believed to exist only on domains with large aspect ratios. We show that one of the newly found steady-state solutions underpins the temporal intermittencies, i.e. high energy dissipation episodes of the flow. More precisely, it is shown that each intermittent episode of a generic turbulent trajectory corresponds to its close passage to this equilibrium solution.}, journal={Journal of Fluid Mechanics}, year={2016}, month={May} }
 @article{haller_hadjighasem_farazmand_huhn_2016, title={Defining coherent vortices objectively from the vorticity}, volume={795}, ISSN={0022-1120 1469-7645}, url={http://dx.doi.org/10.1017/jfm.2016.151}, DOI={10.1017/jfm.2016.151}, abstractNote={Rotationally coherent Lagrangian vortices are formed by tubes of deforming fluid elements that complete equal bulk material rotation relative to the mean rotation of the deforming fluid volume. We show that the initial positions of such tubes coincide with tubular level surfaces of the Lagrangian-averaged vorticity deviation (LAVD), the trajectory integral of the normed difference of the vorticity from its spatial mean. The LAVD-based vortices are objective, i.e. remain unchanged under time-dependent rotations and translations of the coordinate frame. In the limit of vanishing Rossby numbers in geostrophic flows, cyclonic LAVD vortex centres are precisely the observed attractors for light particles. A similar result holds for heavy particles in anticyclonic LAVD vortices. We also establish a relationship between rotationally coherent Lagrangian vortices and their instantaneous Eulerian counterparts. The latter are formed by tubular surfaces of equal material rotation rate, objectively measured by the instantaneous vorticity deviation (IVD). We illustrate the use of the LAVD and the IVD to detect rotationally coherent Lagrangian and Eulerian vortices objectively in several two- and three-dimensional flows.}, journal={Journal of Fluid Mechanics}, publisher={Cambridge University Press (CUP)}, author={Haller, G. and Hadjighasem, A. and Farazmand, M. and Huhn, F.}, year={2016}, month={Apr}, pages={136–173} }
 @article{dynamical indicators for the prediction of bursting phenomena in high-dimensional systems_2016, url={http://dx.doi.org/10.1103/physreve.94.032212}, DOI={10.1103/physreve.94.032212}, abstractNote={Drawing upon the bursting mechanism in slow-fast systems, we propose indicators for the prediction of such rare extreme events which do not require a priori known slow and fast coordinates. The indicators are associated with functionals defined in terms of optimally time-dependent (OTD) modes. One such functional has the form of the largest eigenvalue of the symmetric part of the linearized dynamics reduced to these modes. In contrast to other choices of subspaces, the proposed modes are flow invariant and therefore a projection onto them is dynamically meaningful. We illustrate the application of these indicators on three examples: a prototype low-dimensional model, a body-forced turbulent fluid flow, and a unidirectional model of nonlinear water waves. We use Bayesian statistics to quantify the predictive power of the proposed indicators.}, journal={Physical Review E}, year={2016}, month={Sep} }
 @article{fedele_chandre_farazmand_2016, title={Kinematics of fluid particles on the sea surface: Hamiltonian theory}, volume={801}, ISSN={0022-1120 1469-7645}, url={http://dx.doi.org/10.1017/jfm.2016.453}, DOI={10.1017/jfm.2016.453}, abstractNote={We derive the John–Sclavounos equations, describing the motion of a fluid particle on the sea surface, from first principles using Lagrangian and Hamiltonian formalisms applied to the motion of a frictionless particle constrained on an unsteady surface. This framework leads to a number of new insights into the particle kinematics. The main result is that vorticity generated on a stress-free surface vanishes at a wave crest when the horizontal particle velocity equals the crest propagation speed, which is the kinematic criterion for wave breaking. If this holds for the largest crest, then the symplectic two-form associated with the Hamiltonian dynamics reduces instantaneously to that associated with the motion of a particle in free flight, as if the surface did not exist. Further, exploiting the conservation of the Hamiltonian function for steady surfaces and travelling waves, we show that particle velocities remain bounded at all times, ruling out the possibility of the finite-time blowup of solutions.}, journal={Journal of Fluid Mechanics}, publisher={Cambridge University Press (CUP)}, author={Fedele, F. and Chandre, C. and Farazmand, M.}, year={2016}, month={Jul}, pages={260–288} }
 @article{farazmand_haller_2016, title={Polar rotation angle identifies elliptic islands in unsteady dynamical systems}, volume={315}, ISSN={0167-2789}, url={http://dx.doi.org/10.1016/j.physd.2015.09.007}, DOI={10.1016/j.physd.2015.09.007}, abstractNote={We propose rotation inferred from the polar decomposition of the flow gradient as a diagnostic for elliptic (or vortex-type) invariant regions in non-autonomous dynamical systems. We consider here two- and three-dimensional systems, in which polar rotation can be characterized by a single angle. For this polar rotation angle (PRA), we derive explicit formulas using the singular values and vectors of the flow gradient. We find that closed level sets of the PRA reveal elliptic islands in great detail, and singular level sets of the PRA uncover centers of such islands. Both features turn out to be objective (frame-invariant) for two-dimensional systems. We illustrate the diagnostic power of PRA for elliptic structures on several examples.}, journal={Physica D: Nonlinear Phenomena}, publisher={Elsevier BV}, author={Farazmand, Mohammad and Haller, George}, year={2016}, month={Feb}, pages={1–12} }
 @article{langlois_farazmand_haller_2015, title={Asymptotic Dynamics of Inertial Particles with Memory}, volume={25}, ISSN={0938-8974 1432-1467}, url={http://dx.doi.org/10.1007/s00332-015-9250-0}, DOI={10.1007/s00332-015-9250-0}, abstractNote={Recent experimental and numerical observations have shown the significance of the Basset–Boussinesq memory term on the dynamics of small spherical rigid particles (or inertial particles) suspended in an ambient fluid flow. These observations suggest an algebraic decay to an asymptotic state, as opposed to the exponential convergence in the absence of the memory term. Here, we prove that the observed algebraic decay is a universal property of the Maxey–Riley equation. Specifically, the particle velocity decays algebraically in time to a limit that is $$\mathcal {O}(\epsilon )$$ -close to the fluid velocity, where $$0<\epsilon \ll 1$$ is proportional to the square of the ratio of the particle radius to the fluid characteristic length scale. These results follow from a sharp analytic upper bound that we derive for the particle velocity. For completeness, we also present a first proof of the global existence and uniqueness of mild solutions to the Maxey–Riley equation, a nonlinear system of fractional differential equations.}, number={6}, journal={Journal of Nonlinear Science}, publisher={Springer Science and Business Media LLC}, author={Langlois, Gabriel Provencher and Farazmand, Mohammad and Haller, George}, year={2015}, month={May}, pages={1225–1255} }
 @article{karrasch_farazmand_haller_2015, title={Attraction-based computation of hyperbolic Lagrangian coherent structures}, volume={2}, ISSN={2158-2491}, url={http://dx.doi.org/10.3934/jcd.2015.2.83}, DOI={10.3934/jcd.2015.2.83}, abstractNote={Recent advances enable the simultaneous computation of both attracting 
and repelling families of Lagrangian Coherent Structures (LCS) at 
the same initial or final time of interest. Obtaining LCS positions 
at intermediate times, however, has been problematic, because either 
the repelling or the attracting family is unstable with respect to 
numerical advection in a given time direction. Here we develop a new 
approach to compute arbitrary positions of hyperbolic LCS in a numerically 
robust fashion. Our approach only involves the advection of attracting 
material surfaces, thereby providing accurate LCS tracking at low 
computational cost. We illustrate the advantages of this approach 
on a simple model and on a turbulent velocity data set.}, number={1}, journal={Journal of Computational Dynamics}, publisher={American Institute of Mathematical Sciences (AIMS)}, author={Karrasch, Daniel and Farazmand, Mohammad and Haller, George}, year={2015}, month={Aug}, pages={83–93} }
 @article{beron-vera_olascoaga_haller_farazmand_triñanes_wang_2015, title={Dissipative inertial transport patterns near coherent Lagrangian eddies in the ocean}, volume={25}, ISSN={1054-1500 1089-7682}, url={http://dx.doi.org/10.1063/1.4928693}, DOI={10.1063/1.4928693}, abstractNote={Recent developments in dynamical systems theory have revealed long-lived and coherent Lagrangian (i.e., material) eddies in incompressible, satellite-derived surface ocean velocity fields. Paradoxically, observed drifting buoys and floating matter tend to create dissipative-looking patterns near oceanic eddies, which appear to be inconsistent with the conservative fluid particle patterns created by coherent Lagrangian eddies. Here, we show that inclusion of inertial effects (i.e., those produced by the buoyancy and size finiteness of an object) in a rotating two-dimensional incompressible flow context resolves this paradox. Specifically, we obtain that anticyclonic coherent Lagrangian eddies attract (repel) negatively (positively) buoyant finite-size particles, while cyclonic coherent Lagrangian eddies attract (repel) positively (negatively) buoyant finite-size particles. We show how these results explain dissipative-looking satellite-tracked surface drifter and subsurface float trajectories, as well as satellite-derived Sargassum distributions.}, number={8}, journal={Chaos: An Interdisciplinary Journal of Nonlinear Science}, publisher={AIP Publishing}, author={Beron-Vera, Francisco J. and Olascoaga, María J. and Haller, George and Farazmand, Mohammad and Triñanes, Joaquín and Wang, Yan}, year={2015}, month={Aug}, pages={087412} }
 @article{farazmand_haller_2014, title={How coherent are the vortices of two-dimensional turbulence?}, journal={arXiv preprint arXiv:1402.4835}, author={Farazmand, M. and Haller, G.}, year={2014} }
 @article{farazmand_blazevski_haller_2014, title={Shearless transport barriers in unsteady two-dimensional flows and maps}, volume={278-279}, DOI={10.1016/j.physd.2014.03.008}, abstractNote={We develop a variational principle that extends the notion of a shearless transport barrier from steady to general unsteady two-dimensional flows and maps defined over a finite time interval. This principle reveals that hyperbolic Lagrangian Coherent Structures (LCSs) and parabolic LCSs (or jet cores) are the two main types of shearless barriers in unsteady flows. Based on the boundary conditions they satisfy, parabolic barriers are found to be more observable and robust than hyperbolic barriers, confirming widespread numerical observations. Both types of barriers are special null-geodesics of an appropriate Lorentzian metric derived from the Cauchy–Green strain tensor. Using this fact, we devise an algorithm for the automated computation of parabolic barriers. We illustrate our detection method on steady and unsteady non-twist maps and on the aperiodically forced Bickley jet.}, journal={Physica D}, author={Farazmand, M. and Blazevski, D. and Haller, G.}, year={2014}, pages={44–57} }
 @article{farazmand_haller_2014, title={The Maxey-Riley Equation: Existence, Uniqueness and Regularity of Solutions}, DOI={10.1016/j.nonrwa.2014.08.002}, abstractNote={The Maxey–Riley equation describes the motion of an inertial (i.e., finite-size) spherical particle in an ambient fluid flow. The equation is a second-order, implicit integro-differential equation with a singular kernel, and with a forcing term that blows up at the initial time. Despite the widespread use of the equation in applications, the basic properties of its solutions have remained unexplored. Here we fill this gap by proving local existence and uniqueness of mild solutions. For certain initial velocities between the particle and the fluid, the results extend to strong solutions. We also prove continuous differentiability of the mild and strong solutions with respect to their initial conditions. This justifies the search for coherent structures in inertial flows using the Cauchy–Green strain tensor.}, note={In press}, journal={J. Nonliner Analysis-B}, author={Farazmand, M. and Haller, G.}, year={2014} }
 @article{farazmand_haller_2013, title={Attracting and repelling Lagrangian coherent structures from a single computation}, volume={15}, DOI={10.1063/1.4800210}, abstractNote={Hyperbolic Lagrangian Coherent Structures (LCSs) are locally most repelling or most attracting material surfaces in a finite-time dynamical system. To identify both types of hyperbolic LCSs at the same time instance, the standard practice has been to compute repelling LCSs from future data and attracting LCSs from past data. This approach tacitly assumes that coherent structures in the flow are fundamentally recurrent, and hence gives inconsistent results for temporally aperiodic systems. Here, we resolve this inconsistency by showing how both repelling and attracting LCSs are computable at the same time instance from a single forward or a single backward run. These LCSs are obtained as surfaces normal to the weakest and strongest eigenvectors of the Cauchy-Green strain tensor.}, journal={Chaos}, publisher={AIP}, author={Farazmand, M. and Haller, G.}, year={2013}, pages={023101} }
 @article{hadjighasem_farazmand_haller_2013, title={Detecting invariant manifolds, attractors, and generalized KAM tori in aperiodically forced mechanical systems}, volume={73}, DOI={10.1007/s11071-013-0823-x}, abstractNote={We show how the recently developed theory of geodesic transport barriers for fluid flows can be used to uncover key invariant manifolds in externally forced, one-degree-of-freedom mechanical systems. Specifically, invariant sets in such systems turn out to be shadowed by least-stretching geodesics of the Cauchy-Green strain tensor computed from the flow map of the forced mechanical system. This approach enables the finite-time visualization of generalized stable and unstable manifolds, attractors and generalized KAM curves under arbitrary forcing, when Poincare maps are not available. We illustrate these results by detailed visualizations of the key finite-time invariant sets of conservatively and dissipatively forced Duffing oscillators.}, number={1-2}, journal={Nonlinear Dynamics}, author={Hadjighasem, A. and Farazmand, M. and Haller, G.}, year={2013}, pages={689–704} }
 @article{farazmand_haller_2012, title={Computing Lagrangian Coherent Structures from their variational theory}, volume={22}, DOI={10.1063/1.3690153}, abstractNote={Using the recently developed variational theory of hyperbolic Lagrangian coherent structures (LCSs), we introduce a computational approach that renders attracting and repelling LCSs as smooth, parametrized curves in two-dimensional flows. The curves are obtained as trajectories of an autonomous ordinary differential equation for the tensor lines of the Cauchy-Green strain tensor. This approach eliminates false positives and negatives in LCS detection by separating true exponential stretching from shear in a frame-independent fashion. Having an explicitly parametrized form for hyperbolic LCSs also allows for their further in-depth analysis and accurate advection as material lines. We illustrate these results on a kinematic model flow and on a direct numerical simulation of two-dimensional turbulence.}, journal={Chaos}, author={Farazmand, M. and Haller, G.}, year={2012}, pages={013128} }
 @article{farazmand_haller_2012, title={Erratum and addendum to ``A variational theory of hyperbolic Lagrangian coherent structures, Physica D 240 (2011) 574-598''}, volume={241}, DOI={https://doi.org/10.1016/j.physd.2011.09.013}, abstractNote={This brief note corrects a minor error in the statement of the main result in Haller (2011) [1] on a variational approach to Lagrangian coherent structures. We also show that the corrected formulation leads to a substantial simplification of LCS criteria for two-dimensional flows.}, journal={Physica D}, author={Farazmand, M. and Haller, G.}, year={2012}, pages={439–441} }
 @article{farazmand_kevlahan_protas_2011, title={Controlling the dual cascade of two-dimensional turbulence}, volume={668}, DOI={10.1017/S0022112010004635}, abstractNote={The Kraichnan–Leith–Batchelor (KLB) theory of statistically stationary forced homogeneous isotropic two-dimensional turbulence predicts the existence of two inertial ranges: an energy inertial range with an energy spectrum scaling ofk−5/3, and an enstrophy inertial range with an energy spectrum scaling ofk−3. However, unlike the analogous Kolmogorov theory for three-dimensional turbulence, the scaling of the enstrophy range in the two-dimensional turbulence seems to be Reynolds-number-dependent: numerical simulations have shown that as Reynolds number tends to infinity, the enstrophy range of the energy spectrum converges to the KLB prediction, i.e.E~k−3. The present paper uses a novel optimal control approach to find a forcing that does produce the KLB scaling of the energy spectrum in a moderate Reynolds number flow. We show that the time–space structure of the forcing can significantly alter the scaling of the energy spectrum over inertial ranges. A careful analysis of the optimal forcing suggests that it is unlikely to be realized in nature, or by a simple numerical model.}, journal={J. Fluid Mech.}, author={Farazmand, M. and Kevlahan, N. K.-R. and Protas, B.}, year={2011}, pages={202–222} }