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Mailybaev A.A.,Instituto Nacional Of Matematica Pura E Aplicada Impa | Mailybaev A.A.,State University of Rio de Janeiro
Physical Review E - Statistical, Nonlinear, and Soft Matter Physics | Year: 2012

We show that multiscaling properties of developed turbulence in shell models, which lead to anomalous scaling exponents in the inertial range, are determined exclusively by instanton dynamics. Instantons represent correlated extreme events localized in space-time, whose structure is described by self-similar statistics with a single universal scaling exponent. We show that anomalous scaling exponents appear due to the process of instanton creation. A simplified model of instanton creation is suggested, which adequately describes this anomaly. © 2012 American Physical Society.

Nehab D.,Instituto Nacional Of Matematica Pura E Aplicada Impa | Hoppe H.,Microsoft
Foundations and Trends in Computer Graphics and Vision | Year: 2012

Discretization and reconstruction are fundamental operations in computer graphics, enabling the conversion between sampled and continuous representations. Major advances in signal-processing research have shown that these operations can often be performed more efficiently by decomposing a filter into two parts: a compactly supported continuous-domain function and a digital filter.This strategy of generalized sampling has appeared in a few graphics papers, but is largely unexplored in our community.This survey broadly summarizes the key aspects of the framework,and delves into specific applications in graphics. Using new notation, we concisely present and extend several key techniques. In addition, we demonstrate benefits for prefiltering in image downscaling and supersample-based rendering, and analyze the effect that generalized sampling has on the noise due to Monte Carlo estimation. We conclude with a qualitative and quantitative comparison of traditional and generalized filters. ©c 2014 I. Goldstein and H. Sapra.

Mailybaev A.A.,Instituto Nacional Of Matematica Pura E Aplicada Impa
Multiscale Modeling and Simulation | Year: 2016

We analyze the phenomenon of spontaneous stochasticity in fluid dynamics formulated as the nonuniqueness of solutions resulting from viscosity at infinitesimal scales acting through intermediate scales on large scales of the flow. We study the finite-time onset of spontaneous stochasticity in a real version of the Gledzer Ohkitani Yamada shell model of turbulence. This model allows high-accuracy numerical simulations for a wide range of scales (up to 10 orders of magnitude) and demonstrates nonchaotic dynamics but leads to an infinite number of solutions in the vanishing viscosity limit after the blowup time. We provide the numerical and theoretical description of the system dynamics at all stages. This includes the asymptotic analysis before and after the blowup leading to universal (periodic and quasi-periodic) renormalized solutions, followed by nonunique stationary states at large times. © 2016 Society for Industrial and Applied Mathematics.

Mailybaev A.A.,Instituto Nacional Of Matematica Pura E Aplicada Impa
Nonlinearity | Year: 2016

In this paper, we study the inviscid limit of the Sabra shell model of turbulence, which is considered as a particular case of a viscous conservation law in one space dimension with a nonlocal quadratic flux function. We present a theoretical argument (with a detailed numerical confirmation) showing that a classical deterministic solution before a finite-time blowup, t < t b, must be continued as a stochastic process after the blowup, t > t b, representing a unique physically relevant description in the inviscid limit. This theory is based on the dynamical system formulation written for the logarithmic time , which features a stable traveling wave solution for the inviscid Burgers equation, but a stochastic traveling wave for the Sabra model. The latter describes a universal onset of stochasticity immediately after the blowup. © 2016 IOP Publishing Ltd & London Mathematical Society.

Gilary I.,Technion - Israel Institute of Technology | Mailybaev A.A.,Instituto Nacional Of Matematica Pura E Aplicada Impa | Moiseyev N.,Technion - Israel Institute of Technology
Physical Review A - Atomic, Molecular, and Optical Physics | Year: 2013

We show here that due to nonadiabatic couplings in decaying systems, applying the same time-dependent protocol in the forward and reverse direction to the same mixed initial state leads to different final pure states. In particular, in laser-driven molecular systems, applying a specifically chosen positively chirped laser pulse or an equivalent negatively chirped laser pulse yields entirely different final vibrational states. This phenomenon occurs when the laser frequency and intensity are slowly varied around an exceptional point (EP) in the laser intensity and frequency parameter space where the non-Hermitian spectrum of the problem is degenerate. The protocol implies that a positively chirped laser pulse traces a loop in time in the laser parameters' space whereas a negatively chirped pulse follows the same loop in the opposite direction. According to this protocol one can choose the final pure state from any initial state. The obtained results imply the intrinsic nonadiabaticity of quantum transport around an EP, and offer a way to observe the EP experimentally in time-dependent quantum systems. © 2013 American Physical Society.

De Oliveira W.,Instituto Nacional Of Matematica Pura E Aplicada Impa | Sagastizabal C.,Instituto Nacional Of Matematica Pura E Aplicada Impa
Optimization Methods and Software | Year: 2014

For non-smooth convex optimization, we consider level bundle methods built using an oracle that computes values for the objective function and a subgradient at any given feasible point. For the problems of interest, the exact oracle information is computable, but difficult to obtain. In order to save computational effort the oracle can provide estimations with an accuracy that depends on two additional parameters, informed to the oracle together with the evaluation point. The first of such parameters is a descent target, while the second one is a bound for inexactness. If the oracle can reach the target with its function estimation, then the corresponding error is bounded by the second parameter. Otherwise, if the oracle detects that the target cannot be met, the function and subgradient estimations can be rough and have an unknown accuracy. For some selected iterates the considered methods drive the inexactness parameter to zero, thus ensuring that an exact solution to the optimization problem is asymptotically found. The approach is comprehensive and covers known exact and inexact level methods as well as some novel variants that can handle inaccuracy in an adaptive manner. In particular, when the feasible set is also compact, some of the new on-demand accuracy methods have the same rate of convergence of exact level variants known in the literature. A numerical benchmark on a battery of two-stage stochastic linear programs assesses the interest of the approach, substantially faster than the L-shaped method, without any accuracy loss. © 2014 Taylor & Francis.

Mailybaev A.A.,Instituto Nacional Of Matematica Pura E Aplicada Impa
Nonlinearity | Year: 2013

We analyse the blowup (finite-time singularity) in inviscid shell models of convective turbulence. We show that the blowup exists and its internal structure undergoes a series of bifurcations under a change of shell model parameter. Various blowup structures are observed and explained, which vary from self-similar to periodic, quasi-periodic and chaotic regimes. Though the blowup takes sophisticated forms, its asymptotic small-scale structure is independent of the initial conditions, i.e. universal. Finally, we discuss the implications of the obtained results for the open problems of blowup in inviscid flows and for the theory of turbulence. © 2013 IOP Publishing Ltd & London Mathematical Society.

Mailybaev A.A.,Instituto Nacional Of Matematica Pura E Aplicada Impa
Physical Review E - Statistical, Nonlinear, and Soft Matter Physics | Year: 2013

Since Kolmogorov proposed his phenomenological theory of hydrodynamic turbulence in 1941, the description of the mechanism leading to the energy cascade and anomalous scaling remains an open problem in fluid mechanics. Soon after, in 1949, Onsager noticed that the scaling properties in the inertial range imply nondifferentiability of the velocity field in the limit of vanishing viscosity. This observation suggests that the turbulence mechanism may be related to a finite-time singularity (blowup) of incompressible Euler equations. However, the existence of such blowup is still an open problem too. In this paper, we show that the blowup indeed represents the driving mechanism of the inertial range for a simplified (shell) model of turbulence. Here, blowups generate coherent structures (instantons), which travel through the inertial range in finite time and are described by universal self-similar statistics. The anomaly (deviation of scaling exponents of velocity moments from the Kolmogorov theory) is related analytically to the process of instanton creation using the large deviation principle. The results are confirmed by numerical simulations. © 2013 American Physical Society.

Mailybaev A.A.,Instituto Nacional Of Matematica Pura E Aplicada Impa | Mailybaev A.A.,Moscow State University
Physical Review E - Statistical, Nonlinear, and Soft Matter Physics | Year: 2012

We consider self-similar solutions describing intermittent bursts in shell models of turbulence and study their relationship with blowup phenomena in continuous hydrodynamic models. First, we show that these solutions are very close to self-similar solution for the Fourier transformed inviscid Burgers equation corresponding to shock formation from smooth initial data. Then, the result is generalized to hyperbolic conservation laws in one space dimension describing compressible flows. It is shown that the renormalized wave profile tends to a universal function, which is independent both of initial conditions and of a specific form of the conservation law. This phenomenon can be viewed as a new manifestation of the renormalization group theory. Finally, we discuss possibilities for application of the developed theory for detecting and describing a blowup in incompressible flows. © 2012 American Physical Society.

Mailybaev A.A.,Instituto Nacional Of Matematica Pura E Aplicada Impa
Nonlinearity | Year: 2015

In this work we construct and analyze continuous hydrodynamic models in one space dimension, which are induced by shell models of turbulence. After Fourier transformation, such continuous models split into an infinite number of uncoupled subsystems, which are all identical to the same shell model. The two shell models, which allow such a construction, are considered: the dyadic (Desnyansky-Novikov) model with the intershell ratio λ = 23/2 and the Sabra model of turbulence with σ = √2+√5 2.580. The continuous models allow for understanding of various properties of shell model solutions and provide their interpretation in physical space. We show that the asymptotic solutions of the dyadic model with Kolmogorov scaling correspond to the shocks (discontinuities) for the induced continuous solutions in physical space, and the finite-time blowup together with its viscous regularization follow the scenario similar to the Burgers equation. For the Sabra model, we provide the physical space representation for blowup solutions and intermittent turbulent dynamics. © 2015 IOP Publishing Ltd & London Mathematical Society.

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