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.

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.

SUFFICIENT CONDITIONS FOR THE SOLUBILITY OF ADDITIVE INVERSE EIGENVALUE PROBLEMS

1 IntroductionLet Rn×n be the set of all n×n real matrices.Rn=Rn×1.Cn×ndenotes the set of all n×n complex matrices.We are interested in solving the following inverse eigenvalue prob-lems:Problem A （Additive inverse eigenvalue problem） Given an n×n real matrix A=(aij),and n distinct real numbers λ1,λ2,…,λn,find a real n×n diagonal matrix

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.

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.

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.

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.

On Convergence of MRQI and IMRQI Methods for Hermitian Eigenvalue Problems

Bai et al.proposed the multistep Rayleigh quotient iteration(MRQI)as well as its inexact variant(IMRQI)in a recent work(Comput.Math.Appl.77:2396–2406,2019).These methods can be used to effectively compute an eigenpair of a Hermitian matrix.The convergence theorems of these methods were established under two conditions imposed on the initial guesses for the target eigenvalue and eigenvector.In this paper,we show that these two conditions can be merged into a relaxed one,so the convergence conditions in these theorems can be weakened,and the resulting convergence theorems are applicable to a broad class of matrices.In addition,we give detailed discussions about the new convergence condition and the corresponding estimates of the convergence errors,leading to rigorous convergence theories for both the MRQI and the IMRQI.

EFFICIENT SPECTRAL METHODS FOR EIGENVALUE PROBLEMS OF THE INTEGRAL FRACTIONAL LAPLACIAN ONABALLOFANYDIMENSION

An efficient spectral-Galerkin method for eigenvalue problems of the integral fractional Laplacian on a unit ball of any dimension is proposed in this paper.The symmetric positive definite linear system is retained explicitly which plays an important role in the numerical analysis.And a sharp estimate on the algebraic system's condition number is established which behaves as N4s with respect to the polynomial degree N,where 2s is the fractional derivative order.The regularity estimate of solutions to source problems of the fractional Laplacian in arbitrary dimensions is firstly investigated in weighted Sobolev spaces.Then the regularity of eigenfunctions of the fractional Laplacian eigenvalue problem is readily derived.Meanwhile,rigorous error estimates of the eigenvalues and eigenvectors are ob-tained.Numerical experiments are presented to demonstrate the accuracy and efficiency and to validate the theoretical results.

On a Hybrid Method for Inverse Transmission Eigenvalue Problems

In this paper, we are concerned with the inverse transmission eigenvalue problem to recover the shape as well as the constant refractive index of a penetrable medium scatterer. The linear sampling method is employed to determine the transmission eigenvalues within a certain wavenumber interval based on far-field measurements. Based on a prior information given by the linear sampling method, the neural network approach is proposed for the reconstruction of the unknown scatterer. We divide the wavenumber intervals into several subintervals, ensuring that each transmission eigenvalue is located in its corresponding subinterval. In each such subinterval, the wavenumber that yields the maximum value of the indicator functional will be included in the input set during the generation of the training data. This technique for data generation effectively ensures the consistent dimensions of model input. The refractive index and shape are taken as the output of the network. Due to the fact that transmission eigenvalues considered in our method are relatively small,certain super-resolution effects can also be generated. Numerical experiments are presented to verify the effectiveness and promising features of the proposed method in two and three dimensions.

Parallel computing study for the large-scale generalized eigenvalue problems in modal analysis

In this paper we study the algorithms and their parallel implementation for solving large-scale generalized eigenvalue problems in modal analysis.Three predominant subspace algorithms,i.e.,Krylov-Schur method,implicitly restarted Arnoldi method and Jacobi-Davidson method,are modified with some complementary techniques to make them suitable for modal analysis.Detailed descriptions of the three algorithms are given.Based on these algorithms,a parallel solution procedure is established via the PANDA framework and its associated eigensolvers.Using the solution procedure on a machine equipped with up to 4800processors,the parallel performance of the three predominant methods is evaluated via numerical experiments with typical engineering structures,where the maximum testing scale attains twenty million degrees of freedom.The speedup curves for different cases are obtained and compared.The results show that the three methods are good for modal analysis in the scale of ten million degrees of freedom with a favorable parallel scalability.

Some Uniqueness Results for a Class of Quasilinear Elliptic Eigenvalue Problems

Existence and uniqueness results are obtained for positive radial solutions of a class of quasilinear elliptic equations in a N-ball or an annulus without monotone assumptions on the nonlinear term f.It is also proved that there is no non-radial positive solution.

A Two-Level Method for Nonsymmetric Eigenvalue Problems

A two-level discretization method for eigenvalue problems is studied.Compared to the standard Galerkin finite element discretization technique performed on a fine gridthis method discretizes the eigenvalue problem on a coarse grid and obtains an improved eigenvector(eigenvalue) approximation by solving only a linear problem on the fine grid (or two linear problemsfor the case of eigenvalue approximation of nonsymmetric problems). The improved solution has theasymptotic accuracy of the Galerkin discretization solution. The link between the method and theiterated Galerkin method is established. Error estimates for the general nonsymmetric case arederived.

Sensitivity Analysis of Semi-simple Eigenvalues of Regular Quadratic Eigenvalue Problems

This paper discusses the sensitivity analysis of semisimple eigenvalues and associated eigen-matrix triples of regular quadratic eigenvalue problems analytically dependent on several parameters. The directional derivatives of semisimple eigenvalues are obtained. The average of semisimple eigenvalues and corresponding eigen-matrix triple are proved to be analytic, and their partial derivatives are given. On these grounds, the sensitivities of the semisimple eigenvalues and corresponding eigenvector matrices are defined.

Local a priori/a posteriori error estimates of conforming finite elements approximation for Steklov eigenvalue problems

Based on the work of Xu and Zhou(2000),this paper makes a further discussion on conforming finite elements approximation for Steklov eigenvalue problems,and proves a local a priori error estimate and a new local a posteriori error estimate in ||·||1,Ω0 norm for conforming elements eigenfunction,which has not been studied in existing literatures.

A SMALLEST SINGULAR VALUE METHOD FOR SOLVING INVERSE EIGENVALUE PROBLEMS

Utilizing the properties of the smallest singular value of a matrix, we propose a new, efficient and reliable algorithm for solving nonsymmetric matrix inverse eigenvalue problems, and compare it with a known method. We also present numerical experiments which illustrate our results.

The Discontinuous Galerkin Method by Divergence-Free Patch Reconstruction for Stokes Eigenvalue Problems

The discontinuous Galerkin method by divergence-free patch reconstruction is proposed for Stokes eigenvalue problems.It utilizes the mixed finite element framework.The patch reconstruction technique constructs two categories of approximation spaces.Namely,the local divergence-free space is employed to discretize the velocity space,and the pressure space is approximated by standard reconstruction space simultaneously.Benefit from the divergence-free constraint;the identical element patch serves two approximation spaces while using the element pair Pm+1/Pm.The optimal error estimate is derived under the inf-sup condition framework.Numerical examples are carried out to validate the inf-sup test and the theoretical results.

Inverse Eigenvalue Problems for Exploring the Dynamics of Systems Biology Models

This paper describes inverse eigenvalue problems that arise in studying qualitative dynamics in systems biology models.An algorithm based on lift-andproject iterations is proposed,where the lifting step entails solving a constrained matrix inverse eigenvalue problem.In particular,prior to carrying out the iterative steps,a-priori bounds on the entries of the Jacobian matrix are computed by relying on the reaction network structure as well as the form of the rate law expressions for the model under consideration.Numerical results on a number of models show that the proposed algorithm can be used to computationally explore the possible dynamical scenarios while identifying the important mechanisms via the use of sparsity-promoting regularization.

An Extended Two-Step Method for Inverse Eigenvalue Problems with Multiple Eigenvalues

In recent years,numerical solutions of the inverse eigenvalue problems with multiple eigenvalues have attracted the attention of some researchers,and there have been a few algorithms with quadratic convergence.We propose here an extended two-step method for solving the inverse eigenvalue problems with multiple eigenvalues.Under appropriate assumptions,the convergence analysis of the extended method is presented and the cubic root-convergence rate is proved.Numerical experiments are provided to confirm the theoretical results and comparisons with the inexact Cayley transform method are made.Our extended method and convergence result in the present paper may enrich the results of numerical solutions of the inverse eigenvalue problems with multiple eigenvalues.

A NUMERICAL CALCULATION METHOD FOR EIGENVALUE PROBLEMS OF NONLINEAR INTERNAL WAVES

Generally speaking, the background shear current U（z） must be taken into account in eigenvalue problems of nonlinear internal waves in ocean, as is different from those of linear internal waves. A numerical calculation method for eigenvalue problems of nonlinear internal waves is presented in this paper on the basis of the Thompson-Haskell＇s calculation method. As an application of this method, at a station （21°N, 117°15′E） in the South China Sea, a modal structure and parameters of nonlinear internal waves are calculated, and the results closely agree with the calculated results based on observation by Yang et al..