Matrix Perturbation Theory Research Articles

Mitigating an Epidemic on a Geographic Network Using Vaccination

We consider a mathematical model describing the propagation of an epidemic on a geographical network. The size of the outbreak is governed by the initial growth rate of the disease given by the maximal eigenvalue of the epidemic matrix formed by the susceptibles and the graph Laplacian representing the mobility. We use matrix perturbation theory to analyze the epidemic matrix and define a vaccination strategy, assuming vaccination reduces the susceptibles. When mobility and the local disease dynamics have similar time scales, it is most efficient to vaccinate the whole network because the disease grows uniformly. However, if only a few vertices can be vaccinated, then we show that it is most efficient to vaccinate along an eigenvector corresponding to the largest eigenvalue of the Laplacian. We illustrate these results by calculations on a seven-vertex graph and a realistic example of the French rail network. When mobility is slower than the local disease dynamics, the epidemic grows on the vertex with largest number of susceptibles. The epidemic growth rate is more reduced when vaccinating a larger degree vertex; it also depends on the neighboring vertices. This study and its conclusions provide guidelines for the planning of a vaccination campaign on a network at the onset of an epidemic.

Form-finding of thermal-adaptive pin-bar assemblies based on eigenvalue modification

Lattice structures with tunable expansion properties have been investigated in multidisciplinary fields to control the temperature effects of structures or materials. The expected thermal adaptivity can be achieved by optimizing the structural geometry. A novel method for the form-finding of thermal-adaptive pin-bar assemblies is developed in this paper by considering the control of structural temperature effects as the minimization of the potential energy of the system. Based on the stationarity condition of the potential energy with respect to the nodal coordinates, the compatibility relationship between the thermal elongations of members and the target nodal displacements is proven to be the sufficient and necessary condition for structural thermal adaptivity. The solvability of the compatibility equation is determined by the rank equality between the compatibility matrix and its augmented form, which can be measured by the number of nonzero eigenvalues of its Gramian matrix. The analytical relationship between the eigenvalues of the Gramian matrix and the nodal coordinates is established using the matrix perturbation theory. A numerical strategy based on Newton’s method is proposed in which the eigenvalues are gradually modified by adjusting the nodal coordinates until the rank equality is satisfied. To address the existence of multiple solutions with structural thermal adaptivity, structural symmetry and periodicity constraints are introduced to narrow the solution space. The thermal-adaptive configurations of three illustrative pin-bar assemblies are analyzed using the proposed form-finding method, and the expected thermal deformations are verified for the obtained configurations using the finite element software ABAQUS. Comparing the results obtained by the proposed method with those obtained by nonlinear programming and the genetic algorithm validates the advantages of the proposed method in terms of computational time, optimality of the obtained configuration and applicability to complex structural geometries.

Huizhou resident population, Guangdong resident population and elderly population forecast based on the NAR neural network Markov model

<abstract> <p>We propose a nonlinear auto regressive neural network Markov model (NARMKM) to predict the annual Huizhou resident population, Guangdong resident population and elderly population in China, and improve the accuracy of population forecasting. The new model is built upon the traditional neural network model and utilized matrix perturbation theory to study the natural and response characteristics of a system when the structural parameters change slightly. The delay order and hidden layer number of neurons has a greater effect the prediction result of NAR neural network model. Therefore, we make full use of prior information to constrain and test when making predictions. We choose reasonable parameter settings to obtain more reliable prediction results. Three experiments are conducted to validate the high prediction accuracy of the NARMKM model, with mean absolute percentage error (MAPE), root mean square error (RMSE), <italic>STD</italic> and <italic>R</italic><sup>2</sup>. These results demonstrate the superior fitting performance of the NARMKM model when compared to other six competitive models, including GM (1, 1), ARIMA, Multiple regression, FGM (1, 1), FANGBM and NAR. Our study provides a scientific basis and technical references for further research in the finance as well as population fields.</p> </abstract>

On the stability of the principal ratio

The principal ratio of a connected graph G, γ(G), is the ratio between the largest and smallest coordinates of the principal eigenvector of the adjacency matrix of G. Over all connected graphs on n vertices, γ(G) ranges from 1 to ncn. Moreover, γ(G)=1 if and only if G is regular. This indicates that γ(G) can be viewed as an irregularity measure of G, as first suggested by Cioabă and Gregory (El. J. Lin. Alg. 2007). We are interested in how stable this measure is. In particular, we ask how γ changes when there is a small modification to a regular graph G. We show that this ratio is polynomially bounded if we remove an edge belonging to a cycle of bounded length in G, while the ratio can jump from 1 to exponential if we join a pair of vertices at distance 2. We study the connection between the spectral gap of a regular graph and the stability of its principal ratio. A naive bound shows that given a constant multiplicative spectral gap and bounded degree, the ratio remains polynomially bounded if we add or delete an edge. Using results from matrix perturbation theory, we show that given an additive spectral gap greater than (2+ϵ)n, the ratio stays bounded after adding or deleting an edge.

Dynamic stiffness matrix perturbation theory for time-varying structural analysis

Matrix perturbation theory (MPT) is widely used in sensitivity analysis and reanalysis of structural eigenvalue problems when small changes are imposed on parameters. However, the conventional MPT based on the finite element method is only applicable to the cases with small parameter variations, and it is difficult to avoid cumulative errors caused by multi-step perturbations. A key issue for the application of MPT is how to evaluate the error of perturbation solution and correct the results during the process of multi-step perturbation solution. By invoking the dynamic stiffness method (DSM), Wittrick-Williams algorithm, and the precise integration method (PIM), in this paper, we propose an improved method with controllable accuracy, self-checking and elimination of accumulated errors, referred to as the dynamic stiffness matrix perturbation theory (DSMPT). In present method, the rationality of the results can be estimated preliminarily according to dynamic stiffness equation, which is not present in the original MPT, can greatly enhance the accuracy of the perturbation results and evaluate the errors. Case studies prove that the DSMPT combined with PIM provides a convenient way to realize accuracy self-inspection and rectification of the perturbation results, and the maximum relative error of eigenvalue and response can be controlled to be smaller than 0.2% and 0.025%, respectively. Besides, DSMPT can also be applied to the analysis of coupled and nonlinear dynamic system with time-varying modal characteristics. Finally, a thermally-induced vibration analysis of tensegrity modules during space deployment is taken as an example to further illustrate the advancements of the DSMPT in solving time-varying dynamic problems. Results show that the DSMPT effectuate an almost sixfold reduction in computational time.

Semi-Empirical Shadow Molecular Dynamics: A PyTorch Implementation.

Extended Lagrangian Born-Oppenheimer molecular dynamics (XL-BOMD) in its most recent shadow potential energy version has been implemented in the semiempirical PyTorch-based software PySeQM. The implementation includes finite electronic temperatures, canonical density matrix perturbation theory, and an adaptive Krylov subspace approximation for the integration of the electronic equations of motion within the XL-BOMB approach (KSA-XL-BOMD). The PyTorch implementation leverages the use of GPU and machine learning hardware accelerators for the simulations. The new XL-BOMD formulation allows studying more challenging chemical systems with charge instabilities and low electronic energy gaps. The current public release of PySeQM continues our development of modular architecture for large-scale simulations employing semi-empirical quantum-mechanical treatment. Applied to molecular dynamics, simulation of 840 carbon atoms, one integration time step executes in 4 s on a single Nvidia RTX A6000 GPU.

D-SVM over networked systems with non-ideal linking conditions

This paper considers distributed optimization algorithms, with application in binary classification via distributed support vector machines (D-SVM) over multi-agent networks subject to some link nonlinearities. The agents solve a consensus-constraint distributed optimization cooperatively via continuous-time dynamics, while the links are subject to strongly sign-preserving odd nonlinear conditions. Logarithmic quantization and clipping (saturation) are two examples of such nonlinearities. In contrast to existing literature that mostly considers ideal links and perfect information exchange over linear channels, we show how general sector-bounded models affect the convergence to the optimizer (i.e. the SVM classifier) over dynamic balanced directed networks. In general, any odd sector-bounded nonlinear mapping can be applied to our dynamics. The main challenge is to show that the proposed system dynamics always have one zero eigenvalue (associated with the consensus) and the other eigenvalues all have negative real parts. This is done by recalling arguments from matrix perturbation theory. Then, the solution is shown to converge to the agreement state under certain conditions. For example, the gradient tracking (GT) step size is tighter than the linear case by factors related to the upper/lower sector bounds. To the best of our knowledge, no existing work in distributed optimization and learning literature considers non-ideal link conditions.

Beamspace MIMO Systems With Reduced Beam Selection Complexity

Beamspace multiple-input multiple-output (B-MIMO) systems with proper beam selection (BS) promise to lower the radio-frequency (RF) chains overhead with no noticeable degradation in performance. Most of the existing BS schemes are not practical due to the computational cost arising from iterative search or alternating optimization. Hence, this letter examines the complexity reduction of the existing BS with incremental QR precoder (I-QR-P) and decremental QR precoder (D-QR-P). The proposed two-stage and three-stage algorithms reduce the complexity of D-QR-P and I-QR-P, respectively. Both aim to lower complexity by decreasing the candidate beam size by eliminating the beams with no contribution to any user and using matrix perturbation theory to update QR decompositions. Numerical results reveal that the proposed algorithms considerably reduce the complexity while maintaining a similar sum-rate with baseline algorithms.

An optimal fractional-order accumulative Grey Markov model with variable parameters and its application in total energy consumption

<abstract> <p>In this paper, we propose an optimal fractional-order accumulative Grey Markov model with variable parameters (FOGMKM (1, 1)) to predict the annual total energy consumption in China and improve the accuracy of energy consumption forecasting. The new model is built upon the traditional Grey model and utilized matrix perturbation theory to study the natural and response characteristics of a system when the structural parameters change slightly. The particle swarm optimization algorithm (PSO) is used to determine the number of optimal fractional order and nonlinear parameters. An experiment is conducted to validate the high prediction accuracy of the FOGMKM (1, 1) model, with mean absolute percentage error (MAPE) and root mean square error (RMSE) values of 0.51% and 1886.6, respectively, and corresponding fitting values of 0.92% and 6108.8. These results demonstrate the superior fitting performance of the FOGMKM (1, 1) model when compared to other six competitive models, including GM (1, 1), ARIMA, Linear, FAONGBM (1, 1), FGM (1, 1) and FOGM (1, 1). Our study provides a scientific basis and technical references for further research in the finance as well as energy fields and can serve well for energy market benchmark research.</p> </abstract>

Non-backtracking spectra of weighted inhomogeneous random graphs

We study a model of random graphs where each edge is drawn independently (but not necessarily identically distributed) from the others, and then assigned a random weight. When the mean degree of such a graph is low, it is known that the spectrum of the adjacency matrix A deviates significantly from that of its expected value \mathbb{E} A . In contrast, we show that over a wide range of parameters the top eigenvalues of the non-backtracking matrix B – a matrix whose powers count the non-backtracking walks between two edges – are close to those of \mathbb{E} A , and all other eigenvalues are confined in a bulk with known radius. We also obtain a precise characterization of the scalar product between the eigenvectors of B and their deterministic counterparts derived from the model parameters. This result has many applications, in domains ranging from (noisy) matrix completion to community detection, as well as matrix perturbation theory. In particular, we establish as a corollary that a result known as the Baik–Ben Arous–Péché phase transition, previously established only for rotationally invariant random matrices, holds more generally for matrices A as above under a mild concentration hypothesis.

Exit Time Analysis for Approximations of Gradient Descent Trajectories Around Saddle Points

Abstract This paper considers the problem of understanding the exit time for trajectories of gradient-related first-order methods from saddle neighborhoods under some initial boundary conditions. Given the ‘flat’ geometry around saddle points, first-order methods can struggle to escape these regions in a fast manner due to the small magnitudes of gradients encountered. In particular, while it is known that gradient-related first-order methods escape strict-saddle neighborhoods, existing analytic techniques do not explicitly leverage the local geometry around saddle points in order to control behavior of gradient trajectories. It is in this context that this paper puts forth a rigorous geometric analysis of the gradient-descent method around strict-saddle neighborhoods using matrix perturbation theory. In doing so, it provides a key result that can be used to generate an approximate gradient trajectory for any given initial conditions. In addition, the analysis leads to a linear exit-time solution for gradient-descent method under certain necessary initial conditions, which explicitly bring out the dependence on problem dimension, conditioning of the saddle neighborhood, and more, for a class of strict-saddle functions.

An equivalent circuit model of characteristic modes with lumped loads

Usage of lumped loads is an effective approach for mode manipulation, but the physical insights are not well disclosed. In this paper, an equivalent circuit model (ECM) is proposed to investigate the effects of lumped loads on the characteristic modes (CMs). This model can predict the modal admittance of a loaded antenna over a broad bandwidth. A scaling coefficient K is proposed to describe the discrepancy of the loading effects on different modes, thus providing a clear explanation of the resonant behaviors of CMs with lumped loads. Matrix perturbation theory is adopted to analyze the change of eigencurrents and eigenvalues upon lumped loads. The applicable boundaries of the ECM with regard to different lumped loads are given for the first time. The proposed ECM is potentially useful to assist mode manipulation.

Computational Protocol to Evaluate Electron-Phonon Interactions Within Density Matrix Perturbation Theory.

We present a computational protocol, based on density matrix perturbation theory, to obtain non-adiabatic, frequency-dependent electron–phonon self-energies for molecules and solids. Our approach enables the evaluation of electron–phonon interaction using hybrid functionals, for spin-polarized systems, and the computational overhead to include dynamical and non-adiabatic terms in the evaluation of electron–phonon self-energies is negligible. We discuss results for molecules, as well as pristine and defective solids.

Polarized induced phase grating in a quantized four-level graphene monolayer system.

We discuss the electromagnetically induced grating (EIG) and electromagnetically induced phase grating (EIPG) in a four-level quantized graphene monolayer system. By using the density matrix technique and perturbation theory, we first obtain the self-Kerr nonlinear susceptibility of the graphene system; afterwards, we study the amplitude and phase modulations of the probe light. We discovered that the EIG and EIPG can be found by controlling the elliptically polarized coupling fields that interact with the monolayer graphene system. Owing to the phase modulation of the transmitted light beam, we recognized that the probe strength can also additionally switch from zeroth-order to high-order diffraction. Moreover, we found that the diffraction performance of the grating may be adjusted through tuning the polarization of the coupling light.

Surplus-based accelerated algorithms for distributed optimization over directed networks

This paper investigates a distributed optimization problem based on the framework of a multi-agent system over a directed communication network, where the global cost function is the sum of the local cost functions of agents. The communication network is abstracted as a weight-unbalanced directed graph. First, a surplus-based accelerated algorithm with a fixed stepsize (SAAFS) is proposed by integrating the gradient tracking strategy into the surplus-based consensus protocol to address the problem considered. The matrix norm argument and matrix perturbation theory are employed to prove the linear convergence of SAAFS under the assumption that each local cost function is strongly convex with the Lipschitz continuous gradient. Second, the limitation of the stepsize, which is common to all agents, is relaxed in the cases of different stepsizes for each agent, such that the surplus-based accelerated algorithm with an uncoordinated stepsize (SAAUS) is proposed. It is proven that SAAUS also has a linear convergence rate if the upper bound of the uncoordinated stepsize at each agent is restricted by a sufficiently small positive number. Finally, two simulation examples are provided to evaluate the proposed algorithms and illustrate that both SAAFS and SAAUS achieve acceleration, particularly for ill-conditioned optimization problems.

Randomized Benchmarking beyond Groups

Randomized benchmarking (RB) is the gold standard for experimentally evaluating the quality of quantum operations. The current framework for RB is centered on groups and their representations, but this can be problematic. For example, Clifford circuits need up to $O(n^2)$ gates, and thus Clifford RB cannot scale to larger devices. Attempts to remedy this include new schemes such as linear cross-entropy benchmarking (XEB), cycle benchmarking, and non-uniform RB, but they do not fall within the group-based RB framework. In this work, we formulate the \emph{universal randomized benchmarking (URB) framework} which does away with the group structure and also replaces the recovery gate plus measurement component with a general ``post-processing'' POVM. Not only does this framework cover most of the existing benchmarking schemes, but it also gives the language for and helps inspire the formulation of new schemes. We specifically consider a class of URB schemes called \emph{twirling schemes}. For twirling schemes, the post-processing POVM approximately factorizes into an intermediate channel, inverting maps, and a final measurement. This leads us to study the twirling map corresponding to the gate ensemble specified by the scheme. We prove that if this twirling map is strictly within unit distance of the Haar twirling map in induced diamond norm, the probability of measurement as a function of gate length is a single exponential decay up to small error terms. The core technical tool we use is the matrix perturbation theory of linear operators on quantum channels.

Quantum Perturbation Theory Using Tensor Cores and a Deep Neural Network.

Time-independent quantum response calculations are performed using Tensor cores. This is achieved by mapping density matrix perturbation theory onto the computational structure of a deep neural network. The main computational cost of each deep layer is dominated by tensor contractions, i.e., dense matrix-matrix multiplications, in mixed-precision arithmetics, which achieves close to peak performance. Quantum response calculations are demonstrated and analyzed using self-consistent charge density-functional tight-binding theory as well as coupled-perturbed Hartree-Fock theory. For linear response calculations, a novel parameter-free convergence criterion is presented that is well-suited for numerically noisy low-precision floating point operations and we demonstrate a peak performance of almost 200 Tflops using the Tensor cores of two Nvidia A100 GPUs.

Appointed-Time Average Consensus Over Directed Networks

Current research on average consensus for multi-agent networks mainly focuses on the cases with bidirectional interaction. However, the unidirectional information flow usually exists in practical applications due to different communication ranges of individuals. To achieve state-average over directed networks, this brief introduces an extra variable for each agent, called ‘surplus’, to record the shift in the average. Then, a bank of distributed average consensus control protocols is developed by incorporating appointed-time control framework. It shows that for the multi-agent system with strongly-connected network, the presented protocol ensures that the state-average and the surplus-stability behaviour emerge at an appointed convergence time. Besides, the settling time is independent of any other parameters and can be assigned according to the task requirements. Corresponding techniques for convergence analysis involve time projection method, motion planning technique and matrix perturbation theory. Lastly, the theoretical results are examined by a numerical simulation.

Estimation of Dynamic Grid Flexibility Using Matrix Perturbation Theory

In this letter, a numerically-based method is proposed to estimate the metrics of dynamic grid flexibility, namely the inertial index, and the flexibility index. The novelty of the proposed method lies in using matrix perturbation theory, through which the sensitivity of a system to any perturbation is determined quantitatively. This notion is applied to the power system scenario under small disturbances, through which the inertial and the flexibility indices are calculated directly from the system matrix. This makes the proposed method immune to modeling complexities, and enables to implicitly understand the impact of any used generator or controller model. The applicability of the method is tested for the IEEE 39-bus system.

Formation Control of Multiagent Networks: Cooperative and Antagonistic Interactions

This article studies the formation control problem of second-order multiagent networks, in which cooperative and antagonistic interactions of the agents spontaneously coexist in the communication process. Based on the convex analysis theory, several convex polytopes that do not require some kinds of system constraints are constructed in the presence of these interactions. Then, the matrix perturbation theory and some mathematical techniques are utilized to analyze these convex polytopes. The obtained results show that the agents with cooperative interactions monotonously converge to their own specified formation shape while maintaining the desired relative position of the other agents with antagonistic interactions. Subsequently, two numerical examples are presented to illustrate the obtained results.