Dominant Eigenvalue Research Articles

The quantum version of the shifted power method and its application in quadratic binary optimization

In this paper, we present a direct quantum adaptation of the classical shifted power method. The method is very similar to the iterative phase estimation algorithm; however, it does not require any initial estimate of an eigenvector, and as in the classical case its convergence and the required number of iterations are directly related to the eigengap. If the amount of the gap is in the order of $1/poly(n)$, then the algorithm can converge to the dominant eigenvalue in $O(poly(n))$ time. The method can be potentially used for solving any eigenvalue related problem and finding minimum/maximum of a data set in lieu of Grover's search algorithm. In addition, if the solution space of an optimization problem with $n$ parameters is encoded as the eigenspace of an $2^n$ dimensional unitary operator in $O(poly(n))$ time and the eigengap is not too small, then the solution for such a problem can be found in $O(poly(n))$. As an example, using the quantum gates, we show how to generate the solution space of the quadratic unconstrained binary optimization as the eigenvectors of a diagonal unitary matrix and find the solution for the problem.

Flash Calculation Using Successive Substitution Accelerated by the General Dominant Eigenvalue Method in Reduced-Variable Space: Comparison and New Insights

Summary On the basis of a previously published reduced-variables method, we demonstrate that using these reduced variables can substantially accelerate the conventional successive-substitution iterations in solving two-phase flash (TPF) problems. By applying the general dominant eigenvalue method (GDEM) to the successive-substitution iterations in terms of the reduced variables, we obtained a highly efficient solution for the TPF problem. We refer to this solution as Reduced-GDEM. The Reduced-GDEM algorithm is then extensively compared with more than 10 linear-acceleration and Newton-Raphson (NR)-type algorithms. The initial equilibrium ratio for flash calculation is generated from reliable phase-stability analysis (PSA). We propose a series of indicators to interpret the PSA results. Two new insights were obtained from the speed comparison among various algorithms and the PSA. First, the speed and robustness of the Reduced-GDEM algorithm are of the same level as that of the reduced-variables NR flash algorithm, which has previously been proved to be the fastest flash algorithm. Second, two-side phase-stability-analysis results indicate that the conventional successive-substitution phase-stability algorithm is time consuming (but robust) at pressures and temperatures near the stability-test limit locus in the single-phase region and near the spinodal in the two-phase region.

Disappointing Survival Forecast for a Local Population of Androsace albana in a Random Environment

A local population of Andropace albana, a short-lived perennial plant, has been monitored during 10 years on permanent plots laid down in an alpine lichen heath in 2009. We summarize the outcome of monitoring as a non-autonomous matrix model of stage-structured population dynamics. The model originates from a life cycle graph constructed earlier for the stages of ontogenesis and consists of 9 annual “projection” matrices that are calibrated in a unique way from the observation data. Five of the 9 matrices have their dominant eigenvalues greater than 1, i.e., give favorable forecasts for the local population survival, while the rest four have those values less than 1, i.e., give the negative forecasts. To make the resulting prediction, we apply an original concept of the pattern-geometric averaging of given nonnegative matrices and obtain the dominant eigenvalue, λ1(G9), of the average matrix G9 markedly less than 1, indicating the population decline in the long term. The traditional method to forecast the local population is to estimate λS, the stochastic growth rate of the population in a random environment formed by a random choice from the same 9 annual matrices. Assuming the choice to be independent and equiprobable, we obtain the negative result as well, yet with higher quantitative values of λS. We associate these higher values with the artificial assumption of equal choice probability when forming the random sequence of annual matrices, each of which is indirectly reflecting the habitat conditions that have influenced the growth and development of plants during the year prior the calibration moment. This motivates the task to construct a more adequate model for choosing annual matrices, in which the probability of choice would be related to the dynamics of the real habitat factors for a given local population.

The power method and beyond

For computing the dominant eigenvalue and the corresponding eigenvector of a real and symmetric matrix, inspired by the classic and powerful power method, we construct a general paradigm of nonstationary Richardson methods and gradient descent methods, called also as the parameterized power methods, and establish their convergence theory. This paradigm also includes the power method as a special case. Both theoretical analysis and numerical experiments show that the parameterized power methods can result in iteration methods that may be much more effective than the power method, provided the involved iteration parameters are chosen appropriately.

Effects of rainfall and terracing-vegetation combinations on water erosion in a loess hilly area, China

Terracing and vegetation restoration are the basic measures to protect soil from water erosion and to combat land degradation. However, long-term quantitative evaluation on the erosion control benefits of different terracing techniques and vegetation types are still insufficient, particularly under variable rainfall. The aim of this article, therefore, is to evaluate the coupling effects of different terracing-vegetation combinations and rainfall types (RTs) on runoff retention and erosion reduction in a loess hilly catchment of China. Six types of terracing-vegetation combinations, including leveled benches-C. microphylla (LM), fish-scale pits-P. orientalis (FO), leveled ditches-P. armeniaca (LA), zig terraces-P. orientalis (ZO), fish-scale pits-P. tabulaeformis (FT), zig terraces-P. tabulaeformis (ZT) and the corresponding plots with same vegetation cover and non-terracing measures were designed and monitored. Based on five consecutive years of monitoring data, 69 rainfall events causing runoff and erosion were observed. Rainfall eigenvalues, including rainfall amount (RA), maximum 10-min intensity (I10), maximum 30-min intensity (I30) and rainfall duration (RD) dominated water erosion processes. Surface runoff and sediment reduction benefits differed with different terracing techniques. Mean runoff coefficients (Rc) among all kinds of terracing-vegetation combinations were FT > LM > FO > LA > ZO > ZT, while mean soil loss rates (Em) among all kinds of combinations were FT > FO > LM > LA > ZT > ZO. ZT showed the highest mean runoff reduction (44.03%), while ZO generated the highest sediment reduction (39.08%). The worst performance was observed in FT. With regards to the results, it is necessary to select the optimal terracing-vegetation measures for erosion control based on the dominant rainfall eigenvalues in different areas. Overall, ZT, ZO and LA combinations are recommended, while uncertainty was detected in combinations with fish-scale pits. Suitable terracing-vegetation measures should be selected after considering the micro-relief construction, the optimization of plant disposition and the efficiency of water erosion reduction. Management should focus on the construction standards, the threshold of resisting erosion for each terracing measure, and timely maintenance of the terraces.

Order reduction mechanism for large-scale continuous-time systems using substructure preservation with dominant mode

This work is about a balanced truncation type order reduction method which is developed for stable and unstable large-scale continuous-time systems. In this method, a quantitative measure criterion for choosing the dominant eigenvalues helps in determining the steady-state and transient information of the dynamical system. These dominant eigenvalues are used to form a new substructure matrix that retains the dominant modes (or may desirable mode) of the original system. Retaining the dominant eigenvalues in the reduced mode assures stability and results in greater accuracy as the retained eigenvalues provides a physical link to the real system. In the quest to preserve the dominant eigenvalue of the real system, the proposed technique uses Sylvester equation for system transformation. Having obtained transformed model, the reduced model has been achieved by truncating the non-dominant eigenvalues using the singular perturbation approximation method. The efficiency and accuracy of the proposed method has been demonstrated by the benchmark test systems which were from the state-of-the-art models.

Accelerating the Sinkhorn–Knopp iteration by Arnoldi-type methods

It is shown that the problem of balancing a nonnegative matrix by positive diagonal matrices can be recast as a constrained nonlinear multiparameter eigenvalue problem. Based on this equivalent formulation some adaptations of the power method and Arnoldi process are proposed for computing the dominant eigenvector which defines the structure of the diagonal transformations. Numerical results illustrate that our novel methods accelerate significantly the convergence of the customary Sinkhorn-Knopp iteration for matrix balancing in the case of clustered dominant eigenvalues.

A Numerical Method to Solve Nonsymmetric Eigensystems Applied to Dynamics of Turbomachinery

In this paper, a canonical transformation is proposed to solve the eigenvalue problem related to the dynamics of rotor-bearing systems. In this problem, all matrices are real, but they may not be symmetric, which leads to the appearance of complex eigenvalues and eigenvectors. The bi-iteration method is selected to solve the original eigenproblem whereas the QR algorithm is adopted to solve the reduced or projected problem. A new canonical transformation of the global eigenproblem which reduces the quadratic eigenproblem to a linear eigenproblem, maintaining numerical stability since all that is required is that the stiffness matrix is well-conditioned, which is always true when it comes to applications in dynamic problems. The proposed technique is good for obtaining dominant eigenvalues and corresponding eigenvectors of real nonsymmetric matrices and it possesses the following properties: (i) the matrix is not transformed, therefore sparsity is maintained, (ii) partial eigensolutions can be obtained and (iii) use may be made of good eigenvectors predictions.

Dynamical phase transition in Floquet optical bistable systems: An approach from finite-size quantum systems

We study a dynamical phase transition in optical bistable systems subject to a time-periodic driving field. The phase transition occurs in the structure of limit cycle as a function of the frequency of the driving field. In the thermodynamic limit, a single limit cycle is divided into two separated limit cycles at the transition point. In finite-size systems, however, there is always a single limit cycle due to the quantum tunneling effect. We use a Floquet dissipative map, which is a time-evolution operator over one period in a dynamics given by a quantum master equation, and discuss the decay rate of relaxation dynamics into the limit cycle based on the dominant eigenvalue of the map. We found that the decay rate exhibits qualitatively different system-size dependence before and after the phase transition, and it shows a finite-size scaling of spinodal phenomena around the transition point. The present work provides a systematic way of studying dynamical phase transition observed in time-periodically driven open systems in terms of the Floquet dissipative map.

Controllability-Gramian submatrices for a network consensus model

Principal submatrices of the controllability Gramian and their inverses are examined, for a network-consensus model with inputs at a subset of network nodes. Several properties of the Gramian submatrices and their inverses – including dominant eigenvalues and eigenvectors, diagonal entries, and sign patterns – are characterized by exploiting the special doubly-nonnegative structure of the matrices. In addition, majorizations for these properties are obtained in terms of cutsets in the network’s graph, based on the diffusive form of the model. The asymptotic (long time horizon) structure of the controllability Gramian is also analyzed. The results on the Gramian are used to study metrics for target control of the network-consensus model.

Stability analysis of deep-water waves on a linear shear current using unsteady conformal mapping

This paper describes linear stability analysis of the two-dimensional steady motion of periodic deep-water waves with symmetric non-overhanging profiles propagating on a linear shear current, namely a vertically sheared current with constant vorticity. In order to investigate numerically with high accuracy the stability of large-amplitude waves, we adopt a formulation using conformal mapping, in which the time-varying water surface is always mapped onto the real axis of a complex plane. This formulation allows us to apply numerical methods developed for large-amplitude irrotational waves without a shear current directly to the present problem, and reduces the linear stability problem to a generalized eigenvalue problem. Numerical solutions describe both super- and sub-harmonic instabilities of the periodic waves for a wide range of wave amplitudes and clarify how the behaviours of dominant eigenvalues change with the strength of the shear current. In particular, it is shown that, even in the presence of a linear shear current, the steady periodic waves lose stability due to superharmonic disturbances at the wave amplitude where the wave energy attains an extremum, similarly to the case of irrotational waves without a shear current. It is also found that re-stabilization with an increase in wave amplitude characterizes subharmonic instability for weak shear currents, but the re-stabilization disappears for strong shear currents.

Dynamic modeling, sensitivity assessment, and design of VSC-based microgrids with composite loads

Microgrids are seen as useful for increasing the flexibility of distribution networks and integrating large amounts of distributed generations. Ensuring the dynamic stability of power converter-dominated microgrids that is robust to a range of load conditions is a significant challenge and essential for ensuring reliability. Induction motor (IM) loads are widespread and have substantial impacts on the dynamic behavior and stability characteristics of low-inertia microgrids. The stability assessment and design of microgrids considering composite loads have not been sufficiently addressed in the current literature, where static loads are commonly used to simplify the modeling and analysis. In this paper, the dynamic stability of voltage source converter-based microgrids is investigated, considering composite loads as dynamic element. A complete state-space model of the microgrid with both IM load and static load is developed. Participation factor analysis is conducted to identify the contribution of the composite loads to the dominant oscillatory modes of the microgrid. Furthermore, sensitivity assessment of the dominant eigenvalues is presented to further identify the appropriate ranges of variations in the control parameters, operating conditions and operating points of the microgrid. It is shown that composite loads in a realistic microgrid significantly affect the dominant oscillatory modes and, consequently, the system stability margin. Ignoring the composite load dynamics in microgrid stability studies may lead to misleading analytical results. Simulations based on MATLAB/Simulink and experimental tests based on a laboratory prototype microgrid are implemented to demonstrate the theoretical analysis.

ML Algorithm Performance to Classify MCS Schemes During UACN Link Adaptation

This research classifies the modulation and coding rate for link adaptation in Underwater Acoustic Communications Networks (UACNs). Recently, the UACN has become a promising technology for military, commercial, and civilian applications, as well as scientific research. However, we should minimize the dataset dimension for real-time implementation due to the sensor nodes’ energy limitations in the underwater environment. We used an Incheon sea trial’s measured dataset of 18 features, applying Principal Component Analysis (PCA) to select the dominant eigenvalue components in order to reduce the curse of dimensionality, and then selected 11 parameters. After that, we applied Machine Learning (ML) algorithms with different combinations of the parameters to separately classify the modulation and the coding rate and measured both individual and overall classification accuracy. The findings are compared with two Taean sea trial datasets with 11 features to finalize the selected parameters for link adaptation. For modulation classification, we observed 96.83% accuracy with the K-nearest Neighbors (KNN) algorithm in three-parameter and two-parameter cases. In coding rate classification, we found 100% accuracy with the KNN algorithm using the same three-parameter case. However, for the best fit among the three datasets, we finalized another three parameters at the expense of accuracy. To find the optimum threshold values for all modulation and coding rate labels, we used Rule-based (RB) 2D and 3D analysis. However, with a hard limit on non-overlapping data, at best, 35.51% classification accuracy was found for a 1/3 coding rate (Turbo code) with QPSK modulation, which showed much less reliability for RB analysis in a UACN, so it is not useful in this regard. Besides, our analysis shows data independence in the Doppler Spread (DS) and the Frequency Shift (FS) , mitigating the time-variability channel’s challenge. We use the Gaussian distribution plot, a confusion matrix, multi-dimensional scatter plots, interpolated plots to analyze the data.

Modelling and optimal control analysis of Lassa fever disease

Abstract Lassa fever is a severe hemorrhagic viral infection whose agents belong to Mastomys natelensis. Generally, humans contract Lassa virus through exposure to food or household products that have been contaminated with the excreta of the infected rodents. Lassa fever is endemic in some West African countries including Nigeria. A basic model is proposed to examine the transmission of the disease. The proposed model is subjected to qualitative study via the theory of differential equations and the threshold quantity that denotes the dominant eigenvalue was derived using next-generation matrix approach. The basic model is further extended to an optimal control model with four controls namely, the fumigation of the environment with pesticide, the use of condom to prevent human to human transmission during sexual activities, early treatment and the use of indoor residual spray. The theory of optimal control was explored to establish the necessary conditions for curtailing the transmission of Lassa fever. Numerical simulation was conducted and the results showed that if the Lassa fever transmission and spread were to be reduced significantly in the endemic region, all the control measures must be taken with all seriousness.

Influence of zirconium on cation mobilities in Na2O-CaO-Al2O3-SiO2 melts: A multicomponent diffusion and XANES study

High temperature cation mobilities, based on the multicomponent diffusion method, have been coupled with a structural investigation in the Na2O-CaO-Al2O3-SiO2-ZrO2 (NCASZ) system at 1200 °C and 1250 °C. From structural investigation using X-ray absorption spectroscopy (XANES), zirconium was determined in six-fold coordinated sites in the reduced glass composition range of our study and no significant structural changes have been evidenced compared to a quaternary Na2O-CaO-Al2O3-SiO2 (NCAS) glass with a similar composition. Additionally, the diffusion experiments revealed that adding zirconium in the quaternary melt has no influence on the dominant eigenvectors that correspond to exchange between calcium and sodium. Despite the expected mobility decrease resulting from a viscosity increase upon the addition of zirconium, the dominant eigenvalue calculated in this study is higher when Zr is introduced in the melt structure. This result strongly suggests that the sodium mobility is enhanced by the presence of Zr, which is explained by a change in the structural role of sodium from network modifier associated to Q3(Si) sites in NCAS melts to charge compensator associated with [ZrO6]2− sites in NCASZ melts. This modification generates lower bond strengths between sodium and other cations in the melt, thus favoring an enhancement in mobility. Moreover, the diffusion matrix was applied to predict diffusion profiles between zirconium-bearing crystals and melts. We observed that even far from the composition field used for the matrix determination, predictions of zirconium diffusion profiles may be relevant and demonstrate the potential of this approach to evaluate crystal/melt dissolution behavior.

On the Event-Triggered Controller Design

The majority of event-triggered control (ETC) literature concentrates on designing triggering rule while assuming control input to emulate an analog design. In this article, however, both state and output feedback laws are jointly synthesized with the triggering law for nonlinear Lipschitz systems. In the proposed method, the dominant eigenvalues of the linear stability matrices are assigned according to desired performance and triggering specifications. The results serve as a local framework for stability of general nonlinear ETC systems. The efficiency of design is validated through compelling examples.

Receptance-Based Dominant Eigenvalues Computation of Controlled Vibrating Systems with Multiple Time-Delays Using a Contour Integral Method

The computation of dominant eigenvalues of second-order linear control systems with multiple time-delays is tackled by using a contour integral method. The proposed approach depends on a reduced characteristic function and the associated characteristic matrix comprised of measured open-loop receptances. This reduced characteristic function is derived from the original characteristic function of the second-order time delayed systems based on the reasonable assumption that eigenvalues of the closed-loop system are distinct from those of the open-loop system, and has the same eigenvalues as those of the original. Then, the eigenvalues computation is equivalent to solve a nonlinear eigenvalue problem of the associated characteristic matrix by using a contour integral method. The proposed approach also utilizes the spectrum distribution features of the retarded time-delay systems. Finally, numerical examples are given to illustrate the effectiveness of the proposed approach.

Stability and steady state of complex cooperative systems: a diakoptic approach.

Cooperative dynamics are common in ecology and population dynamics. However, their commonly high degree of complexity with a large number of coupled degrees of freedom renders them difficult to analyse. Here, we present a graph-theoretical criterion, via a diakoptic approach (divide-and-conquer) to determine a cooperative system’s stability by decomposing the system’s dependence graph into its strongly connected components (SCCs). In particular, we show that a linear cooperative system is Lyapunov stable if the SCCs of the associated dependence graph all have non-positive dominant eigenvalues, and if no SCCs which have dominant eigenvalue zero are connected by a path.

Another look at the eigenvalues of a population matrix model.

Population matrix models are important tools in resource management, in part because they are used to calculate the finite rate of growth (“dominant eigenvalue”). But understanding how a population matrix model converts life history traits into the finite rate of growth can be tricky. We introduce interactive software (“IsoPOPd”) that uses the characteristic equation to display how vital rates (survival and fertility) contribute to the finite rate of growth. Higher-order interactions among vital rates complicate the linkage between a management intervention and a population’s growth rate. We illustrate the use of the software for investigating the consequences of three management interventions in a 3-stage model of white-tailed deer (Odocoileus virginianus). The software is applicable to any species with 2- or 3-stages, but the mathematical concepts underlying the software are applicable to a population matrix model of any size. The IsoPOPd software is available at: https://cwhl.vet.cornell.edu/tools/isopopd.

Implicit Floquet analysis for parametric instabilities in a variable angle tow composite panel

In the literature dealing with the stability of time-periodic structures, Bolotin’s method is widely used for generating the approximate stability charts. Further, increasing the order of approximation in Bolotin’s approach requires solving several eigenvalue problems and may not lead to the rapid convergence of stability charts. Floquet theory can be used to calculate accurate stability charts for time-periodic systems. In Floquet theory, a Floquet transition matrix (FTM) is computed, the dominant eigenvalue of the FTM determine the stability of the time-periodic system. For a large degree of freedom systems like finite element models of VAT panels, the calculation of FTM becomes computationally expensive. However, an implicit Floquet analysis can significantly reduce the computational load. In this technique, the dominant eigenvalue of the FTM is computed without calculating the full FTM matrix. In this work, the stability regions obtained from the implicit Floquet analysis are compared with the results from Bolotin’s approach and verified using the time response of the panel obtained from numerical integration. Also, unlike Bolotin’s method, Floquet analysis provides information about the damping present in different areas of stability charts and the nature of bifurcation through which the stability is lost.