Perturbation Framework Research Articles

Quantifying the impact of environmental changes on migratory species: a model perturbation framework

Migratory species use different habitats and pathways across their migratory route. Pathway contribution metrics are transient metrics of population growth, derived from population models, and quantify the predicted contribution of an individual, travelling along a specified migratory route, to the total population over a specified length of time. Environmental disturbances or management actions may occur temporally or spatially throughout the process of migration. The impact that a given perturbation may have on pathway contribution metrics is not always obvious owing to the propagation of the perturbation through the migratory cycle. Here, we develop a general modelling framework that incorporates perturbations into a class of matrix migratory population models, and which quantifies the effect that perturbations to the model, in terms of the transition rates of habitats and pathways, have on pathway contribution metrics. We also detail how to calculate the sensitivity of pathway contribution metrics to the perturbations considered. Our framework may be used to provide insights into the impact that environmental disturbances or management actions have on migratory populations. These insights may be used to inform management actions which either buffer against possible deleterious disturbances or increase the population size through targeted interventions. Our theoretical results are illustrated via hypothetical examples and a model inspired by the monarch butterfly; we uncover results that are not clear from the calculation of the pathway contribution metrics alone.

Stability of the composite wave of two planar viscous shock waves for the compressible Navier–Stokes system

We are concerned with the nonlinear stability of the composite wave consisting of two planar viscous shock waves to the three-dimensional compressible Navier–Stokes system. It is shown that if the shock strengths are suitably small but not necessarily of the same order of magnitude, and the initial perturbations are suitably small without the zero-mass conditions, then there exists a unique, globally strong solution in time to the compressible Navier–Stokes system, which asymptotically approaches the corresponding composite wave up to time-dependent shifts in L^{\infty} norm. The proof employs the weighted relative entropy method, an L^{2} -contraction technique with time-dependent shifts to the shocks developed by Kang and Vasseur [J. Eur. Math. Soc. (JEMS) 23 (2021), 585–638; Invent. Math. 224 (2021), 55–146]. We perform the stability analysis within the original H^{2} -perturbation framework instead of using the anti-derivative technique. Compared with the previous work of the author and Wang [J. Eur. Math. Soc. (JEMS) (2023), DOI 10.4171/JEMS/1486] for the single shock case, a major difficulty is the construction of shifts to ensure that the two shock waves are well separated.

A joint lightly perturbation and synthesizing optimization attack framework for point cloud perturbation

Due to the inherent characteristic of deep learning models, prior researchers have focused on studying adversarial strategies aimed at uncovering their vulnerabilities. However, these strategies have been primarily been confined to the realm of image processing. When considering 3D vision structures, such as point clouds and meshes, the robustness of learning models also diminishes significantly in the presence of discernible outliers, and these strategies often fail to generalize across different DNN architectures. In this paper, a simple adversarial perturbation and optimization attack framework, Tangent Plane Perturbation and Synthesizing Optimization Attack (TPSOA) is presented. TPSOA provides a point cloud projection direction for point clouds, subtly inducing imperceptible perturbations on their tangent planes. Furthermore, acknowledging the geometry form of adversarial generation and its potential for 3D printing in physical scenarios, we reconstruct the adversarial mesh alongside the adversarial point cloud. A synthesized loss, incorporating both the Sobolev loss and the Chamfer loss, is utilized to optimize the distances between the adversarial generation and ground truth. Through a combination of visualization and quantitative analysis experiments, we validate the imperceptible nature of TPSOA, the necessity of the loss terms, and the magnitude of perturbations. These experiments that TPSOA enables targeted point cloud attacks that are imperceptible and of minor magnitude. In real-world adversarial attack and defense scenarios, as evidenced by both white-box and black-box experiments, the results show that TPSOA surpasses other state-of-the-art adversarial attack models, effectively manipulating a greater number of victim models while maintaining high efficiency, all without compromising its original structural integrity.

Nonlinear stability of planar viscous shock wave to three-dimensional compressible Navier–Stokes equations

We prove the nonlinear stability of the planar viscous shock up to a time-dependent shift for the three-dimensional (3D) compressible Navier–Stokes equations under the generic perturbations, in particular, without zero mass conditions. Moreover, the time-dependent shift function keeps the shock profile shape time-asymptotically. Our stability result is unconditional for the weak planar Navier–Stokes shock. Our proof is motivated by the a -contraction method (a kind of weighted L^{2} -relative entropy method) with time-dependent shift for the stability of viscous shock in the one-dimensional (1D) case. Instead of the classical anti-derivative techniques, we perform the stability analysis of the planar Navier–Stokes shock in the original H^{2} -perturbation framework and therefore zero mass conditions are not necessarily needed, which, in turn, brings out the essential difficulties due to the compressibility of viscous shock. Furthermore, compared with 1D case, there are additional difficulties coming from the wave propagations along the multi-dimensional transverse directions and their interactions with the viscous shock. To overcome these difficulties, a multi-dimensional version of the sharp weighted Poincaré inequality, a -contraction techniques with time-dependent shift, and some essential physical structures of the multi-dimen­sional Navier–Stokes system are fully used.

A spatial perturbation framework to validate implantation of the epileptogenic zone

Stereo-electroencephalography (SEEG) is the gold standard to delineate surgical targets in focal drug-resistant epilepsy. SEEG uses electrodes placed directly into the brain to identify the seizure-onset zone (SOZ). However, its major constraint is limited brain coverage, potentially leading to misidentification of the ‘true’ SOZ. Here, we propose a framework to assess adequate SEEG sampling by coupling epileptic biomarkers with their spatial distribution and measuring the system’s response to a perturbation of this coupling. We demonstrate that the system’s response is strongest in well-sampled patients when virtually removing the measured SOZ. We then introduce the spatial perturbation map, a tool that enables qualitative assessment of the implantation coverage. Probability modelling reveals a higher likelihood of well-implanted SOZs in seizure-free patients or non-seizure free patients with incomplete SOZ resections, compared to non-seizure-free patients with complete resections. This highlights the framework’s value in sparing patients from unsuccessful surgeries resulting from poor SEEG coverage.

Physically constrained eigenspace perturbation for turbulence model uncertainty estimation

Aerospace design is increasingly incorporating design under uncertainty-based approaches to lead to more robust and reliable optimal designs. These approaches require dependable estimates of uncertainty in simulations for their success. The key contributor of predictive uncertainty in computational fluid dynamics (CFD) simulations of turbulent flows are the structural limitations of Reynolds-averaged Navier–Stokes models, termed model-form uncertainty. Currently, the common procedure to estimate turbulence model-form uncertainty is the eigenspace perturbation framework (EPF), involving perturbations to the modeled Reynolds stress tensor within physical limits. The EPF has been applied with success in design and analysis tasks in numerous prior works from the industry and academia. Owing to its rapid success and adoption in several commercial and open-source CFD solvers, in-depth verification and validation of the EPF is critical. In this work, we show that under certain conditions, the perturbations in the EPF can lead to Reynolds stress dynamics that are not physically realizable. This analysis enables us to propose a set of necessary physics-based constraints, leading to a realizable EPF. We apply this constrained procedure to the illustrative test case of a converging-diverging channel, and we demonstrate that these constraints limit physically implausible dynamics of the Reynolds stress tensor, while enhancing the accuracy and stability of the uncertainty estimation procedure.

Downstream network transformations dissociate neural activity from causal functional contributions

Neuroscientists rely on distributed spatio-temporal patterns of neural activity to understand how neural units contribute to cognitive functions and behavior. However, the extent to which neural activity reliably indicates a unit's causal contribution to the behavior is not well understood. To address this issue, we provide a systematic multi-site perturbation framework that captures time-varying causal contributions of elements to a collectively produced outcome. Applying our framework to intuitive toy examples and artificial neural networks revealed that recorded activity patterns of neural elements may not be generally informative of their causal contribution due to activity transformations within a network. Overall, our findings emphasize the limitations of inferring causal mechanisms from neural activities and offer a rigorous lesioning framework for elucidating causal neural contributions.

Cosmological complexity in K-essence

We calculate the cosmological complexity under the framework of scalar curvature perturbations for a K-essence model with constant potential. In particular, the squeezed quantum states are defined by acting a two-mode squeezed operator which is characterized by squeezing parameters rk and ϕk on vacuum state. The evolution of these squeezing parameters are governed by the Schrödinger equation, in which the Hamiltonian operator is derived from the cosmological perturbative action. With aid of the solutions of rk and ϕk, one can calculate the quantum circuit complexity between unsqueezed vacuum state and squeezed quantum states via the wave-function approach. One advantage of K-essence is that it allows us to explore the effects of varied sound speeds on evolution of cosmological complexity. Besides, this model also provides a way for us to distinguish the different cosmological phases by extracting some basic information, like the scrambling time and Lyapunov exponent etc, from the evolution of cosmological complexity.

Asymptotic analysis of detonation development at SI engine conditions using computational singular perturbation

The occurrence and intensity of the detonation phenomenon at spark-ignition (SI) engine conditions is investigated, with the objective to successfully predict super-knock and to elucidate the effect of kinetics and transport at the ignition front. The computational singular perturbation (CSP) framework is employed in order to investigate the chemical and transport mechanisms of deflagration and detonation cases in the context of 2D high-fidelity numerical simulations. The analysis revealed that the detonation development is characterised by: (i) stronger explosive dynamics and (ii) enhanced role of convection. The role of chemistry was also found to be pivotal to the detonation development which explained the stronger explosive character of the system, the latter being an indication of the system's reactivity. The role of convection was found to be enhanced at the edge of the detonating front. Moreover, the increased contribution of convection was found to be related mainly to heat convection. Remarkably, the detonation front was mainly characterised by dissipative and not explosive dynamics. Finally, diffusion was found to have negligible role to both examined cases.

On Stability of Homogeneous Systems in Presence of Parasitic Dynamics

In this paper, the effect of a homogeneous parasitic dynamics on the stability of a homogeneous system, when homogeneity degrees are possibly different, is studied via ISS approach in the framework of singular perturbations. Thus, the possibilities to reduce the order of the interconnected system considering only the reduced-order dynamics and neglecting the parasitic ones are examined. Proposed analysis discovers three kinds of stability in the behavior of such an interconnection by assuming that both, reduced-order and unforced parasitic dynamics, are globally asymptotically stable. In the first case, global asymptotic stability of the interconnection can be concluded, when the homogeneity degrees of both systems coincide and the singular perturbation parameter is small enough. In the second case, only local stability of interconnection can be ensured, when the homogeneity degree of reduced-order dynamics is greater than the homogeneity degrees of the parasitic ones. Helpfully, the proposed approach allows to estimate the domain of attraction for the system's trajectories as a function of the singular perturbation parameter. In the third case, only practical stability can be guaranteed, when the homogeneity degree of reduced-order dynamics is smaller than the homogeneity degrees of the parasitic ones. Additionally, the proposed approach provides an estimation of the asymptotic bound of the system's trajectories in terms of the singular perturbation parameter.

A Perturbation Framework for Convex Minimization and Monotone Inclusion Problems with Nonlinear Compositions

We introduce a framework based on Rockafellar’s perturbation theory to analyze and solve general nonsmooth convex minimization and monotone inclusion problems involving nonlinearly composed functions as well as linear compositions. Such problems have been investigated only from a primal perspective and only for nonlinear compositions of smooth functions in finite-dimensional spaces in the absence of linear compositions. In the context of Banach spaces, the proposed perturbation analysis serves as a foundation for the construction of a dual problem and of a maximally monotone Kuhn–Tucker operator, which is decomposable as the sum of simpler monotone operators. In the Hilbertian setting, this decomposition leads to a block-iterative primal-dual algorithm that fully splits all the components of the problem and appears to be the first proximal splitting algorithm for handling nonlinear composite problems. Various applications are discussed. Funding: The work of L. M. Briceño-Arias was supported by Agencia Nacional de Investigación y Desarrollo-Chile [Grant Fondo Nacional de Desarrollo Científico y Tecnológico 1190871, Grant Centro de Modelamiento Matemático ACE210010, Grant Centro de Modelamiento Matemático FB210005, and basal Funds for Centers of Excellence], and the work of P. L. Combettes was supported by the National Science Foundation [Grant DMS-1818946].

Improved self-consistency of the Reynolds stress tensor eigenspace perturbation for uncertainty quantification

The limitations of turbulence closure models in the context of Reynolds-averaged Navier–Stokes (RANS) simulations play a significant part in contributing to the uncertainty of computational fluid dynamics (CFD). Perturbing the spectral representation of the Reynolds stress tensor within physical limits is common practice in several commercial and open-source CFD solvers, in order to obtain estimates for the epistemic uncertainties of RANS turbulence models. Recent research revealed that there is a need for moderating the amount of perturbed Reynolds stress tensor to be considered due to upcoming stability issues of the solver. In this paper, we point out that the consequent common implementation can lead to unintended states of the resulting perturbed Reynolds stress tensor. The combination of eigenvector perturbation and moderation factor may actually result in moderated eigenvalues, which are not linearly dependent on the originally unperturbed and fully perturbed eigenvalues anymore. Hence, the computational implementation is no longer in accordance with the conceptual idea of the Eigenspace Perturbation Framework. We verify the implementation of the conceptual description with respect to its self-consistency. Adequately representing the basic concept results in formulating a computational implementation to improve self-consistency of the Reynolds stress tensor perturbation.

Global bounded solutions to the Boltzmann equation for a polyatomic gas

In this paper, we consider the Boltzmann equation modeling the motion of a polyatomic gas where the integration collision operator in comparison with the classical one involves an additional internal energy variable [Formula: see text] and a parameter [Formula: see text] standing for the number of internal degrees of freedom. In perturbation framework, we establish the global well-posedness for bounded mild solutions near global equilibria on torus. The proof is based on the [Formula: see text] approach. Precisely, we first study the [Formula: see text] decay property for the linearized equation, then use the iteration technique for the linear integral operator to get the linear weighted [Formula: see text] decay, and in the end obtain [Formula: see text] bounds as well as exponential time decay of solutions for the nonlinear problem with the help of the Duhamel’s principle. Throughout the proof, we present a careful analysis for treating the extra effect of internal energy variable I and the parameter [Formula: see text].

Evaluation of physics constrained data-driven methods for turbulence model uncertainty quantification

In order to achieve a virtual certification process and robust designs for turbomachinery, the uncertainty bounds for Computational Fluid Dynamics have to be known. The formulation of turbulence closure models implies a major source of the overall uncertainty of Reynolds-averaged Navier–Stokes simulations. We discuss the common practice of applying a physics constrained eigenspace perturbation of the Reynolds stress tensor in order to account for the model form uncertainty of turbulence models. Since the basic methodology often leads to overly generous uncertainty estimates, we extend a recent approach of adding a machine learning strategy. The application of a data-driven method is motivated by striving for the detection of flow regions, which are prone to suffer from a lack of turbulence model prediction accuracy. In this way any user input related to choosing the degree of uncertainty is supposed to become obsolete. This work especially investigates an approach, which tries to determine an a priori estimation of prediction confidence, when there is no accurate data available to judge the prediction. The flow around the NACA 4412 airfoil at near-stall conditions demonstrates the successful application of the data-driven eigenspace perturbation framework. Furthermore, we especially highlight the objectives and limitations of the underlying methodology.

MODIFIED HOMOTOPY PERTURBATION METHOD BASED SOLUTION OF LINEARLY DAMPED DUFFING OSCILLATOR AND COMPARISON WITH SIMULATED SOLUTION*

The Dung oscillator provides a basis for studying nonlinear dynamics as its phase space trajectory is fairly complex and depends on the parameter of the system viz., initial amplitude, phase, frequency, linear damping coefficient and non-linearity parameter. In order to understand the complexity of the system, three variable effective expansions have been introduced in the usual homotopy perturbation framework to obtain the solution of damped Dung system which nds application in several areas in engineering sciences such as vibration of bars, plates and electronic circuits, etc. The necessity of the extended homotopy frame work has been further discussed for non-conservative system. Simulation results for different parameters of the systems, such as, linear damping coefficient (μ), amplitude (α) and nonlinearity parameter (ε) are compared with the corresponding results based on perturbative homotopy analysis up to third order by changing (i) the magnitude of linear damping coefficient (μ), (ii) the magnitude of the nonlinearity of the system (ε). Even though the simulated result matches satisfactorily with the perturbative solution over the entire evolutionary time scale, noticeable divergence and phase shift are observed only lately for increased value μ and ε, respectively.

Perturbations in the interacting vacuum

In this study, we present the evolution of cosmological perturbations in a universe consisting of standard matter and interacting vacuum. We use the 1 + 3 covariant formalism in perturbation framework and consider two different models for the interacting vacuum; namely, a linear interacting model and interaction with creation pressure model. For both models, we derive the evolution equations governing the growth of linear perturbations for both radiation- and dust-dominated universe. We find numerical solutions in appropriate limits, namely long and short wavelengths. For both models, the perturbations grow with time (decay with redshift), showing that structure formation is possible in an accelerated cosmic background. The perturbation amplitudes — and their relative scalings with those of [Formula: see text]CDM — depend on the values of the interaction parameters considered, and in a way that can be used to constrain the models using existing and future large-scale structure data. In the vanishing limits of the coupling parameters of the interaction, we show that standard [Formula: see text]CDM cosmology, both background and perturbed, is recovered.

Improved local search for the minimum weight dominating set problem in massive graphs by using a deep optimization mechanism

The minimum weight dominating set (MWDS) problem is an important generalization of the minimum dominating set problem with various applications. In this work, we develop an efficient local search scheme that can dynamically adjust the number of added and removed vertices according to the information of the candidate solution. Based on this scheme, we further develop three novel ideas to improve performance, resulting in our so-called DeepOpt-MWDS algorithm. First, we use a new construction method with five reduction rules to significantly reduce massive graphs and construct an initial solution efficiently. Second, an improved configuration checking strategy called CC2V3+ is designed to reduce the cycling phenomenon in local search. Third, a general perturbation framework called deep optimization mechanism (DeepOpt) is proposed to help the algorithm avoid local optima and to converge to a new solution quickly.Extensive experiments based on eight popular benchmarks of different scales are carried out to evaluate the proposed algorithm. Compared to seven state-of-the-art heuristic algorithms, DeepOpt-MWDS performs better on random and classic benchmarks and obtains the best solutions on almost all massive graphs. We investigate three main algorithmic ingredients to understand their impacts on the performance of the proposed algorithm. Moreover, we adapt the proposed general framework DeepOpt to another NP-hard problem to verify its generality and achieve good performance.

Black-hole perturbation theory with post-Newtonian theory: Towards hybrid waveforms for neutron-star binaries

We consider the motion of nonspinning, compact objects orbiting around a Kerr black hole with tidal couplings. The tide-induced quadrupole moment modifies both the orbital energy and outgoing fluxes, so that over the inspiral timescale there is an accumulative shift in the orbital and gravitational wave phase. Previous studies on compact object tidal effects have been carried out in the Post-Newtonian (PN) and Effective-One-Body (EOB) formalisms. In this work, within the black hole perturbation framework, we propose to characterize the tidal influence in the expansion of mass ratios, while higher-order PN corrections are naturally included. For the equatorial and circular orbit, we derive the leading order, frequency dependent tidal phase shift which agrees with the Post-Newtonian result at low frequencies but deviates at high frequencies. We also find that such phase shift has weak dependence ($\le 10\%$) on the spin of the primary black hole. Combining this black hole perturbation waveform with the Post-Newtonian waveform, we propose a frequency-domain, hybrid waveform that shows comparable accuracy as the EOB waveform in characterizing the tidal effects, as calibrated by numerical relativity simulations. Further improvement is expected as the next-leading order in mass ratio and the higher-PN tidal corrections are included. This hybrid approach is also applicable for generating binary black hole waveforms.

Ediacaran carbon cycling and Shuram excursion recorded in the Tarim Block, northwestern China

The largest perturbation of the carbon cycle in geological history, named as the ‘Shuram Carbon Isotope Excursion (SCIE)’, has been widely reported in the Ediacaran Period in different continents over the world. In the Tarim Block, a complete Ediacaran carbonate succession was deposited; however, whether it recorded the SCIE or not still remains controversial, hindering establishment of a reliable stratigraphic framework and understanding of environmental perturbations and evolution during the Ediacaran Period. Here, we present two high-resolution carbon isotopic stratigraphy profiles from the upper Sugetbrak through the Qigebrak formations in the northwestern Tarim Block. Two negative excursions (TN1, TN2), and two positive excursions (TP1, TP2) are identified from both sections and supposed to be of the primary origin. Of these, the TN1 was recorded in a storm-dominated inner-to-middle ramp beginning with the marine transgression. It marks the largest negative excursion with a nadir down to ca. −12‰ which can be correlated to the SCIE of late Ediacaran in terms of its similar magnitude, variation pattern, time constraints and depositional characteristics. Other carbon isotope anomalies can also be correlated with coeval anomalies elsewhere around the world. The temporal coincidence of TN1 (or SCIE) with onset of transgressive, storm-dominated successions suggests that enhanced oceanic upwelling and storm agitations together may have greatly increased the depth of ventilated seafloor and extent/volume of oxygenated DOC pool in shallow marine realms, leading to the extremely δ13C excursion of carbonates. Our new carbon isotopic chemostratigraphic profiles of upper Ediacaran from the Tarim Block thus not only provide a more reliable time stratigraphic framework helpful for documenting depositional history and environment-life evolution, but also promote comparisons with other continental blocks globally.

A Distributed Active Perception Strategy for Source Seeking and Level Curve Tracking

Algorithms for multiagent systems to locate a source or to follow a desired level curve of spatially distributed scalar fields generally require sharing field measurements among the agents for gradient estimation. Yet, in this article, we propose a distributed active perception strategy that enables swarms of various sizes and graph structures to perform source seeking and level curve tracking without the need to explicitly estimate the field gradient or explicitly share measurements. The proposed method utilizes a consensus-like principal component analysis perception algorithm that does not require explicit communication in order to compute a local body frame. This body frame is used to design a distributed control law where each agent modulates its motion based only on its instantaneous field measurement. Several stability results are obtained within a singular perturbation framework that justifies the convergence and robustness of the strategy. Additionally, efficiency is validated through robots experiments.