arXiv subjects

Themistoklis P. Sapsis

Publications and source records attributed to Themistoklis P. Sapsis.

46 records · Page 3Linked to original sources

arXiv2017

Reduced-order Description of Transient Instabilities and Computation of Finite-Time Lyapunov Exponents

High-dimensional chaotic dynamical systems can exhibit strongly transient features. These are often associated with instabilities that have 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.

math.DS↗

arXiv2017

Reduced-order prediction of rogue waves in two-dimensional deep-water waves

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.

physics.ao-ph↗

arXiv2016

Reduced-space Gaussian Process Regression for Data-Driven Probabilistic Forecast of Chaotic Dynamical Systems

We formulate a reduced-order strategy for efficiently forecasting complex high-dimensional dynamical systems entirely based on data streams. The first step of our method involves reconstructing the dynamics in a reduced-order subspace of choice using Gaussian Process Regression (GPR). GPR simultaneously allows for reconstruction of the vector field and more importantly, estimation of local uncertainty. The latter is due to i) local interpolation error and ii) truncation of the high-dimensional phase space. This uncertainty component can be analytically quantified in terms of the GPR hyperparameters. In the second step we formulate stochastic models that explicitly take into account the reconstructed dynamics and their uncertainty. For regions of the attractor which are not sufficiently sampled for our GPR framework to be effective, an adaptive blended scheme is formulated to enforce correct statistical steady state properties, matching those of the real data. We examine the effectiveness of the proposed method to complex systems including the Lorenz 96, the Kuramoto-Sivashinsky, as well as a prototype climate model. We also study the performance of the proposed approach as the intrinsic dimensionality of the system attractor increases in highly turbulent regimes.

physics.data-an↗

arXiv2015

A probabilistic decomposition-synthesis method for the quantification of rare events due to internal instabilities

We consider the problem of probabilistic quantification of dynamical systems that have heavy-tailed characteristics. These heavy-tailed features are associated with rare transient responses due to the occurrence of internal instabilities. Here we develop a computational method, a probabilistic decomposition-synthesis technique, that takes into account the nature of internal instabilities to inexpensively determine the non-Gaussian probability density function for any arbitrary quantity of interest. Our approach relies on the decomposition of the statistics into a `non-extreme core', typically Gaussian, and a heavy-tailed component. This decomposition is in full correspondence with a partition of the phase space into a `stable' region where we have no internal instabilities, and a region where non-linear instabilities lead to rare transitions with high probability. We quantify the statistics in the stable region using a Gaussian approximation approach, while the non-Gaussian distributions associated with the intermittently unstable regions of the phase space are inexpensively computed through order-reduction methods that take into account the strongly nonlinear character of the dynamics. The probabilistic information in the two domains is analytically synthesized through a total probability argument. The proposed approach allows for the accurate quantification of non-Gaussian tails at more than 10 standard deviations, at a fraction of the cost associated with the direct Monte-Carlo simulations. We demonstrate the probabilistic decomposition-synthesis method for rare events for two dynamical systems exhibiting extreme events: a two-degree-of-freedom system of nonlinearly coupled oscillators, and in a nonlinear envelope equation characterizing the propagation of unidirectional water waves.

physics.comp-ph↗

arXiv2015

A moment-equation-copula-closure method for nonlinear vibrational systems subjected to correlated noise

We develop a moment equation closure minimization method for the inexpensive approximation of the steady state statistical structure of nonlinear systems whose potential functions have bimodal shapes and which are subjected to correlated excitations. Our approach relies on the derivation of moment equations that describe the dynamics governing the two-time statistics. These are combined with a non-Gaussian pdf representation for the joint response-excitation statistics that has i) single time statistical structure consistent with the analytical solutions of the Fokker-Planck equation, and ii) two-time statistical structure with Gaussian characteristics. Through the adopted pdf representation, we derive a closure scheme which we formulate in terms of a consistency condition involving the second order statistics of the response, the closure constraint. A similar condition, the dynamics constraint, is also derived directly through the moment equations. These two constraints are formulated as a low-dimensional minimization problem with respect to unknown parameters of the representation, the minimization of which imposes an interplay between the dynamics and the adopted closure. The new method allows for the semi-analytical representation of the two-time, non-Gaussian structure of the solution as well as the joint statistical structure of the response-excitation over different time instants. We demonstrate its effectiveness through the application on bistable nonlinear single-degree-of-freedom energy harvesters with mechanical and electromagnetic damping, and we show that the results compare favorably with direct Monte-Carlo Simulations.

nlin.CD↗

arXiv2015

The unsteady evolution of localized unidirectional deep water wave groups

We study the evolution of localized wave groups in unidirectional water wave envelope equations (nonlinear Schrodinger (NLS) and modified NLS (MNLS)). These localizations of energy can lead to disastrous extreme responses (rogue waves). Previous studies have focused on the role of energy distribution in the frequency domain in the formation of extreme waves. We analytically quantify the role of spatial localization, introducing a novel technique to reduce the underlying PDE dynamics to a simple ODE for the wave packet amplitude. We use this reduced model to show how the scale- invariant symmetries of NLS break down when the additional terms in MNLS are included, inducing a critical scale for the occurrence of extreme waves.

nlin.PS↗

arXiv2015

Reduced order prediction of rare events in unidirectional nonlinear water waves

We consider the problem of short-term prediction of rare, extreme water waves in unidirectional fields, a critical topic for ocean structures and naval operations. One possible mechanism for the occurrence of such rare, unusually-intense waves is nonlinear wave focusing. Recent results have demonstrated that random localizations of energy, induced by the dispersive mixing of different harmonics, can grow significantly due to localized nonlinear focusing. Here we show how the interplay between i) statistical properties captured through linear information such as the waves power spectrum and ii) nonlinear dynamical properties of focusing localized wave groups defines a critical length scale associated with the formation of extreme events. The energy that is locally concentrated over this length scale acts as the "trigger" of nonlinear focusing for wave groups and the formation of subsequent rare events. We use this property to develop inexpensive, short-term predictors of large water waves. Specifically, we show that by merely tracking the energy of the wave field over the critical length scale allows for the robust, inexpensive prediction of the location of intense waves with a prediction window of 25 wave periods. We demonstrate our results in numerical experiments of unidirectional water wave fields described by the Modified Nonlinear Schrodinger equation. The presented approach introduces a new paradigm for understanding and predicting intermittent and localized events in dynamical systems characterized by uncertainty and potentially strong nonlinear mechanisms.

nlin.PS↗

arXiv2014

Probabilistic description of extreme events in intermittently unstable systems excited by correlated stochastic processes

In this work, we consider systems that are subjected to intermittent instabilities due to external stochastic excitation. These intermittent instabilities, though rare, have a large impact on the probabilistic response of the system and give rise to heavy-tailed probability distributions. By making appropriate assumptions on the form of these instabilities, which are valid for a broad range of systems, we formulate a method for the analytical approximation of the probability distribution function (pdf) of the system response (both the main probability mass and the heavy-tail structure). In particular, this method relies on conditioning the probability density of the response on the occurrence of an instability and the separate analysis of the two states of the system, the unstable and stable state. In the stable regime we employ steady state assumptions, which lead to the derivation of the conditional response pdf using standard methods for random dynamical systems. The unstable regime is inherently transient and in order to analyze this regime we characterize the statistics under the assumption of an exponential growth phase and a subsequent decay phase until the system is brought back to the stable attractor. The method we present allows us to capture the statistics associated with the dynamics that give rise to heavy-tails in the system response and the analytical approximations compare favorably with direct Monte Carlo simulations, which we illustrate for two prototype intermittent systems: an intermittently unstable mechanical oscillator excited by correlated multiplicative noise and a complex mode in a turbulent signal with fixed frequency, where multiplicative stochastic damping and additive noise model interactions between various modes.

nlin.CD↗

arXiv2014

Performance measures for single-degree-of-freedom energy harvesters under stochastic excitation

We develop performance criteria for the objective comparison of different classes of single-degree-of-freedom oscillators under stochastic excitation. For each family of oscillators, these objective criteria take into account the maximum possible energy harvested for a given response level, which is a quantity that is directly connected to the size of the harvesting configuration. We prove that the derived criteria are invariant with respect to magnitude or temporal rescaling of the input spectrum and they depend only on the relative distribution of energy across different harmonics of the excitation. We then compare three different classes of linear and nonlinear oscillators and using stochastic analysis tools we illustrate that in all cases of excitation spectra (monochromatic, broadband, white-noise) the optimal performance of all designs cannot exceed the performance of the linear design. Subsequently, we study the robustness of this optimal performance to small perturbations of the input spectrum and illustrate the advantages of nonlinear designs relative to linear ones.

nlin.CD↗

arXiv2014

Quantification and prediction of extreme events in a one-dimensional nonlinear dispersive wave model

The aim of this work is the quantification and prediction of rare events characterized by extreme intensity in nonlinear waves with broad spectra. We consider a one-dimensional non- linear model with deep-water waves dispersion relation, the Majda-McLaughlin-Tabak (MMT) model, in a dynamical regime that is characterized by broadband spectrum and strong non- linear energy transfers during the development of intermittent events with finite-lifetime. To understand the energy transfers that occur during the development of an extreme event we perform a spatially localized analysis of the energy distribution along different wavenumbers by means of the Gabor transform. A stochastic analysis of the Gabor coefficients reveals i) the low-dimensionality of the intermittent structures, ii) the interplay between non-Gaussian statis- tical properties and nonlinear energy transfers between modes, as well as iii) the critical scales (or critical Gabor coefficients) where a critical amount of energy can trigger the formation of an extreme event. We analyze the unstable character of these special localized modes directly through the system equation and show that these intermittent events are due to the interplay of the system nonlinearity, the wave dispersion, and the wave dissipation which mimics wave breaking. These localized instabilities are triggered by random localizations of energy in space, created by the dispersive propagation of low-amplitude waves with random phase. Based on these properties, we design low-dimensional functionals of these Gabor coefficients that allow for the prediction of the extreme event well before the nonlinear interactions begin to occur.

nlin.CD↗