Approximate Eigenvalues Research Articles

Estimates for eigenvalues of the Neumann and Steklov problems

Abstract We prove Li-Yau-Kröger-type bounds for Neumann-type eigenvalues of the biharmonic operator on bounded domains in a Euclidean space. We also prove sharp estimates for lower order eigenvalues of a biharmonic Steklov problem and of the Laplacian, which directly implies two sharp Reilly-type inequalities for the corresponding first nonzero eigenvalue.

Pattern-Multiplicative Average of Nonnegative Matrices Revisited: Eigenvalue Approximation Is the Best of Versatile Optimization Tools

Given several nonnegative matrices with a single pattern of allocation among their zero/nonzero elements, the average matrix should have the same pattern, too. This is the first tenet of the pattern-multiplicative average (PMA) concept, while the second one suggests the multiplicative (or geometric) nature of averaging. The original concept of PMA was motivated by the practice of matrix population models as a tool to assess the population viability from long-term monitoring data. The task has reduced to searching for an approximate solution to an overdetermined system of polynomial equations for unknown elements of the average matrix (G), and hence to a nonlinear constrained minimization problem for the matrix norm. Former practical solutions faced certain technical problems, which required sophisticated algorithms but returned acceptable estimates. Now, we formulate (for the first time in ecological modeling and nonnegative matrix theory) the PMA problem as an eigenvalue approximation one and reduce it to a standard problem of linear programing (LP). The basic equation of averaging also determines the exact value of λ1(G), the dominant eigenvalue of matrix G, and the corresponding eigenvector. These are bound by the well-known linear equations, which enable an LP formulation of the former nonlinear problem. The LP approach is realized for 13 fixed-pattern matrices gained in a case study of Androsace albana, an alpine short-lived perennial, monitored on permanent plots over 14 years. A standard software routine reveals the unique exact solution, rather than an approximate one, to the PMA problem, which turns the LP approach into ‘’the best of versatile optimization tools”. The exact solution turns out to be peculiar in reaching zero bounds for certain nonnegative entries of G, which deserves modified problem formulation separating the lower bounds from zero.

Numerical spectra of the Laplacian for line bundles on Calabi-Yau hypersurfaces

We give the first numerical calculation of the spectrum of the Laplacian acting on bundle-valued forms on a Calabi-Yau three-fold. Specifically, we show how to compute the approximate eigenvalues and eigenmodes of the Dolbeault Laplacian acting on bundle-valued (p, q)-forms on Kähler manifolds. We restrict our attention to line bundles over complex projective space and Calabi-Yau hypersurfaces therein. We give three examples. For two of these, ℙ3 and a Calabi-Yau one-fold (a torus), we compare our numerics with exact results available in the literature and find complete agreement. For the third example, the Fermat quintic three-fold, there are no known analytic results, so our numerical calculations are the first of their kind. The resulting spectra pass a number of non-trivial checks that arise from Serre duality and the Hodge decomposition. The outputs of our algorithm include all the ingredients one needs to compute physical Yukawa couplings in string compactifications.

Eigenvalue estimates for a class of elliptic differential operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space

Eigenvalue estimates for a class of elliptic differential operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space

Numerical approximation based on a decoupled dimensionality reduction scheme for Maxwell eigenvalue problem

We present a high‐accuracy numerical method based on a decoupled dimensionality reduction scheme for Maxwell eigenvalue problem in spherical domains. Using the orthogonality of vector spherical harmonics and the variable separation approach, we decompose the original problem into two classes of decoupled one‐dimensional TE mode and TM mode. For the TE mode, we establish a variational formulation and its discrete scheme and give the error estimations of the approximate eigenvalues and eigenfunctions. For the TM mode, it is different from TE mode which naturally meets the divergence‐free condition and will not generate some spurious eigenvalues. We design a numerical algorithm based on a parameterized method to filter out the spurious eigenvalues. Finally, some numerical results are presented to confirm the theoretical results and validate the algorithms.

Free Vibration Response of a Steel Liquid Storage Tank with Porous and Perforated Columns via an Exact Continuum Method

The presence of predictable cavities inside a structure is often preferable as it will reduce the vibration amplitude. In this article, inspired by the traditional mass attached to the beam system, in order to solve the problem of free vibration of the steel liquid storage tank with a column made of porous and perforated materials using an exact analytical continuum model is formed. The free vibration response of the mass attached column made of porous materials and holes is established by using a new analytical method known as the Fourier series with Stokes’ transform. The free vibration frequencies are obtained by using an eigen-value approximation, and the influences of the number of holes, filling ratio, porosity and other parameters on the free vibration response are explored. It is shown that an increase in the mass parameter, filling ratio and porosity parameter of the column with an attached mass system could significantly affect the free vibration response of the system.

Unique continuation for the gradient of eigenfunctions and Wegner estimates for random divergence-type operators

We prove a scale-free quantitative unique continuation estimate for the gradient of eigenfunctions of divergence-type operators, i.e., operators of the form −divA∇, where the matrix function A is uniformly elliptic. The proof uses a unique continuation principle for elliptic second-order operators and a lower bound on the L2-norm of the gradient of eigenfunctions corresponding to strictly positive eigenvalues.As an application, we prove an eigenvalue lifting estimate that allows us to prove a Wegner estimate for random divergence-type operators. Here our approach allows us to get rid of a restrictive covering condition that was essential in previous proofs of Wegner estimates for such models.

A Small-Signal Stability Criterion for DC Microgrid Based on Extended-Gershgorin Theorem

Recent years have witnessed emerging oscillation stability issues in DC microgrid via power electronic converters. Considering the confidentiality of the parameters and structure of power electronic devices, the impedance method is widely used to study the small signal stability. However, the stability assessment based on traditional impedance model and Gershgorin Theorem have some shortcomings for multiple converters DC microgrid. This may lead to the problem that the stability analysis results are not accurate enough or the criteria cannot be applied. In this article, the system is modeled as a medium and high dimensional negative feedback system. Further, a sufficient stability criterion based on extended Gershgorin Theorem is developed through reshaping the range of eigenvalues estimation for the non-diagonal advantage system. Finally, the validity of the criterion is verified by a 6 converters system. Test results indicate the proposed method has the potential to assess the high-order system stability with low computational complexity. Moreover, its low conservatism as a sufficient criterion is helpful to guide the system design.

Factor modeling of multivariate time series: A frequency components approach

A frequency domain factor model method for multivariate stationary time series is proposed. The dimension reduction framework aims to find a lower-dimensional multivariate stationary factor series. Frequency components of the observed series are assumed to be linearly generated by the corresponding frequency components of a latent factor series using frequency-specific factor loadings matrices. These loadings matrices are then estimated using an eigendecomposition of symmetric non-negative definite matrices involving the real and imaginary parts of the spectral matrix. The factor dimension is estimated using nonparametric bootstrap tests. Consistency results concerning the estimation of eigenvalues, eigenvectors, factor loadings matrices and the factor dimension are provided. The numerical performance of the proposed method is illustrated through simulation examples. An application to modeling resting-state fMRI time series from autism individuals is demonstrated where a frequency-specific factor analysis helps understand functional connectivity.

Asymptotically exact a posteriori error estimates for the BDM finite element approximation of mixed Laplace eigenvalue problems

We derive optimal and asymptotically exact a posteriori error estimates for the approximation of the eigenfunction of the Laplace eigenvalue problem. To do so, we combine two results from the literature. First, we use the hypercircle techniques developed for mixed eigenvalue approximations with Raviart-Thomas finite elements. In addition, we use the post-processings introduced for the eigenvalue and eigenfunction based on mixed approximations with the Brezzi-Douglas-Marini finite element. To combine these approaches, we define a novel additional local post-processing for the fluxes that appropriately modifies the divergence without compromising the approximation properties. Consequently, the new flux can be used to derive optimal and asymptotically exact upper bounds for the eigenfunction, and optimal upper bounds for the corresponding eigenvalue. Numerical examples validate the theory and motivate the use of an adaptive mesh refinement.

Radial solution of Schrödinger equation with Hulthen-Yukawa-Inverse quadratic potential in a Point-Like defect under AB-flux field

In this paper, we determine the approximate eigenvalue solution of the non-relativistic wave equation in the presence of the Aharonov-Bohm flux field with Hulthen-Yukawa-Inverse Quadratic potential in a topological defect via point-like global monopole (PGM) geometry. We use the Greene-Aldrich improved approximation scheme into the centrifugal and reciprocal terms appear in the radial Schrödinger equation. We then solve this radial equation using the parametric Nikiforov-Uvarov method and analyze the effects on the eigenvalue solution. We see that the energy levels and the radial wave functions get modified by the topological defect of a point-like global monopole and the magnetic flux field that shows an analogue of the Aharonov-Bohm effect for the bound state. Finally, we utilize the eigenvalue solution to some potential models, such as Hulthen potential, Hulthen plus Yukawa potential, and Hulthen plus inverse quadratic potential and discuss the results.

Weighted isoperimetric inequalities in warped product manifolds

We prove some sharp isoperimetric type inequalities in warped product manifolds, or more generally, multiply warped product manifolds. We also relate them to inequalities involving the higher order mean-curvature integrals. Applications include some sharp eigenvalue estimates, Pólya-Szegö inequality, Faber-Krahn inequality, Sobolev inequality and some sharp geometric inequalities in some warped product spaces.

Cheeger estimates of Dirichlet-to-Neumann operators on infinite subgraphs of graphs

In this paper, we study the Dirichlet-to-Neumann operators on infinite subgraphs of graphs. For an infinite subgraph, we prove Cheeger-type estimates for the bottom spectrum of the Dirichlet-to-Neumann operator, and the higher order Cheeger estimates for higher order eigenvalues of the Dirichlet-to-Neumann operator.

Genetic parameters and selection of mango genotypes using the FAI-BLUP multitrait index

Mango (Mangifera indica L.) presents phenotypic diversity in relation to fruit quality. Due to this diversity, studies on performance and evaluation of genotypes help in the selection of new cultivars. The aim of this study was to select superior mango genotypes for fruit quality using a mixed model selection index. The fruit quality of two hundred and forty six different hybrids and four commercial cultivars (Ataulfo, Kent, Palmer and Tommy Atkins) was evaluated in three consecutive harvests. Estimates of variance components and predictions of phenotypic and genotypic values were obtained, as well as principal component analysis, factor analysis and genotype-ideotype distance (FAI-BLUP). Permanent variance among plants represented the largest portion of phenotypic variance for most traits. Estimates of the repeatability coefficient ranged from 0.09 to 0.96, being high for most variables. The selective accuracy predicted by the restricted maximum likelihood model (REML) ranged from 0.48 to 0.99, revealing a good degree of confidence in the inferences. Eigenvalue estimates revealed five principal components responsible for 77.72% of the total genetic variability. The greatest positive genetic gains were for fruit weight (37.29%), soluble solids/titratable acidity ratio (35.02%), b* (23.02%), pulp firmness (20.99%), ventral diameter (12.18%), transverse diameter (11.80%), skin luminosity (8.08%), soluble solids content (6.97%), length (4.28%), c* (3.08%), pulp luminosity (2.18%) and h* (0.88%). Negative genetic gains were observed for titratable acidity (-26.04%) and a* (-12.82%). Twenty-five genotypes were selected as the most promising based on factor analysis and FAI-BLUP index, promoting improvement and genetic gain in several traits. Between commercial cultivars, Ataulfo was the highest ranked genotype (29th), followed by Kent (36th), Palmer (243th) and Tommy Atkins (247th). The genotypes selected should be included in the next stages of the mango breeding program for high-quality fruits under irrigated cultivation in the Brazilian semiarid region.

Sensitivity Analysis in Coupled Monte Carlo Radiation Transport Simulations

Sensitivity analysis methods have found extensive use in nuclear criticality safety applications for understanding the impact of uncertain nuclear data on eigenvalue estimates. Significant uncertainty exists in nuclear data and nuclear physics models for photon and electron transport applications, and the goal of this work is to explore whether recently developed adjoint-based, first-order generalized perturbation theory reaction rate sensitivity methods can be extended to coupled Monte Carlo radiation transport simulations. This paper presents a rigorous theoretical derivation for this extended sensitivity analysis method, which is then implemented in a one-dimensional test Monte Carlo code. The adjoint-based sensitivity coefficients are found to agree well with reference direct perturbation and deterministic SENSMG sensitivity coefficients for a simple test problem.

Computing Eigenvalues of Sturm--Liouville Operators with a Family of Trigonometric Polynomial Potentials

We provide estimates for the periodic and antiperiodic eigenvalues of non-self-adjoint Sturm--Liouville operators with a family of complex-valued trigonometric polynomial potentials. We even approximate complex eigenvalues by the roots of some polynomials derived from some iteration formulas. Moreover, we give a numerical example with error analysis.

An hp-mixed discontinuous Galerkin method for the biharmonic eigenvalue problem

The biharmonic equation/eigenvalue problem is one of the fundamental model problems in mathematics and physics, and has wide applications. In this paper, we study an hp-mixed discontinuous Galerkin method for the biharmonic eigenvalue problem. Based on the work of Gudi et al. [J Sci Comput, 37 (2008)], using piecewise polynomials of degree p≥3, we derive the a priori error estimates of the approximate eigenfunction in the broken H1 norm and L2 norm which are optimal in h and suboptimal in p. When p=2, the approximate eigenfunctions converge but with only suboptimal convergence rate. When p≥2 and the eigenfunction u∈Hs(Ω)(s≥p+1), we prove that the convergence rate of approximate eigenvalues reaches 2p−2 in h and p−(2s−7) in p. We also discuss the a posterior error estimates of the approximate eigenvalues and implement the adaptive calculation. Numerical experiments show that the methods are easy to implement and can efficiently compute biharmonic eigenvalues.

Numeric estimates of the principal eigenvalue of the p-Laplacian using interval arithmetic

We present a numerical algorithm for computing rigorous upper and lower estimates of the principal eigenvalue of the p-Laplacian. To control all possible errors including the rounding errors of the computer arithmetic, we use the interval arithmetic. We implement our algorithm in the Julia programming language using IntervalArithmetic.jl package [12]. See also https://ejde.math.txstate.edu/special/02/b3/abstr.html

Priority ranking for the best-worst method

The best-worst method is a recently developed approach that addresses pairwise comparisons in multi-criteria decision-making problems. It has attracted the interest of many academics and has gained widespread use for its effectiveness in reducing the pairwise comparison times and for its strong performance in maintaining consistency between judgments. However, the existing best-worst method prioritization methods don't consider the indirect judgments and fail to meet several established criteria. This paper first develops two prioritization methods, the approximate eigenvalue method and the logarithmic least squares method, to the best-worst method. Then, the inconsistency thresholds of the proposed prioritization methods are given. Finally, the Monte Carlo simulation method is applied to generate random matrixes to compare the performance of various prioritization methods on the selected criteria and analyze the relationship between different consistency measures. It is found that the logarithmic least squares method is the best prioritization method because of simple calculation, considering indirect judgments and containing minimum violations.

Finite element modeling of straight pipeline with partially attached viscoelastic damping patch based on variable thickness laminated element

To successfully implement the viscoelastic damping patch (VDP) for vibration reduction of the straight pipeline, it is necessary to perform dynamic modeling to guide vibration reduction design. The nonlinear vibration characteristic of the straight pipeline with partially attached VDP is studied by the FEM. A modeling method of laminated shell elements with variable thickness is proposed. Additionally, the local coordinate transformation is executed to facilitate the integration calculation of each layer. The nonlinear dynamic equation of the straight pipeline with VDP is established by considering the frequency dependence of VDP and the non-uniform continuous variable stiffness distribution of the clamp. The Vector Iteration method based on Improved Approximate Eigenvalue (VIM/IAE) is proposed to solve the nonlinear eigenvalues further. The rationality of the FE model is verified by constructing an experimental test system and compared with the ANSYS simulation and references. Finally, the vibration reduction mechanism of the clamp-straight pipeline with partially attached VDP considering frequency dependence is analyzed under different nonlinear iterative algorithms, VDP parameters (thickness, elastic modulus, and loss modulus), and boundary stiffness.