Optimal Perturbation Research Articles

Localisation of optimal perturbations in variable viscosity channel flow

We discuss how a variable fluid viscosity affects the nonmodal stability characteristics of the pressure driven flow between two parallel walls maintained at different temperatures. In this work, we specify the fluid viscosity to be a function of the fluid temperature. We employ an Arrhenius model to model the viscosity of water, and Sutherland’s law to model the viscosity of air. We impose a stable density stratification, and find that strong density stratification can suppress optimal transient growth regardless of how strong the viscosity variation is. Some studies have been inclined to neglect viscosity stratification, since the changes in levels of optimal growth, when compared to the uniform viscosity case, are often not too significant. In this article, we show significant localisation of optimal perturbation energy in the less viscous region, a feature that is not observed in uniform viscosity flows. This can have a bearing on the route to turbulence in these systems.

Impacts of parameter uncertainties on deep chlorophyll maximum simulation revealed by the CNOP-P approach

Parameter uncertainty is a primary source of uncertainty in ocean ecosystem simulations. The deep chlorophyll maximum (DCM) is a ubiquitous ecological phenomenon in the ocean. Using a theoretical nutrients-phytoplankton model and the conditional nonlinear optimal perturbation approach related to parameters, we investigated the effects of parameter uncertainties on DCM simulations. First, the sensitivity of single parameter was analyzed. The sensitivity ranking of 10 parameters was obtained by analyzing the top four specifically. The most sensitive parameter (background turbidity) affects the light supply for DCM formation, whereas the other three parameters (nutrient content of phytoplankton, nutrient recycling coefficient, and vertical turbulent diffusivity) control nutrient supply. To explore the interactions among different parameters, the sensitivity of multiple parameters was further studied by examining combinations of four parameters. The results show that background turbidity is replaced by the phytoplankton loss rate in the optimal parameter combination. In addition, we found that interactions among these parameters are responsible for such differences. Finally, we found that reducing the uncertainties of sensitive parameters could improve DCM simulations remarkably. Compared with the sensitive parameters identified in the single parameter analysis, reducing parameter uncertainties in the optimal combination produced better model performance. This study shows the importance of nonlinear interactions among various parameters in identifying sensitive parameters. In the future, the conditional nonlinear optimal perturbation approach related to parameters, especially optimal parameter combinations, is expected to greatly improve DCM simulations in complex ecosystem models.

Optimal Perturbation Iteration Method for Solving Fractional Model of Damped Burgers’ Equation

The newly constructed optimal perturbation iteration procedure with Laplace transform is applied to obtain the new approximate semi-analytical solutions of the fractional type of damped Burgers’ equation. The classical damped Burgers’ equation is remodeled to fractional differential form via the Atangana–Baleanu fractional derivatives described with the help of the Mittag–Leffler function. To display the efficiency of the proposed optimal perturbation iteration technique, an extended example is deeply analyzed.

On the Kahan-Parlett-Jiang theorem—A globally optimal backward perturbation error for two-sided invariant subspaces

A crucial question in large non-Hermitian eigenproblems is whether or not any residual bounds can be very useful in practical situations. For this problem, the famous Kahan-Parlett-Jiang theorem Kahan et al. (1982) [15] seems to be the best compromise between generality and sharpness. In essence, it presents an a posteriori error bound on approximate solutions to eigenproblems, which provides a powerful tool to evaluate quality of the computed two-sided invariant subspaces of large-scale non-Hermitian matrices. However, the perturbation error determined by this theorem is only locally optimal rather than globally optimal. In this work, we revisit this problem and derive a globally optimal backward perturbation error for given two-sided approximate invariant subspaces. Our theorem enhances the Kahan-Parlett-Jiang theorem, and numerical experiments demonstrate the effectiveness of our theoretical results.

Spot evasion attacks: Adversarial examples for license plate recognition systems with convolutional neural networks

Recent studies have shown that convolutional neural networks (CNNs) for image recognition are vulnerable to evasion attacks with carefully manipulated adversarial examples. Previous work primarily focused on how to generate adversarial examples closed to source images, by introducing pixel-level perturbations into the whole or specific part of images. In this paper, we propose an evasion attack on CNN classifiers in the context of License Plate Recognition (LPR), which adds predetermined perturbations to specific regions of license plate, simulating some sort of naturally formed spots (such as sludge, etc.). Therefore, the problem is modeled as an optimization process searching for optimal perturbation positions, which is different from previous work that consider pixel values as decision variables. Notice that this is a complex nonlinear optimization problem, and we use a genetic-algorithm based approach to obtain optimal perturbation positions. In experiments, we use the proposed algorithm to generate various adversarial examples in the form of rectangle, circle, ellipse and spots cluster. Experimental results show that these adversarial examples are almost ignored by human eyes, but can fool HyperLPR with high attack success rate over 93%. Therefore, we believe that this kind of spot evasion attacks would pose a great threat to current LPR systems, and needs to be investigated further by the security community.

Quantum scars as embeddings of weakly broken Lie algebra representations

We present an interpretation of scar states and quantum revivals as weakly "broken" representations of Lie algebras spanned by a subset of eigenstates of a many-body quantum system. We show that the PXP model, describing strongly-interacting Rydberg atoms, supports a "loose" embedding of multiple $\mathrm{su(2)}$ Lie algebras corresponding to distinct families of scarred eigenstates. Moreover, we demonstrate that these embeddings can be made progressively more accurate via an iterative process which results in optimal perturbations that stabilize revivals from arbitrary charge density wave product states, $|\mathbb{Z}_n\rangle$, including ones that show no revivals in the unperturbed PXP model. We discuss the relation between the loose embeddings of Lie algebras present in the PXP model and recent exact constructions of scarred states in related models.

Modulation instability in lossy and nonlinear thulium-doped fiber amplifiers in S and near-C bands: analytical and numerical approach

We present the effect of losses and nonlinearity on the behavior of modulation instability (MI) in thulium-doped fiber amplifier (TDFA) operational in S and near-C bands. Based on a derived analytical model, the relative change in maximum integrated gain at the respective perturbation frequency was observed and compared with lossless TDFAs. At an estimated small-signal attenuation of 0.36 (corresponding to small-signal attenuation coefficient α = 0.01 / m), a frequency shift of 1.44 THz with a reduction in integrated gain of 45 dB was obtained for an input signal power of 250 mW. For the same set of parameters, the effect of nonlinear and loss factor was calculated and compared with present day state-of-the-art erbium-doped fiber amplifiers. The analysis helped in the selection of optimum values for small-signal gain coefficient (g0 = 0.21 / m) and TDFA length (L = 100 m) to maximize the integrated gain of the perturbations for different input signal powers. The special case pertaining to nonideal conditions with respect to the large amplitude of input perturbations was solved numerically. Dipole relaxation time was estimated (450 fs) from the results and compared with the theoretically calculated value (318 fs) for TDFAs. The results obtained can be used constructively in the selection of optimum parameters and perturbation frequency value for the design of TDFA lasers in which MI acts as an active saturable absorber.

Optimal error analysis of MJO prediction associated with uncertainties in sea surface temperature over Indian Ocean

In this study, the predictability of the Madden–Julian Oscillation (MJO) is investigated using the coupled Community Earth System Model (CESM) and the climatically relevant singular vector (CSV) method. The CSV method is an ensemble-based strategy to calculate the optimal growth of the initial error on the climate scale. We focus on the CSV analysis of MJO initialized at phase II, facilitating the investigation of the effect of the initial errors of the sea surface temperature (SST) in the Indian Ocean on it. Six different MJO events are chosen as the study cases to ensure the robustness of the results. The results indicate that for all the study cases, the optimal perturbation structure of the SST, denoted by the leading mode of the singular vectors (SVs), is a meridional dipole-like pattern between the Bay of Bengal and the southern central Indian Ocean. The MJO signal tends to be more converged and significant in the Eastern Hemisphere while the model is perturbed by leading SV. The moist static energy analysis results indicate that the eastward propagation is much more evident in the terms of vertical advection and radiation flux than others. Therefore, the SV perturbation can strengthen and converge the MJO signal mostly by increasing the vertical advection of the moist static energy. Further, the sensitivity studies indicate that the structure of the leading SV is not sensitive to the initial states, which suggests that we might not need to calculate SVs for each initial time in constructing the ensemble prediction, significantly saving computational time in the operational forecast systems.

Optimal suppression of a separation bubble in a laminar boundary layer

By means of nonlinear optimization, we seek the velocity disturbances at a given upstream position that suppress a laminar separation bubble as effectively as possible. Both steady and unsteady disturbances are examined and compared. For steady disturbances, an informed guess based on linear analysis of transient perturbation growth leads to significant delay of separation and serves as a starting point for the nonlinear optimization algorithm. It is found that the linear analysis largely captures the suppression of the separation bubble attained by the nonlinear optimal perturbations. The mechanism of separation delay is the generation of a mean flow distortion by nonlinear interactions during the perturbation growth. The mean flow distortion enhances the momentum close to the wall, counteracting the deceleration of the flow in that region. An examination of the effect of the disturbance spanwise wavenumber reveals that perturbations maximizing the mean flow distortion also approximately maximize the peak wall pressure, which is beneficial for lowering form drag. The optimal spanwise wavenumber leading to maximal peak wall pressure is significantly larger than the one maximizing the shift in separation onset. For unsteady disturbances, the mechanism of separation delay relies on enhancing wall-normal momentum transfer by triggering instabilities of the separated shear layer. It is found that Tollmien–Schlichting waves obtained from linear stability theory provide accurate estimates of the nonlinearly optimal disturbances. Comparison of optimal steady and unsteady perturbations reveals that the latter are able to obtain a higher time-averaged peak wall pressure.

Influence of the Arctic on the Predictability of Eurasian Winter Extreme Weather Events

The linkage between the Arctic and midlatitudes has received much attention recently due to the rapidly changing climate. Many investigations have been conducted to reveal the relationship between the Arctic and Eurasian extreme events from the perspective of climatological statistics. As a prediction source for extreme events in Eurasia, Arctic conditions are crucial for extreme event predictions. Therefore, it is urgent to explore the Arctic influence on the predictability of Eurasian extreme events due to the large uncertainties in Arctic conditions. Considering the sensitivity and nonlinearity of the atmospheric circulations in midlatitude to Arctic conditions, it is necessary to investigate the Arctic influences on Eurasian extreme weather events in case studies at weather time scales. Previous studies indicate that only perturbations in specific patterns have fast growth. Thus, the conditional nonlinear optimal perturbation approach is recommended for exploring the uncertainties in Arctic initial and boundary conditions and their synergistic effect on Eurasian extreme events. Moreover, the mechanism for extreme event formation may differ in different cases. Therefore, more extreme cases should be investigated to reach robust conclusions.

Perturbation analysis for the QX factorization for centrosymmetric matrices

The QX factorization of the centrosymmetric matrix is a structure-preserving QR factorization proposed by Konrad Burnik. In this paper, we first propose a sufficient condition for the uniqueness of this decomposition. Then, using the refined matrix equation approach, we derive the first-order normwise perturbation bounds convenient for computation for the QX factorization. Using the matrix-vector equation approach and the modified matrix-vector equation approach, we obtain the optimal first-order normwise perturbation bounds. Moreover, the normwise condition numbers for the QX factorization are also presented. Some numerical examples are given to illustrate our theoretical results.

The Optimal Precursor of El Niño in the GFDL CM2p1 Model

AbstractBy applying the principal component analysis‐based particle swarm optimization algorithm, the conditional nonlinear optimal perturbation is firstly calculated in the Geophysical Fluid Dynamics Laboratory Climate Model version 2p1 (GFDL CM2p1) to identify the optimal precursor (OPR) of El Niño. Specifically, through optimizing the initial perturbation, the OPRs that have the largest nonlinear evolution (i.e., mature state of El Niño) for two reference states are obtained, which are then confirmed according to the validation test. The results indicate that both OPRs show positive sea surface temperature perturbation in the west (2°N–2°S, 135.5–165.5°E). For the subsurface component, they exhibit positive subsurface temperature perturbation (STP) in the whole mixed layer of the west and negative STP in the upper layer of the east (i.e., 0‐ to 85‐m depth, 2°N–2°S, 79.5–109.5°W). Further analyses of the evolution of the sea surface temperature perturbation, STP, and surface wind perturbation suggest that the development of the OPRs in the model is consistent with the recognized mechanism for El Niño‐Southern Oscillation development, that is, through the Bjerknes positive feedback. The results indicate that the model can realistically capture the dominant processes for El Niño development, and the principal component analysis‐based particle swarm optimization algorithm is a practical solution for calculating the conditional nonlinear optimal perturbation in a complicated numerical model such as the GFDL CM2p1. They both shed a light on guiding the realistic observing systems.

Optimal perturbation iteration technique for solving nonlinear Volterra‐Fredholm integral equations

In this work, the optimal perturbation iteration method is briefly presented and employed for solving nonlinear Volterra‐Fredholm integral equations. The classical form of the optimal perturbation iteration method is modified, and new algorithms are constructed for integral equations. Comparing our new algorithms with some earlier papers proved the excellent accuracy of the newly proposed technique.

HYPERFINE STRUCTURE PARAMETERS FOR Li-LIKE MULTICHARGED IONS WITHIN RELATIVISTIC MANY-BODY PERTURBATION THEORY

The relativistic many-body perturbation theory with the optimized Dirac-Kohn-Sham zeroth approximation is applied to calculation of the hyperfine structure parameters for some Li-like multicharged ions. The relativistic, exchange-correlation and other corrections are accurately taken into account. The optimized relativistic orbital basis set is generated in the optimal many-body perturbation theory approximation with fulfilment of the gauge invariance principle. The obtained data on the hyperfine structure parameters of the Li-like multicharged ions are analyzed and compared with alternative theoretical and experimental results.

Optimal one-bit perturbation in Boolean networks based on cascading aggregation

We investigate the problem of finding optimal one-bit perturbation that maximizes the size of the basin of attractions (BOAs) of desired attractors and minimizes the size of the BOAs of undesired attractors for large-scale Boolean networks by cascading aggregation. First, via the aggregation, a necessary and sufficient condition is given to ensure the invariance of desired attractors after one-bit perturbation. Second, an algorithm is proposed to identify whether the one-bit perturbation will cause the emergence of new attractors or not. Next, the change of the size of BOAs after one-bit perturbation is provided in an algorithm. Finally, the efficiency of the proposed method is verified by a T-cell receptor network.

Linear two-dimensional stability of a Lamb-Oseen dipole as an aircraft wake model

The dynamics of perturbed aircraft wake models is investigated in the two-dimensional limit by means of a linear stability analysis. The base flow, computed by a direct numerical simulation of the incompressible Navier-Stokes equations, is a viscous counter-rotating vortex dipole obtained from an initial condition which is either a superposition of two Lamb-Oseen vortices or a vorticity sheet with elliptical vorticity distribution. The former, referred to as the Lamb-Oseen dipole (LOD), is a model for the far field of the wake and gives rise to a family of quasisteady dipoles parametrized by their aspect ratio only. The later approaches the near-field wake of a wing during the rolling phase which eventually converges towards the LOD model at later times. First, a modal stability analysis of the LOD under the assumption of a frozen base flow is performed for aspect ratios a/b ranging from 0.05 to 0.36 at various Reynolds numbers. Several families of unstable antisymmetric and symmetric modes are observed. The maximal growth rates are reached at low Reynolds numbers. The results are consistent with those obtained by Brion et al. [Phys. Fluids 26, 064103 (2014)] for the higher aspect ratio inviscid Lamb-Chaplygin dipole (LCD). However, the a posteriori verification of the validity of the frozen base flow assumption shows that, except for a few modes occurring at the highest aspect ratios and large Reynolds numbers, these two-dimensional instabilities do not survive the base flow unsteadiness due to viscous diffusion. They are thus not likely to develop in the flow. Second, we focus on the transient dynamics of the dipoles by looking for the optimal perturbations through a nonmodal stability analysis based on a direct-adjoint approach. The observed energy gains are substantial and indicate the potential of transient mechanisms. In the short time dynamics, the optimal perturbation consists of intertwined vorticity layers located within each vortex core and leading to a m=2 deformation Kelvin wave excited by the Orr mechanism. For moderate to large horizon times, the optimal perturbation takes the form of vorticity layers localized outside the vortex core which eventually give rise to the two-dimensional unstable mode unveiled by the modal analysis through a combination of Orr mechanism and velocity induction. The robustness of these modes is examined by considering the initial stage of the development of aircraft wakes. The optimal perturbations developing on an elliptic and a double-elliptic vorticity sheet present a similar structure and rely on the same mechanisms as the ones observed for the LOD but come with lower energy gains. It is concluded that the rolling-up of the vorticity sheet in the near-field of the wake does not influence significantly the linear development of these two-dimensional perturbations.

Transient growth and symmetrizability in rectilinear miscible viscous fingering

The influence of dispersion or equivalently of the Peclet number (Pe) on miscible viscous fingering in a homogeneous porous medium is examined. The linear optimal perturbations maximizing finite-time energy gain is demonstrated with the help of the propagator matrix approach based non-modal analysis (NMA). We show that onset of instability is a monotonically decreasing function of Pe and the onset time determined by NMA emulates the non-linear simulations. Our investigations suggest that perturbations will grow algebraically at early times, contrary to the well-known exponential growth determined from the quasi-steady eigenvalues. One of the over-arching objective of the present work is to determine whether there are alternative mechanisms which can describe the mathematical understanding of the spectrum of the time-dependent stability matrix. Good agreement between the NMA and non-linear simulations is observed. It is shown that within the framework of $$L^2$$-norm, the non-normal stability matrix can be symmetrizable by a similarity transformation and thereby we show that the non-normality of the linearized operator is norm dependent. A framework is thus presented to analyze the exchange of stability which can be determined from the eigenmodes.

Semi-analytical investigation of modified Boussinesq-Burger equations

In this paper, a new analysis of nonlinear modified Boussinesq-Burger equation is revisited via optimal perturbation iteration technique. We first consider artificial parameters and perturbation theory and combine them to deal with nonlinear partial differential equations. After that, the recommended theory is employed to get new semi-analytical solutions of nonlinear partial differential equations. As will be seen from the results, this technique needs no discretization or linearization and can be directly applied to many nonlinear differential equations.

An optimal perturbation bound for the partial isometry associated to the generalized polar decomposition

In terms of singular values and determinants, a new perturbation bound for partial isometry polar factor is derived and is proved to be optimal. The sharpness of this newly obtained perturbation bound is illustrated by numerical tests.

OPTIMAL ITERATIVE PERTURBATION TECHNIQUE FOR SOLVING JEFFERY–HAMEL FLOW WITH HIGH MAGNETIC FIELD AND NANOPARTICLE

In this research paper, a different semi-analytical analysis of modified magnetohydrodynamic Jeffery–Hamel flow is conducted via the newly developed technique. We use the optimal iterative perturbation method with multiple parameters to see the effects of the magnetic field and nanoparticle on the Jeffery–Hamel flow. Comparing our new approximate solutions with some earlier works proved the excellent accuracy of the newly proposed technique. Convergence analysis of the proposed method is also discussed and error estimation is given to anticipate the accuracy of higher-order approximate solutions.