Solution Of Eigenvalue Problem Research Articles

Numerical solution of eigenvalue problems for a compact integral operator with Green’s kernels

Numerical solution of eigenvalue problems for a compact integral operator with Green’s kernels

Bernstein spectral method for quasinormal modes and other eigenvalue problems

Spectral methods are now common in the solution of ordinary differential eigenvalue problems in a wide variety of fields, such as in the computation of black hole quasinormal modes. Most of these spectral codes are based on standard Chebyshev, Fourier, or some other orthogonal basis functions. In this work we highlight the usefulness of a relatively unknown set of non-orthogonal basis functions, known as Bernstein polynomials, and their advantages for handling boundary conditions in ordinary differential eigenvalue problems. We also report on a new user-friendly package, called SpectralBP, that implements Berstein-polynomial-based pseudospectral routines for eigenvalue problems. We demonstrate the functionalities of the package by applying it to a number of model problems in quantum mechanics and to the problem of computing scalar and gravitational quasinormal modes in a Schwarzschild background. We validate our code against some known results and achieve excellent agreement. Compared to continued-fraction or series methods, global approximation methods are particularly well-suited for computing purely imaginary modes such as the algebraically special modes for Schwarzschild gravitational perturbations.

Passive damping of vibrations of plates and cylindrical shells interacting with a flowing fluid using piezoelectric elements

The paper presents a mathematical formulation of the problem of natural and forced harmonic vibrations of electro-elastic thin-walled structures interacting with a flowing fluid. The numerical algorithm based on the finite element method is used to choose optimal parameters of a shunt electric circuit for problems of passive vibration damping. The key idea behind the proposed approach is a solution of eigenvalue problem. It allows us to obtain higher damping rate of free vibrations compared to those evaluated by the commonly used analytical expressions and leads to the smallest difference in the natural frequencies of the structure and the electric circuit.

Extracting connectivity paths in digital core images using solution of partial minimum eigenvalue problem

Abstract We show theoretically and numerically that the lowest non-trivial eigenvector function for a specific eigenproblem has almost constant values in high conductivity channels, which are different in separate channels. Therefore, based on these distinct values, all separate connected clusters of open pores can be identified in digital cores.

Application of machine learning regression models to inverse eigenvalue problems

In this work, we study the numerical solution of inverse eigenvalue problems from a machine learning perspective. Two different problems are considered: the inverse Sturm-Liouville eigenvalue problem for symmetric potentials and the inverse transmission eigenvalue problem for spherically symmetric refractive indices. Firstly, we solve the corresponding direct problems to produce the required eigenvalues datasets in order to train the machine learning algorithms. Next, we consider several examples of inverse problems and compare the performance of each model to predict the unknown potentials and refractive indices respectively, from a given small set of the lowest eigenvalues. The supervised regression models we use are k-Nearest Neighbours, Random Forests and Multi-Layer Perceptron. Our experiments show that these machine learning methods, under appropriate tuning on their parameters, can numerically solve the examined inverse eigenvalue problems.

Distributed Transfer Function Formulations for Vibration Analysis of Cracked Beam and Frame Structures

In this paper, based on distributed transfer function method (DTFM), the closed-form analytical solutions for vibrations of Euler–Bernoulli beam and frame structures with arbitrary number of cracks are studied. First, generalized DTFM is employed to characterize the dynamical model for a single cracked beam and its analytical solutions for eigenvalue problem and frequency response are obtained. Then, a new DTFM cracked element that encapsulates one crack of arbitrary location inside the beam is proposed. Using the DTFM cracked element and global dynamic stiffness matrix assembly technique, damaged frame structures of arbitrary form can be modeled for vibration analysis. Previous analytical methods only addressed low-frequency vibration of simple cracked beam structures, the proposed method can yield analytical solutions in the medium- and high-frequency regions, which is critical for the small crack detection in complex frames. Lastly, three numerical examples are given to illustrate the correctness and effectiveness of the DTFM in analyzing natural frequencies, modal shapes and frequency responses for cracked structures. By comparing with the Finite Element Method (FEM) and benchmarks from literatures, we proved that the DTFM has better performances in terms of accuracy and efficiency.

Bloch mode synthesis for band-structure computation of unit cell with non-matching boundaries

Bloch mode synthesis is an efficient method to calculate the band property of periodic structures, which involves the repeated solution of many eigenvalue problems on each wave vector around the Brillouin zone. It is based on projecting internal degrees of freedom (DOFs) of the unit cell to the modal domain and keeping the boundary DOFs in the physical domain. While this transformation greatly reduces the model order of the unit cell, it leaves the question of how to impose the Floquet-Bloch boundary conditions when the boundary nodes are non-matching. In this paper, an interpolation-based method is proposed to address the non-conformity problem, which consists of building a smooth function by interpolating the variables on the master boundary and determining the variables on the slave boundary by sampling that smooth function. This procedure creates an interpolation matrix to connect the master and slave variables, which can be integrated into the original Bloch mode synthesis formulation to absolve the incompatibility caused by non-matching boundaries. The accuracy of the proposed method is verified using one non-conforming unit cell example in the end.

Learning adaptive coarse basis functions of FETI-DP

Domain decomposition methods are successful and highly parallel scalable iterative solution methods for discretized partial differential equations. Nevertheless, for many classes of problems, for example, elliptic partial differential equations with arbitrary coefficient distributions, adaptive coarse spaces are necessary to obtain robustness or, in other words, to guarantee a reliable and fast convergence. Adaptive coarse spaces are usually computed by solving many localized eigenvalue problems related to edges or faces of the domain decomposition. This results in a computationally expensive setup of the domain decomposition preconditioner or system operator. In earlier work, to which the authors have contributed, a deep learning based classification model was used to classify the critical edges or faces where eigenvalue problems have to be solved. In the present paper, we suggest to directly learn the adaptive constraints using a deep feedforward neural network regression model and thus completely skip the computationally most expensive part of the setup, i.e., the solution of local eigenvalue problems. We consider a specific adaptive FETI-DP (Finite Tearing and Interconnecting – Dual Primal) approach and concentrate on stationary diffusion problems in two dimensions with arbitrary coefficient functions with large jumps. As an input for the neural network, we use an image representation of the coefficient function which resolves the structure of the coefficient distribution but is not necessarily identical to the discretization of the partial differential equation. Therefore, our approach is independent of the finite element mesh and can, in principle, be easily extended to other adaptive coarse spaces, problems, and domain decomposition methods. We show the robustness of our method for different problems and the generalization property of our trained neural networks by considering different coefficient distributions not contained in the training set. We also combine the learned constraints with computationally cheap frugal constraints to further improve our approach.

Solution of boundary eigenvalue problems and IBVP involving a system of PDEs using the successive linearization method

AbstractThe paper illustrates a numerical technique to solve a system of three partial differential equations that govern the problem of Rayleigh‐Bénard‐Brinkman convection in a two‐dimensional porous rectangular box. As a result of linear and weakly nonlinear stability analyses of the system a boundary eigenvalue problem (BEVP) and an initial boundary value problem (IBVP) arise. Spatial information on the periodicity of the convection cells is first used in the system of PDEs to make it possible for the successive linearization method (SLM) to be applied. The resulting much‐simplified versions of BEVP and the IVP are then solved by direct and time multi‐stepping versions of SLM, respectively. The SLM solution of the BEVP is compared with that obtained through MATLAB routine bvp4c and the multi‐stepping‐SLM solution of the IVP is validated with that of the Runge‐Kutta‐Fehlberg (RKF45) method (using MATLAB routine ode45). The present numerical technique is found to have quadratic convergence for any desired accuracy.

Multiple solution for a fourth-order nonlinear eigenvalue problem with singular and sublinear potential

"Let $(M,g)$ be a Cartan-Hadamard manifold. For certain positive numbers $\mu$ and $\lambda$, we establish the multiplicity of solutions to the problem $$\Delta_g^2 u-\Delta_g u+u=\mu \frac{u}{d_g(x_0,x)^4}+\lambda \alpha(x)f(u),\ \mbox{ in } M,$$ where $x_0\in M$, while $f:\R\to\R$ is continuous function, superlinear at zero and sublinear at infinity."

Multiple solutions for eigenvalue problems involving the (p,q)-Laplacian

"This paper is devoted to a subject that Professor Csaba Varga suggested during his frequent visits to the University of Perugia and in my regular stays at the ""Babe s-Bolyai"" University. More speci cally, continuing the work started in [7] jointly with Professor Varga, here we establish the existence of two nontrivial (weak) solutions of some one parameter eigenvalue (p,q)-Laplacian problems under homogeneous Dirichlet boundary conditions in bounded domains of R^N."

A Two-Level Preconditioned Helmholtz Subspace Iterative Method for Maxwell Eigenvalue Problems

In this paper, based on a domain decomposition method, we shall propose a two-level preconditioned Helmholtz subspace iterative (PHSI) method for solving algebraic eigenvalue problems resulting from edge element approximation of Maxwell eigenvalue problems. The two-level PHSI method may compute simple eigenpairs, multiple eigenpairs, and clustered eigenpairs. It is very efficient and robust for solving Maxwell eigenvalue problems since the method reduces solutions of fine eigenvalue problems to a series of solutions of parallel preconditioned systems and Helmholtz projected systems. After solving a small dimensional eigenvalue problem in a trial subspace, we finally get the total error reduction of the first several discrete eigenvalues as , where is a positive parameter, is the overlapping size among subdomains, and is the diameter of subdomains. The positive number is independent of the mesh size and internal gaps among the first several Maxwell eigenvalues, i.e., the method is optimal corresponding to the degrees of freedom and cluster robust. The number increases with increasing, and decreases with increasing. The -dependent constant decreases monotonically to 1 as , which means the more the number of subdomains, the better the convergence rate. Numerical results supporting our theory are given.

The instability onset of generalized isoflux mean flow using Brinkman-Darcy-Bénard model in a fluid saturated porous channel

Infinitely wide single-cell basic flow formation within a porous channel in the context of isoflux boundary conditions and its instability onset were determined in a recent work (Brandao et al., 2021). In the present analysis, the scope is to extend the latter work to incorporate the effective viscosity term within the transitional flow in the frame of Brinkman-Darcy-Bénard convection. The accounted model entails a physical parameter, so-called as the Darcy number whose limiting values result in either the Darcy-Bénard convection of Brandao et al. (2021) or the Rayleigh-Bénard convection, which is the first time studied in the literature. Apart from the Darcy number, the Rayleigh number and the horizontal temperature parameter are the essential physical parameters. Neutral stability curves affected by these parameters are numerically simulated to explore the onset of instability within the linear stability analysis. The solution of eigenvalue problem fully conforms to the one in the above published in the absence of Darcy number. Comparatively larger Rayleigh numbers are derived within the Brinkman’s model rather than the published Darcy model.

Singular quadratic eigenvalue problems: linearization and weak condition numbers

The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of $$\delta $$ -weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases $$\delta $$ -weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.

Damage detection in the T-welded joint using Rayleigh-like feature guided wave

Steel T-welded joint structures have been largely applied in aerospace and marine infrastructures. The weld length in T-weld joint structures is usually long, and it is difficult to access the weld conditions for damage detection purposes. This is due to the T-welded joint structure possessing a more complex cross-section than the conventional welded plate-like structures. Rather than formed by two plates and joined with a butt weld, the joint regions in the T-welded joint structure are connected to the sides of three plates to form a T-welded joint, and hence cannot be considered as plate-like structures. This makes guided wave modes difficult to be identified and exploited. This paper assesses the feasibility to detect weld defects on a steel T-welded joint structure using the feature guided wave (FGW) technique. FGW technique can provide a fast and long-range inspection for local topographical features, such as welded joints, offering great potential for effective screening of damages. Semi-Analytical Finite Element (SAFE) method is applied to acquire modal solutions in the waveguide. A new algorithm is developed to obtain FGW modes with both high energy concentration and low attenuation from the solutions of SAFE eigenvalue problem. One particular FGW mode with similar mode shape as the Rayleigh surface wave is identified, of which the propagation occurs only in the joint region with small attenuation. This FGW, labelled Rayleigh T-welded wave (RTW), is explored in comprehensive experimental and numerical studies to investigate the propagation and backward scattering characteristics regarding optimal excitation frequency and different sizes of weld defect. The numerical and experimental results are in good agreement. The outcome of the study confirms the capability and robustness of RTW in detecting small weld defects, which allows a long-distance and sensitive screening for the small weld defects at the joint region in the T-welded joint structure.

Convergence Properties of a Linear Diffusion-Acceleration Method for k-Eigenvalue Transport Problems

Most methods that use low-order operators to accelerate the iterative solution of transport eigenvalue problems employ nonlinear functionals of the transport solution (such as Eddington tensors) in their low-order equations, which are themselves standard eigenvalue problems. Here, we discuss linear diffusion synthetic acceleration (DSA) for -eigenvalue problems, which belongs to a family of methods that has received less attention than its nonlinear counterparts. We review the history of these linear methods as far as we know it and describe theoretical questions that to our knowledge have remained unanswered. With these methods, a low-order problem is solved after each transport step for an updated eigenvalue and an additive correction to the eigenfunction. These low-order problems are not standard eigenvalue problems, for they contain residuals as fixed sources. The low-order problems admit infinitely many solutions (updated and additive correction to the eigenfunction), and the solution that is obtained depends on the initial guess and iterative method chosen for the low-order problems. Experience has shown that when the low-order problems are solved with a powerlike iteration method and certain initial guesses, they yield solutions that cause rapid convergence to the correct high-order solution. We study the convergence properties of this algorithm applied to two model problems: an infinite homogeneous medium and a one-cell problem. For the infinite homogeneous problem, we present a Fourier analysis of the linear DSA method, which demonstrates that when the low-order problems are solved using a powerlike iteration scheme, the linear DSA scheme provides immediate convergence of the -eigenvalue and rapid convergence of the eigenfunction (much like DSA applied to scattering iterations in fixed-source problems). For the one-cell problems, we find that the linear scheme for -eigenvalue problems performs approximately as well as DSA for fixed-source problems. The latter analysis reveals a quantitative bound on the consistency between low- and high-order operators that is necessary and sufficient for convergence of those problems. With some theoretical foundations for the linear methods now established, we turn to numerical testing. We find, as others have before us using different low-order operators, that the method works well in practice. We provide numerical results from reactor problems in which our linear DSA is approximately as effective as the more widely used nonlinear methods. Our theoretical and numerical results add to the body of evidence that the linear methodology offers a simple path to rapid convergence of -eigenvalue problems, especially for codes that already employ linear low-order operators to converge scattering iterations.

On the $$R$$-matrix identities related to elliptic anisotropic spin Ruijsenaars–Macdonald operators

We propose and prove a set of identities for ${\rm GL}_M$ elliptic $R$-matrix (in the fundamental representation). In the scalar case ($M=1$) these are elliptic function identities derived by S.N.M. Ruijsenaars as necessary and sufficient conditions for his kernel identity underlying construction of integral solutions to quantum spinless Ruijsenaars-Schneider model. In this respect the result of the present paper can be considered as the first step towards constructing solutions of quantum eigenvalue problem for the anisotropic spin Ruijsenaars model.

Sharp Morrey–Sobolev inequalities and eigenvalue problems on Riemannian–Finsler manifolds with nonnegative Ricci curvature

Combining the sharp isoperimetric inequality established by Z. Balogh and A. Kristály [Math. Ann., in press, doi:10.1007/s00208-022-02380-1] with an anisotropic symmetrization argument, we establish sharp Morrey–Sobolev inequalities on [Formula: see text]-dimensional Finsler manifolds having nonnegative [Formula: see text]-Ricci curvature. A byproduct of this method is a Hardy–Sobolev-type inequality in the same geometric setting. As applications, by using variational arguments, we guarantee the existence/multiplicity of solutions for certain eigenvalue problems and elliptic PDEs involving the Finsler–Laplace operator. Our results are also new in the Riemannian setting.

Refined isogeometric analysis of quadratic eigenvalue problems

Certain applications that analyze damping effects require the solution of quadratic eigenvalue problems (QEPs). We use refined isogeometric analysis (rIGA) to solve quadratic eigenproblems. rIGA discretization, while conserving desirable properties of maximum-continuity isogeometric analysis (IGA), reduces the interconnection between degrees of freedom by adding low-continuity basis functions. This connectivity reduction in rIGA’s algebraic system results in faster matrix LU factorizations when using multifrontal direct solvers. We compare computational costs of rIGA versus those of IGA when employing Krylov eigensolvers to solve quadratic eigenproblems arising in 2D vector-valued multifield problems. For large problem sizes, the eigencomputation cost is governed by the cost of LU factorization, followed by costs of several matrix–vector and vector–vector multiplications, which correspond to Krylov projections. We minimize the computational cost by introducing C0 and C1 separators at specific element interfaces for our rIGA generalizations of the curl-conforming Nédélec and divergence-conforming Raviart–Thomas finite elements. Let p be the polynomial degree of basis functions; the LU factorization is up to O((p−1)2) times faster when using rIGA compared to IGA in the asymptotic regime. Thus, rIGA theoretically improves the total eigencomputation cost by O((p−1)2) for sufficiently large problem sizes. Yet, in practical cases of moderate-size eigenproblems, the improvement rate deteriorates as the number of computed eigenvalues increases because of multiple matrix–vector and vector–vector operations. Our numerical tests show that rIGA accelerates the solution of quadratic eigensystems by O(p−1) for moderately sized problems when we seek to compute a reasonable number of eigenvalues.

Approximate solutions of the Fourth-Order Eigenvalue Problem

In this paper, the differential transformation (DTM) and the Adomian decomposition (ADM) methods are proposed for solving fourth order eigenvalue problem. This fourth order eigenvalue problem has nonstrongly regular boundary conditions. This the fourth order problem has been examined for p(t) = t, B = 0, a = 0,01 where p(t) ≠ 0 is a complex valued and a ≠ 0 The differential transformation and the Adomian decomposition methods are briefly described. An approximate solution is obtained by performing seven iterations with the Adomian decomposition method. The same number of iterations have been made in the differential transformation method. The approximation results obtained by both methods have been compared with each other. These data have been presented in table. The ADM and the DTM approximation solutions have been shown by plotting in Figure 1. Here, the approaches obtained by using the two methods are found to be in high agreement. Consequently, highly accurate approximate solutions of fourth order eigenvalue problem are obtained. Such good results also revealed that the Adomian decomposition and the differential transformation methods are fast, economical and motivating. The exact solution of the fourth order eigenvalue problem for nonstrongly regular can not be found in the literature. Therefore, this study will give an important idea to determine approximate solution behavior of this fourth order problem.