Eigenvalue Perturbation Research Articles

Criticality of the O(N) universality via global solutions to nonperturbative fixed-point equations

Fixed-point equations in the functional renormalization group approach are integrated from large to vanishing field, where an asymptotic potential in the limit of large field is implemented as initial conditions. This approach allows us to obtain a global fixed-point potential with high numerical accuracy, that incorporates the correct asymptotic behavior in the limit of large field. Our calculated global potential is in good agreement with the Taylor expansion in the region of small field, and it also coincides with the Laurent expansion in the regime of large field. Laurent expansion of the potential in the limit of large field for general case, that the spatial dimension d is a continuous variable in the range 2≤d≤4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\le d \\le 4$$\\end{document}, is obtained. Eigenfunctions and eigenvalues of perturbations near the Wilson–Fisher fixed point are computed with the method of eigenperturbations. Critical exponents for different values of d and N of the O(N) universality class are calculated.

Mode-wise principal subspace pursuit and matrix spiked covariance model

Abstract This paper introduces a novel framework called Mode-wise Principal Subspace Pursuit (MOP-UP) to extract hidden variations in both the row and column dimensions for matrix data. To enhance the understanding of the framework, we introduce a class of matrix-variate spiked covariance models that serve as inspiration for the development of the MOP-UP algorithm. The MOP-UP algorithm consists of two steps: Average Subspace Capture (ASC) and Alternating Projection. These steps are specifically designed to capture the row-wise and column-wise dimension-reduced subspaces which contain the most informative features of the data. ASC utilizes a novel average projection operator as initialization and achieves exact recovery in the noiseless setting. We analyse the convergence and non-asymptotic error bounds of MOP-UP, introducing a blockwise matrix eigenvalue perturbation bound that proves the desired bound, where classic perturbation bounds fail. The effectiveness and practical merits of the proposed framework are demonstrated through experiments on both simulated and real datasets. Lastly, we discuss generalizations of our approach to higher-order data.

Spatio-temporal patterns and global bifurcation of a nonlinear cross-diffusion predator–prey model with prey-taxis and double Beddington–DeAngelis functional responses

The aim of this article is investigating the spatio-temporal patterns of a nonlinear cross-diffusion predator–prey model with prey-taxis and double Beddington–DeAngelis functional responses. First, by utilizing user-friendly version of Crandall–Rabinowitz bifurcation theory as an analytical method, the spatio-temporal patterns of positive steady state are obtained. Then, by regarding the cross-diffusion coefficient d1 as a bifurcation parameter, we will derive a bifurcation theorem for the corresponding nonlinear cross-diffusion model. Moreover, by applying spectrum theory, perturbation of simple eigenvalues and linearized stability, it is discovered that the bifurcation solutions possess local stability near the bifurcating point in proper conditions. These theoretical results mean that the cross-diffusion mechanism can create a coexistence environment for the preys and predator under some circumstances. Finally, a numerical example is proposed to verify our results.

Existence and stability of steady-state bifurcations in a prey–predator system with a nonmonotonic functional response

A cross-diffusion prey–predator system exhibiting the prey group defense under homogeneous Neumann boundary conditions is studied. By considering the diffusive rate of the prey as a bifurcation parameter, we investigate sudden changes in the population dynamics of the prey and predator which can have a substantial effect on population size of the species. First a priori estimate for positive steady states is obtained. Next we prove the existence of a pitchfork bifurcation of positive steady states at a simple eigenvalue. The structure of the global steady-state bifurcation is discussed. We also investigate the stability of the trivial solution line and nontrivial steady-state solutions via the eigenvalue perturbation theory. To illustrate our theoretical results some numerical simulations are given. Numerical examples contain a supercritical and a subcritical pitchfork bifurcation.

First order structure-preserving perturbation theory for eigenvalues of symplectic matrices: Part II

The first order eigenvalue perturbation theory for structure-preserving perturbations of real or complex symplectic matrices started in [21] is continued and completed. While in [21] all eigenvalues were considered that qualitatively behave similarly under structure-preserving and structure-ignoring perturbations, the focus of this paper is on one particular case when the behavior under structure-preserving perturbations differs significantly from the one under structure-ignoring perturbations. This case occurs for the exceptional eigenvalues ±1 and is caused by restrictions in the possible Jordan structure of symplectic matrices. The main result gives asymptotic expansions of specific perturbed eigenvalues that are not yet covered by the results from [21]. While the main theorem requires that the rank of the perturbation is equal to the rank of its first order term, it is also discussed what can be said if this hypothesis is not satisfied.

Zonostrophic instabilities in magnetohydrodynamic Kolmogorov flow

A classic stability problem relevant to many applications in geophysical and astrophysical fluid mechanics is that of Kolmogorov flow, a unidirectional purely sinusoidal velocity field written here as u = ( 0 , sin ⁡ x ) in the infinite ( x , y ) -plane. Near onset, instabilities take the form of large-scale transverse flows, in other words flows in the x-direction with a small wavenumber k in the y-direction. This is similar to the phenomenon known as zonostrophic instability, found in many examples of randomly forced fluid flows modelling geophysical and planetary systems. The present paper studies the effect of incorporating a magnetic field B 0 , in particular a y-directed “vertical” field or an x-directed “horizontal” field. The linear stability problem is truncated to determining the eigenvalues of finite matrices numerically, allowing exploration of the instability growth rate p as a function of the wavenumber k in the y-direction and a Bloch wavenumber ℓ in the x-direction, with − 1 / 2 < ℓ ≤ 1 / 2 . In parallel, asymptotic approximations are developed, valid in the limits k → 0 , ℓ → 0 , using matrix eigenvalue perturbation theory. Results are presented showing the robust suppression of the hydrodynamic Kolmogorov flow instability as the imposed magnetic field B 0 is increased from zero. However with increasing B 0 , further branches of instability become evident. For vertical field there is a strong-field branch of destabilised Alfvén waves present when the magnetic Prandtl number P m < 1 , as found recently by A.E. Fraser, I.G. Cresswell and P. Garaud (J. Fluid Mech. 949, A43, 2022), and a further branch for 1 $ ]]> P m > 1 in the presence of an additional imposed x-directed fluid flow U 0 . For horizontal magnetic field, a branch of field-driven, tearing mode instabilities emerges as B 0 increases. The above instabilities are present for Bloch wavenumber ℓ = 0 ; however allowing ℓ to be non-zero gives rise to a further branch of instabilities in the case of horizontal field. In some circumstances, even when the system is hydrodynamically stable arbitrarily weak magnetic fields can give growing modes, via the instability taking place on large scales in x and y. Detailed comparisons are given between theory for small k and ℓ, and numerical results.

Topology uniformity pinning control for multi-agent flocking

The optimal selection of pinning nodes for multi-agent flocking is a challenging NP-hard problem. Current pinning node selection strategies mainly rely on centrality measures of complex networks, which lack rigorous mathematical proof for effective flocking control. This paper proposes a pinning node selection strategy based on matrix eigenvalue theory. First, the effect of the pinning node on the eigenvalue of the Laplacian matrix is analyzed. Then, a synchronization index representing the topology uniformity of the multi-agent system is proposed to exert maximum influence on the system synchronizability. A practical optimal pinning node selection method based on the synchronization index is proposed and analyzed using the eigenvalue perturbation method. Finally, simulations demonstrate that the convergence rate of the system obtained using the optimal synchronizability pinning node selection method is better than that achieved with the maximum degree centrality node selection strategy.

Asymptotic behavior for eigenvalues and eigenfunctions associated to Stokes operator in the presence of a rapidly oscillating boundary

In this paper, we analyze rigorously the asymptotic behaviors for perturbation of eigenvalues and eigenfunctions associated to the Stokes eigenvalue problem in the presence of a rapidly oscillating boundary. The aim is to construct asymptotic approximations, as , of the eigenvalues and corresponding eigenfunctions for the case where the eigenvalue of the reference problem is simple or multiple. Taking advantage of small oscillations, we use the method of matching of asymptotic expansions to construct the associated leading terms. We believe that our results are ambitious tools for determining shape and/or size of the small perturbed part of the domain by taking eigenvalue measurements.

A distributed consensus based algorithm for economic dispatch over time‐varying digraphs

AbstractIn this paper, a consensus based fully distributed optimization algorithm is proposed for solving economic dispatch problem (EDP) in smart grid. Since the incremental cost of all buses reach consensus when the optimal solution is achieved, it is selected as a consensus variable. An additional variable at each bus, called “surplus” is added to record the local power mismatch, which is used as a feedback variable to purse the balance between power supply and demand. Different from most of the existing distributed methods which require the communication network to be balanced, the algorithm uses a row random matrix and a column random matrix to precisely steer all the agents to asymptotically converge to a global optimal solution over a time‐varying directed communication network. Due to the use of a fixed step size, the proposed algorithm also outperforms other algorithms in terms of convergence speed. The graph and eigenvalue perturbation theories are employed for the algorithm convergence analysis, and the upper bound of the parameters required for convergence is given theoretically. Finally, the performance and scalability of the proposed distributed algorithm are verified by several case studies conducted on the IEEE 14‐bus power system and a 200‐node test system.

An optimal spatial-filtering method derived from eigenvalue perturbation for extending the Courant-Friedrichs-Lewy stability limit

Explicit time-marching schemes are widely used in numerical simulations. However, their maximum time step is constrained by the Courant-Friedrichs-Lewy (CFL) stability limit. Two methods have been developed to extend this limit: (1) the eigenvalue perturbation method, which exhibits high numerical accuracy but incurs unaffordable memory demand and computational costs, even for middle-scale models, and (2) the spatial-filtering method, which can be implemented easily but results in significant numerical errors under large time steps. However, the intrinsic relationship between these two methods remains unknown. We reveal the intrinsic relationship between these two methods by considering the eigenvalue perturbation method for a homogeneous model. Using this relationship, we derive an analytical spatial filter for the homogeneous model and develop an optimal spatial filter for heterogeneous models. Compared to the classical spatial-filtering method, which removes all high-wavenumber components, our method retains all high-wavenumber components that contribute to wave propagation while eliminating other high-wavenumber components that cause instability. For the same numerical accuracy, the maximum time step allowed by the proposed method is approximately twice that of the classical spatial-filtering method. Compared to the eigenvalue perturbation method, our method can be used in large-scale models without additional memory consumption and computational cost. This can significantly accelerate the pseudospectral method with significantly larger time steps beyond the CFL stability limit, which is particularly promising for large-scale models with a fine grid.

Perturbed eigenvalues of polyharmonic operators in domains with small holes

We study singular perturbations of eigenvalues of the polyharmonic operator on bounded domains under removal of small interior compact sets. We consider both homogeneous Dirichlet and Navier conditions on the external boundary, while we impose homogeneous Dirichlet conditions on the boundary of the removed set. To this aim, we develop a notion of capacity which is suitable for our higher-order context, and which permits to obtain a description of the asymptotic behaviour of perturbed simple eigenvalues in terms of a capacity of the removed set, in dependence of the respective normalized eigenfunction. Then, in the particular case of a subset which is scaling to a point, we apply a blow-up analysis to detect the precise convergence rate, which turns out to depend on the order of vanishing of the eigenfunction. In this respect, an important role is played by Hardy–Rellich inequalities in order to identify the appropriate functional space containing the limiting profile. Remarkably, for the biharmonic operator this turns out to be the same, regardless of the boundary conditions prescribed on the exterior boundary.

A Mathematical Tool to Investigate the Stability Analysis of Structured Uncertain Dynamical Systems with M-Matrices

The μ-value or structured singular value is a prominent mathematical tool to analyze and synthesize both the robustness and performance of time-invariant systems. We establish and analyze new results concerning structured singular values for the Hadamard product of real square M-matrices. The new results are obtained for structured singular values while considering a set of block diagonal uncertainties. The targeted uncertainties are of two types, that is, pure real scalar block uncertainties and real full-block uncertainties. The eigenvalue perturbation result is utilized in order to determine the behavior of the spectrum of perturbed matrices (A∘B)Δ(t) and ((A∘B)TΔ(t)+Δ(t)(A∘B)).

Distributed algorithm for economic dispatch with transmission losses under digraph

By taking transmission losses and directed communication into considered, a more practical scenario of economic dispatch problem is studied in the paper. It is formulated as a non-convex optimization problem. Using convex relaxation, the non-convex optimization problem is transformed into convex optimization problem, and the conditions to ensure that they have the same solution are given. Then, a consensus-based distributed algorithm with time-varying feedback gains is presented, which can be used to the general directed communication networks. Especially, the convergence and optimality of the proposed algorithm are proved by using multi-parameter eigenvalue perturbation theory and graph theory. Finally, simulation results validate the theoretical results and illustrate the effectiveness and advantages of the proposed algorithm.

A Systematic Stability Enhancement Method for Microgrids With Unknown-Parameter Inverters

With massive electronic inverters, microgrids are threatened by the instability problems caused by the impedance interactions among inverters and the network. For the microgrids with black-box inverters (whose parameters are unknown due to industry secrets), it is hard to assess, much less enhance, the stability of such systems. This article proposes a systematic impedance-based stability assessment and enhancement method for the microgrids with black-box inverters. First, the return-ratio matrix <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">net</sub> of the system with both current-controlled and voltage-controlled inverters is formulated based on the nodal admittance matrix. And then, the sensitivities of the critical eigenvalues of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">net</sub> are calculated with respect to individual admittances/impedances of inverters, which can identify the “trouble maker(s).” Moreover, the low voltage active damper (LVAD) is proposed for the stability enhancement of the system. An eigenvalue perturbation sensitivity analysis method is presented to calculate the sensitivities of the critical eigenvalues with respect to nodal parallel admittances, which identifies the optimal installation position for LVAD, and accordingly provides the guidance for the design of LVAD. The effectiveness of the proposed method is verified using a modified IEEE 6-bus system in PSACD/EMTDC and RT-Lab platforms.

Noise equalization scheme based on complex-valued ANN for multiple-eigenvalue modulated nonlinear frequency division multiplexing systems.

In multiple-eigenvalue modulated nonlinear frequency division multiplexing (NFDM) systems, the noise degrades the accuracy of the nonlinear Fourier transform (NFT) algorithm, resulting in perturbations in the received eigenvalues and the corresponding discrete spectrum. Moreover, with the increase in the number of eigenvalues and the order of the modulation formats, the impact of noise on the performance of the system is even more. A noise equalization scheme based on complex-valued artificial neural network (c-ANN) for multiple-eigenvalue modulated NFDM systems is proposed. This sceheme inputs the eigenvalues perturbation and the impaired discrete spectrum corresponding to the eigenvalues into the c-ANN in complex form. The scheme constructs a complex-valued logic structure with both amplitude and phase information, overlapping reuse input features and, thereby, effectively reducing the effect of noise on the multiple-eigenvalue NFDM system. The effectiveness of the scheme is verified in long-haul seven-eigenvalue modulated single-polarization NFDM simulation systems with 1GBaud 16APSK/16QAM/64APSK/64QAM modulation formats, and the results show that the scheme outperforms the NFT receiving without equalization by 1 to 2 orders of magnitude in terms of bit error rate (BER). Among them, the transmission distance of the 64APSK signal after equalization exceeds 800km while the BER meets 7% forward error correction (FEC) threshold, which is 600km longer than that of the disequilibrium case, and the spectral efficiency (SE) can reach 1.85bit/s/Hz. Compared with other schemes, the proposed scheme has better equalization performance under the same complexity, and the complexity can be reduced by half or even under the same performance.

Approach to computing spectral shifts for black holes beyond Kerr

Recent measurements of gravitational-wave ringdown following the merger of binary black holes raise the prospect of precision black hole spectroscopy in the near future. To perform the most sensitive tests of the nature of black holes using ringdown measurements, it is critical to compute the deviations to the spectrum of black holes in particular extensions of relativity. These spectral shifts are also needed to interpret any violations of the predictions of relativity that may be detected during ringdown. Here we present a first step towards computing the shifts to the spectrum of Kerr black holes with arbitrary spins, by deriving a modified Teukolsky equation governing the perturbations of black holes in theories beyond GR. Our approach applies to a class of theories which includes dynamical Chern-Simons gravity and shift-symmetric scalar Gauss-Bonnet gravity, in the case where the deviations from relativity are small. This allows for a perturbative approach to solving the equations of motion. Further, we show how to use the modified equation to compute the leading-order spectral shifts of Kerr black holes, using eigenvalue perturbation methods. Our formalism provides a practical approach to predicting ringdown for black holes in a range of promising extensions to relativity, enabling future precision searches for their signatures in black hole ringdown.

Model-form uncertainty quantification of Reynolds-averaged Navier–Stokes modeling of flows over a SD7003 airfoil

Reynolds-averaged Navier–Stokes (RANS) models are known to be inaccurate in complex flows, for instance, laminar-turbulent transition, and RANS uncertainty quantification (UQ) is essential to estimate the uncertainty in their predictions. In this study, a recent physics-based UQ framework that introduces eigenvalue, eigenvector, and turbulence kinetic energy perturbations to the modeled Reynolds stress tensor has been used to estimate the uncertainty in the flow field. We introduce a regression-based marker function that focuses on the turbulence kinetic energy perturbation for the simulation of laminar-turbulent transitional flows over an Selig–Donovan 7003 airfoil. We observed a monotonic behavior of the magnitude of the predicted uncertainty bounds varying with the turbulence kinetic energy perturbation. Importantly, the predicted uncertainty bounds show a synergy behavior that dramatically increases the size of uncertainty bounds and can successfully encompass the reference data when the eigenvalue perturbations are augmented with the marker function.

Quantification of Reynolds-averaged-Navier–Stokes model-form uncertainty in transitional boundary layer and airfoil flows

It is well known that the Boussinesq turbulent-viscosity hypothesis can introduce uncertainty in predictions for complex flow features such as separation, reattachment, and laminar-turbulent transition. This study adopts a recent physics-based uncertainty quantification (UQ) approach to address such model-form uncertainty in Reynolds-averaged Naiver–Stokes (RANS) simulations. Thus far, almost all UQ studies have focused on quantifying the model-form uncertainty in turbulent flow scenarios. The focus of the study is to advance our understanding of the performance of the UQ approach on two different transitional flow scenarios: a flat plate and a SD7003 airfoil, to close this gap. For the T3A (flat-plate) flow, most of the model-form uncertainty is concentrated in the laminar-turbulent transition region. For the SD7003 airfoil flow, the eigenvalue perturbations reveal a decrease as well as an increase in the length of the separation bubble. As a consequence, the uncertainty bounds successfully encompass the reattachment point. Likewise, the region of reverse flow that appears in the separation bubble is either suppressed or bolstered by the eigenvalue perturbations. This is the first successful RANS UQ study for transitional flows.

Solution approaches for evaporation-driven density instabilities in a slab of saturated porous media

This work considers the gravitational instability of a saline boundary layer formed by an evaporation-induced flow through a fully saturated porous slab. Evaporation of saline waters can result in the formation of salt lakes as salt accumulates near the top surface. As gravitational instabilities can impede the accumulation of salt, establishing a relation between their occurrence and the value of physical parameters such as evaporation rate is crucial. One step toward determining when gravitational instabilities can arise is to compute the ground-state salinity. In this work, this ground state is analytically derived within the framework of Sturm–Liouville theory. Then, a linear stability analysis in conjunction with the quasi-steady state approach is employed to investigate the occurrence of instabilities. These instabilities can develop and grow over time depending on the Rayleigh number and the dimensionless height of the porous medium. To calculate the critical Rayleigh number, which is used to determine the stability of a particular system, the eigenvalues of the linear perturbation equations have to be computed. Here, a novel fundamental matrix method is proposed to solve this perturbation eigenvalue problem and shown to coincide with an established Chebyshev–Galerkin method in their shared range of applicability. Finally, a two-dimensional direct numerical simulation of the full equation system via the finite volume method is employed to validate the time of onset of convective instabilities predicted by the linear theory. Moreover, the fully nonlinear convection patterns are analyzed.

Detecting and Classifying Multiple Class Events in a Power Distribution System Based on an Eigenvalue Fluctuation Model

Detecting and classifying the occurrence of multiple class events is important in enhancing the wide-area situational awareness ability of power distribution systems. Multiple class event classification based on voltage magnitude measurement has not been largely deployed due to the inadequate elucidation between the occurrence of events and the corresponding eigenvalue perturbations in the measurement data. Herein, a detection and classification method for multiple class events in power distribution systems is proposed based on an eigenvalue fluctuation model. We first elucidated the interrelation between event occurrence and the eigenvalue fluctuations in the system voltage magnitude measurement data. Based on the discovery that different classes of events lead to their respective data eigenvalue behavior, we proposed detection criteria C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SRL</sub> , C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">MLP1</sub> , and C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">MLP2</sub> to quantitatively assess these behaviors. Since the different value changes in C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SRL</sub> , C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">MLP1</sub> and C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">MLP2</sub> can indicate the occurrence of different classes of events, we trained and utilized a supervised classifier to achieve multiple event detection and classification. Simulation tests are performed in 7 different test feeders. Eight classes of events can be accurately detected and classified with only voltage magnitude measurement data. An 80% correct detection rate can be obtained with only a 20% measurement device penetration rate.