Highly Accurate Golden Section Search Algorithms and Fictitious Time Integration Method for Solving Nonlinear Eigenvalue Problems

This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues.Simultaneously,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly.The real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very fast.All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.

Some Trace Formulas of a Eigenvalue Problem

In present paper, using some methods of approximation theory, the trace formulas for eigenvalues of a eigenvalue problem are calculated under the periodic condition and the decaying condition at x∞.

THE EIGENVALUE PROBLEM FOR THE LAPLACIAN EQUATIONS

This article studies the Dirichlet eigenvalue problem for the Laplacian equations △u = -λu, x ∈Ω , u = 0, x ∈δΩ, where Ω belong to R^n is a smooth bounded convex domain. By using the method of appropriate barrier function combined with the maximum principle, authors obtain a sharp lower bound of the difference of the first two eigenvalues for the Dirichlet eigenvalue problem. This study improves the result of S.T. Yau et al.

Mode decomposition of nonlinear eigenvalue problems and application in flow stability

Direct numerical simulations are carried out with different disturbance forms introduced into the inlet of a flat plate boundary layer with the Mach number 4.5. According to the biorthogonal eigenfunction system of the linearized Navier-Stokes equations and the adjoint equations, the decomposition of the direct numerical simulation results into the discrete normal mode is easily realized. The decomposition coefficients can be solved by doing the inner product between the numerical results and the eigenfunctions of the adjoint equations. For the quadratic polynomial eigenvalue problem, the inner product operator is given in a simple form, and it is extended to an Nth-degree polynomial eigenvalue problem. The examples illustrate that the simplified mode decomposition is available to analyze direct numerical simulation results.

THE TWO-LEVEL STABILIZED FINITE ELEMENT METHOD BASED ON MULTISCALE ENRICHMENT FOR THE STOKES EIGENVALUE PROBLEM

In this paper,we first propose a new stabilized finite element method for the Stokes eigenvalue problem.This new method is based on multiscale enrichment,and is derived from the Stokes eigenvalue problem itself.The convergence of this new stabilized method is proved and the optimal priori error estimates for the eigenfunctions and eigenvalues are also obtained.Moreover,we combine this new stabilized finite element method with the two-level method to give a new two-level stabilized finite element method for the Stokes eigenvalue problem.Furthermore,we have proved a priori error estimates for this new two-level stabilized method.Finally,numerical examples confirm our theoretical analysis and validate the high effectiveness of the new methods.

Physics-constrained neural network for solving discontinuous interface K-eigenvalue problem with application to reactor physics

Machine learning-based modeling of reactor physics problems has attracted increasing interest in recent years.Despite some progress in one-dimensional problems,there is still a paucity of benchmark studies that are easy to solve using traditional numerical methods albeit still challenging using neural networks for a wide range of practical problems.We present two networks,namely the Generalized Inverse Power Method Neural Network(GIPMNN)and Physics-Constrained GIPMNN(PC-GIPIMNN)to solve K-eigenvalue problems in neutron diffusion theory.GIPMNN follows the main idea of the inverse power method and determines the lowest eigenvalue using an iterative method.The PC-GIPMNN additionally enforces conservative interface conditions for the neutron flux.Meanwhile,Deep Ritz Method(DRM)directly solves the smallest eigenvalue by minimizing the eigenvalue in Rayleigh quotient form.A comprehensive study was conducted using GIPMNN,PC-GIPMNN,and DRM to solve problems of complex spatial geometry with variant material domains from the fleld of nuclear reactor physics.The methods were compared with the standard flnite element method.The applicability and accuracy of the methods are reported and indicate that PC-GIPMNN outperforms GIPMNN and DRM.

Two-level stabilized finite element method for Stokes eigenvalue problem

A two-level stabilized finite element method for the Stokes eigenvalue problem based on the local Gauss integration is considered. This method involves solving a Stokes eigenvalue problem on a coarse mesh with mesh size H and a Stokes problem on a fine mesh with mesh size h -- O（H2）, which can still maintain the asymptotically optimal accuracy. It provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution, which involves solving a Stokes eigenvalue problem on a fine mesh with mesh size h. Hence, the two-level stabilized finite element method can save a large amount of computational time. Moreover, numerical tests confirm the theoretical results of the present method.

Eigenvalue Problem of Doubly Stochastic Hamiltonian Systems with Boundary Conditions

In this paper, we investigate the eigenvalue problem of forward-backward doubly stochastic dii^erential equations with boundary value conditions. We show that this problem can be represented as an eigenvalue problem of a bounded continuous compact operator. Hence using the famous Hilbert-Schmidt spectrum theory, we can characterize the eigenvalues exactly.

Uniqueness theorem,theorem of reciprocity,and eigenvalue problems in linear theory of porous piezoelectricity

The uniqueness theorem and the theorem of reciprocity in the linearized porous piezoelectricity are established under the assumption of positive definiteness of elastic and electric fields. General theorems in the linear theory of porous piezoelectric materials are proved for the quasi-static electric field approximation. The uniqueness theorem is also proved without using the positive definiteness of the elastic field. An eigenvalue problem associated with free vibrations of a porous piezoelectric body is stud- ied using the abstract formulation. Some properties of operators are also proved. The problem of frequency shift due to small disturbances, based on an abstract formulation, is studied using a variational and operator approach. A perturbation analysis of a special ease is also given.

A NEW MATRIX PERTURBATION METHOD FOR ANALYTICAL SOLUTION OF THE COMPLEX MODAL EIGENVALUE PROBLEM OF VISCOUSLY DAMPED LINEAR VIBRATION SYSTEMS

A new matrix perturbation analysis method is presented for efficient approximate solution of the complex modal quadratic generalized eigenvalue problem of viscously damped linear vibration systems. First, the damping matrix is decomposed into the sum of a proportional-and a nonproportional-damping parts, and the solutions of the real modal eigenproblem with the proportional dampings are determined, which are a set of initial approximate solutions of the complex modal eigenproblem. Second, by taking the nonproportional-damping part as a small modification to the proportional one and using the matrix perturbation analysis method, a set of approximate solutions of the complex modal eigenvalue problem can be obtained analytically. The result is quite simple. The new method is applicable to the systems with viscous dampings-which do not deviate far away from the proportional-damping case. It is particularly important that the solution technique be also effective to the systems with heavy, but not over, dampings. The solution formulas of complex modal eigenvlaues and eigenvectors are derived up to second-order perturbation terms. The effectiveness of the perturbation algorithm is illustrated by an exemplar numerical problem with heavy dampings. In addition, the practicability of approximately estimating the complex modal eigenvalues, under the proportional-damping hypothesis, of damped vibration systems is discussed by several numerical examples.

A BIHARMONIC EIGENVALUE PROBLEM AND ITS APPLICATION

In this article, we focus on the eigenvalue problem of the following linear biharmonic equation in R^N：△^2u-αu＋λg（x）u=0 with u ∈H^2（R^N）,u≠0,N≥5Note that there are two parameters α and λ in it, which is different from the usual eigenvalue problems. Here, we consider λ as an eigenvalue and seek Ior a sulble range of parameter α, which ensures that problem （＊） has a maximal eigenvalue. As the loss of strong maximum principle for our problem, we can only get the existence of non-trivial solutions, not positive solutions, in this article. As an application, by using these results, we studied also the existence of non-trivial solutions for an asymptotically linear biharmonic equation in R^N.

Solving Fully Fuzzy Nonlinear Eigenvalue Problems of Damped Spring-Mass Structural Systems Using Novel Fuzzy-Affine Approach

The dynamic analysis of damped structural system by using finite element method leads to nonlinear eigenvalue problem(NEP)(particularly,quadratic eigenvalue problem).In general,the parameters of NEP are considered as exact values.But in actual practice because of different errors and incomplete information,the parameters may have uncertain or vague values and such uncertain values may be considered in terms of fuzzy numbers.This article proposes an efficient fuzzy-affine approach to solve fully fuzzy nonlinear eigenvalue problems(FNEPs)where involved parameters are fuzzy numbers viz.triangular and trapezoidal.Based on the parametric form,fuzzy numbers have been transformed into family of standard intervals.Further due to the presence of interval overestimation problem in standard interval arithmetic,affine arithmetic based approach has been implemented.In the proposed method,the FNEP has been linearized into a generalized eigenvalue problem and further solved by using the fuzzy-affine approach.Several application problems of structures and also general NEPs with fuzzy parameters are investigated based on the proposed procedure.Lastly,fuzzy eigenvalue bounds are illustrated with fuzzy plots with respect to its membership function.Few comparisons are also demonstrated to show the reliability and efficacy of the present approach.

THE BI-SELF-CONJUGATE AND NONNEGATIVE DEFINITE SOLUTIONS TO THE INVERSE EIGENVALUE PROBLEM OF QUATERNION MATRICES

The main aim of this paper is to discuss the following two problems: Problem I: Given X ∈ Hn×m (the set of all n×m quaternion matrices), A = diag(λ1,…, λm) EEEEE Hm×m, find A ∈ BSHn×n≥such that AX = X(?), where BSHn×n≥ denotes the set of all n×n quaternion matrices which are bi-self-conjugate and nonnegative definite. Problem Ⅱ2= Given B ∈ Hn×m, find B ∈ SE such that ||B-B||Q = minAE∈=sE ||B-A||Q, where SE is the solution set of problem I , || ·||Q is the quaternion matrix norm. The necessary and sufficient conditions for SE being nonempty are obtained. The general form of elements in SE and the expression of the unique solution B of problem Ⅱ are given.

Multiple Solutions for the Eigenvalue Problem of Nonlinear Fractional Differential Equations

In this article, multiple solutions for the eigenvalue problem of nonlinear fractional differential equation is considered. We obtain the existence and multiplicity results of positive solutions by using some fixed point theorems.

ARNOLDI REDUCTION ALGORITHM FOR LARGE SCALE GYROSCOPIC EIGENVALUE PROBLEM

Based on Arnoldi's method, a version of generalized Arnoldi algorithm has been developed for the reduction of gyroscopic eigenvalue problems. By utilizing the skew symmetry of system matrix, a very simple recurrence scheme, named gyroscopic Arnoldi reduction algorithm has been obtained, which is even simpler than the Lanczos algorithm for symmetric eigenvalue problems. The complex number computation is completely avoided. A restart technique is used to enable the reduction algorithm to have iterative characteristics. It has been found that the restart technique is not only effective for the convergence of multiple eigenvalues but it also furnishes the reduction algorithm with a technique to check and compute missed eigenvalues. By combining it with the restart technique, the algorithm is made practical for large-scale gyroscopic eigenvalue problems. Numerical examples are given to demonstrate the effectiveness of the method proposed.

Structured Eigenvalue Problems in Electronic Structure Methods from a Unified Perspective

In(relativistic)electronic structure methods,the quaternion matrix eigenvalue problem and the linear response(Bethe-Salpeter)eigenvalue problem for excitation energies are two frequently encoun-tered structured eigenvalue problems.While the former problem was thoroughly studied,the later problem in its most general form,namely,the complex case without assuming the positive definiteness of the electronic Hessian,was not fully understood.In view of their very similar mathematical structures,we examined these two problems from a unified point of view.We showed that the identification of Lie group structures for their eigenvectors provides a framework to design diagonalization algorithms as well as numerical optimizations techniques on the corresponding manifolds.By using the same reduction algorithm for the quaternion matrix eigenvalue problem,we provided a necessary and sufficient condition to characterize the different scenarios,where the eigenvalues of the original linear response eigenvalue problem are real,purely imaginary,or complex.The result can be viewed as a natural generalization of the well-known condition for the real matrix case.

AN INVERSE EIGENVALUE PROBLEM FOR JACOBI MATRICES

is gained by deleting the kth row and the kth column （k=1,2,...,n） from Tn.We put for-ward an inverse eigenvalue problem to be that:If we don’t know the matrix T1,n,but weknow all eigenvalues of matrix T1,k-1,all eigenvalues of matrix Tk+1,k,and all eigenvaluesof matrix T1,n could we construct the matrix T1,n.Let μ1,μ2,…,μk-1,μk,μk+1,…,μn-1,

Generalized Inverse Eigenvalue Problem for Centrohermitian Matrices

In this paper we first consider the existence and the general form of solution to the following generalized inverse eigenvalue problem(GIEP): given a set of n-dimension complex vectors {x j}m j=1 and a set of complex numbers {λ j}m j=1, find two n×n centrohermitian matrices A,B such that {x j}m j=1 and {λ j}m j=1 are the generalized eigenvectors and generalized eigenvalues of Ax=λBx, respectively. We then discuss the optimal approximation problem for the GIEP. More concretely, given two arbitrary matrices, , ∈C n×n, we find two matrices A and B such that the matrix (A*,B*) is closest to (,) in the Frobenius norm, where the matrix (A*,B*) is the solution to the GIEP. We show that the expression of the solution of the optimal approximation is unique and derive the expression for it.

Inverse Eigenvalue Problems for a Structure with Linear Parameters

The inverse design method of a dynamic system with linear parameters has been studied. For some specified eigenvalues and eigenvectors, the design parameter vector which is often composed of whole or part of coefficients of spring and mass of the system can be obtained and the rigidity and mass matrices of an initially designed structure can be reconstructed through solving linear algebra equations. By using implicit function theorem, the conditions of existence and uniqueness of the solution are also deduced. The theory and method can be used for inverse vibration design of complex structure system.

THE SOLUTION OF RANDOM EIGENVALUE PROBLEM WITH SMALL STOCHASTIC PROCESSES

This paper considers an eigenvalue problem containing small stochastic processes. For every fixed is, we can use the Prufer substitution to prove the existence of the random solutions lambda(n) and u(n) in the meaning of large probability. These solutions can be expanded in epsilon regularly, and their correction terms can be obtained by solving some random linear differential equations.