Approximating eigenvectors and eigenvalues across a wide range of design

This paper concerns optimization of the design of large structures in which the solution to a finite-element-based vibration problem is part of the objective function. A method is presented where a spline-fit is used as an approximation of the eigenvalues and eigenvectors across a given design range. subroutines facilitate nonlinear sensitivity calculations which provide the eigenvalue and eigenvector derivatives used in the approximation. This paper presents an example that illustrates good approximations to frequency and mode shapes valid across a wide range of design changes. The method provides the potential for changing mode shapes interactively, and using that information as part of the optimization process.     

A Riemann solver for RANS

An exact expression for a system of both eigenvalues and right/left eigenvectors of a Jacobian matrix for a convective two-equation differential closure RANS operator split along a curvilinear coordinate is derived. It is shown by examples of numerical modeling of supersonic flows over a flat plate and a compression corner with separation that application of the exact system of eigenvalues and eigenvectors to the Roe approach for approximate solution of the Riemann problem gives rise to an increase in the convergence rate, better stability and higher accuracy of a steady-state solution in comparison with those in the case of an approximate system.     

Positive eigenvector of nonlinear perturbations of nonsymmetric M-matrix and Newton iterative solution

Positive eigenvector of nonlinear perturbations of nonsymmetric M-matrix and its Newton iterative solution are studied. It is shown that any number greater than the smallest positive eigenvalue of the M-matrix is an eigenvalue of the nonlinear problem and that the corresponding positive eigenvector is unique and the Newton iteration of the positive eigenvector is convergent. Moreover, such positive eigenvectors form a monotone increasing and continuous function of the corresponding eigenvalues.     

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Approximation of the eigenvalues of a fourth order differential equation with non-smooth coefficients

The eigenvalues of a fourth order, generalized eigenvalue problem in one dimension, with non-smooth coefficients are approximated by a finite element method, introduced in an earlier work by the author and A. Lutoborski, in the context of a similar source problem with non-smooth coefficients. Error estimates for the approximate eigenvalues and eigenvectors are obtained, showing a better performance of this method, when applied to eigenvalue approximation, compared to a standard finite element method with arbitrary mesh.     

Backward errors for eigenvalues and eigenvectors of Hermitian,skew-Hermitian,H-even and H-odd matrix polynomials

We discuss the perturbation analysis for eigenvalues and eigenvectors of structured homogeneous matrix polynomials with Hermitian, skew-Hermitian, H-even and H-odd structure. We construct minimal structured perturbations (structured backward errors) such that an approximate eigenvalue and eigenvector pair (finite or infinite eigenvalues) is an exact eigenvalue eigenvector pair of an appropriately perturbed structured matrix polynomial. We present various comparisons with unstructured backward errors and previous backward errors constructed for the non-homogeneous case and show that our results generalize previous results.     

The distance spectrum of the path Pn and The First Distance Eigenvector of Connected Graphs

Forumulas are given for all of the eigenvalues and eigenvectors of the distance matrix of the path P_n on n vertices. It is shown that P_n has the maximum distance spectral radius among all connected graphs of order n, and an ordering property of the entries of the Perron-Frobenius eigenvector is presented.     

Enhanced algebraic substructuring for symmetric generalized eigenvalue problems

This article proposes a new substructuring algorithm to approximate the algebraically smallest eigenvalues and corresponding eigenvectors of a symmetric positive-definite matrix pencil $(A,M)$. The proposed approach partitions the graph associated with $(A,M)$ into a number of algebraic substructures and builds a Rayleigh–Ritz projection subspace by combining spectral information associated with the interior and interface variables of the algebraic domain. The subspace associated with interior variables is built by computing substructural eigenvectors and truncated Neumann series expansions of resolvent matrices. The subspace associated with interface variables is built by computing eigenvectors and associated leading derivatives of linearized spectral Schur complements. The proposed algorithm can take advantage of multilevel partitionings when the size of the pencil. Experiments performed on problems stemming from discretizations of model problems showcase the efficiency of the proposed algorithm and verify that adding eigenvector derivatives can enhance the overall accuracy of the approximate eigenpairs, especially those associated with eigenvalues located near the origin.     

Robust Solution to Fuzzy Identification Problem with Uncertain Data by Regularization

This study considers the robust identification of the parameters describing a Sugeno type fuzzy inference system with uncertain data. The objective is to minimize the worst-case residual error using a numerically efficient algorithm. The Sugeno type fuzzy systems are linear in consequent parameters but nonlinear in antecedent parameters. The robust consequent parameters identification problem can be formulated as second-order cone programming problem. The optimal solution of this second-order cone problem can be interpreted as solution of a Tikhonov regularization problem with a special choice of regularization parameter which is optimal for robustness (Ghaoui and Lebret (1997). SAIM Journal of Matrix Analysis and Applications 18, 1035–1064). The final regularized nonlinear optimization problem allowing simultaneous identification of antecedent and consequent parameters is solved iteratively using a generalized Gauss–Newton like method. To illustrate the approach, several simulation studies on numerical examples including the modelling of a spectral data function (one-dimensional benchmark example) is provided. The proposed robust fuzzy identification scheme has been applied to approximate the physical fitness of patients with a fuzzy expert system. The identified fuzzy expert system is shown to be capable of capturing the decisions (experiences) of a medical expert.     

Nowhere-zero eigenvectors of graphs

A vector is called nowhere-zero if it has no zero entry. In this article we search for graphs with nowhere-zero eigenvectors. We prove that distance-regular graphs and vertex-transitive graphs have nowhere-zero eigenvectors for all of their eigenvalues and edge-transitive graphs have nowhere-zero eigenvectors for all non-zero eigenvalues. Among other results, it is shown that a graph with three distinct eigenvalues has a nowhere-zero eigenvector for its smallest eigenvalue.     

Pseudoeigenvector bases and deflated GMRES for highly nonnormal matrices

Pseudoeigenvalues have been extensively studied for highly nonnormal matrices. This paper focuses on the corresponding pseudoeigenvectors. The properties and uses of pseudoeigenvector bases are investigated. It is shown that pseudoeigenvector bases can be much better conditioned than eigenvector bases. We look at the stability and the varying quality of pseudoeigenvector bases. Then applications are considered including the exponential of a matrix. Several aspects of GMRES convergence are looked at, including why using approximate eigenvectors to deflate eigenvalues can be effective even when there is not a basis of eigenvectors.     

Weighting method for bi-level linear fractional programming problems

In this paper we consider the solution of a bi-level linear fractional programming problem (BLLFPP) by weighting method. A non-dominated solution set is obtained by this method. In this article decision makers (DMs) provide their preference bounds to the decision variables that is the upper and lower bounds to the decision variables they control. We convert the hierarchical system into scalar optimization problem (SOP) by finding proper weights using the analytic hierarchy process (AHP) so that objective functions of both levels can be combined into one objective function. Here the relative weights represent the relative importance of the objective functions.     

Newton's method for the common eigenvector problem

In El Ghazi et al. [Backward error for the common eigenvector problem, CERFACS Report TR/PA/06/16, Toulouse, France, 2006], we have proved the sensitivity of computing the common eigenvector of two matrices A and B, and we have designed a new approach to solve this problem based on the notion of the backward error.If one of the two matrices (saying A) has n eigenvectors then to find the common eigenvector we have just to write the matrix B in the basis formed by the eigenvectors of A. But if there is eigenvectors with multiplicity >1, the common vector belong to vector space of dimension >1 and such strategy would not help compute it.In this paper we use Newton's method to compute a common eigenvector for two matrices, taking the backward error as a stopping criteria.We mention that no assumptions are made on the matrices A and B.     

Continuous methods for extreme and interior eigenvalue problems

In this paper, continuous methods are introduced to compute both the extreme and interior eigenvalues and their corresponding eigenvectors for real symmetric matrices. The main idea is to convert the extreme and interior eigenvalue problems into some optimization problems. Then a continuous method which includes both a merit function and an ordinary differential equation (ODE) is introduced for each resulting optimization problem. The convergence of each ODE solution is proved for any starting point. The limit of each ODE solution for any starting point is fully studied. Both the extreme and the interior eigenvalues and their corresponding eigenvectors can be easily obtained under a very mild condition. Promising numerical results are also presented.     

A curvilinear optimization method based upon iterative estimation of the eigensystem of the Hessian matrix

An algorithm was recently presented that minimizes a nonlinear function in several variables using a Newton-type curvilinear search path. In order to determine this curvilinear search path the eigenvalue problem of the Hessian matrix of the objective function has to be solved at each iteration of the algorithm. In this paper an iterative procedure requiring gradient information only is developed for the approximation of the eigensystem of the Hessian matrix. It is shown that for a quadratic function the approximated eigenvalues and eigenvectors tend rapidly to the actual eigenvalues and eigenvectors of its Hessian matrix. The numerical tests indicate that the resulting algorithm is very fast and stable. Moreover, the fact that some approximations to the eigenvectors of the Hessian matrix are available is used to get past saddle points and accelerate the rate of convergence on flat functions.     

<Emphasis Type="Italic">α</Emphasis>-Conservative approximation for probabilistically constrained convex programs

In this paper, we address an approximate solution of a probabilistically constrained convex program (PCCP), where a convex objective function is minimized over solutions satisfying, with a given probability, convex constraints that are parameterized by random variables. In order to approach to a solution, we set forth a conservative approximation problem by introducing a parameter α which indicates an approximate accuracy, and formulate it as a D.C. optimization problem.     

The eigenvalue problem for elliptic partial differential equation with multi-point nonlocal conditions

We study the spectral problem for the system of difference equations of a two-dimensional elliptic partial differential equation with nonlocal conditions. A new form of two-point nonlocal conditions that involve interior points is proposed. The matrix of the difference system is nonsymmetric thus different types of eigenvalues occur. The conditions for the existence of the eigenvalues and their corresponding eigenvectors are presented for the one dimensional problem. Then, these relations are generalized to the two-dimensional problem by the separation of variables technique.     

Eigenvectors and ratio limit theorems for Markov chains and their relatives

Associated to classes of countable discrete Markov chains or, more generally, column-finite nonnegative infinite matrices, and a finite subset of the state space, is a dimension group. In many cases, this dimension group gives information about the nonnegative eigenvectors of the process. Moreover, the study of the nonnegative eigenvectors is, equivalent to the traces on an analytic one parameter family of dimension groups. We pay particular attention to the case that there is at most one nonnegative eigenvector per eigenvalue, giving a number of sufficient conditions. Using the techniques developed here, we also show that under a reasonable set of conditions (principle among them that there be just one nonnegative eigenvector for the spectral radius), a (one-sided) ratio limit theorem holds. Supported in part by an operating grant from NSERC (Canada) and an Isaac Walton Killam Fellowship (Canada Council).     

Galerkin eigenvector approximations

How close are Galerkin eigenvectors to the best approximation available out of the trial subspace? Under a variety of conditions the Galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace--and this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors. Both orthogonal-Galerkin and Petrov-Galerkin methods are considered here with a special emphasis on nonselfadjoint problems, thus extending earlier studies by Chatelin, Babuska and Osborn, and Knyazev. Consequences for the numerical treatment of elliptic PDEs discretized either with finite element methods or with spectral methods are discussed. New lower bounds to the of a pair of operators are developed as well.

     

Simulation-based parametric optimization for long-term asset allocation using behavioral utilities

This article presents a simulation-based methodology for finding optimal investment strategy for long-term financial planning. The problem becomes intractable due to its size or the properties of the utility function of the investors. One approach is to make simplifying assumptions regarding the states of the world and/or utility functions in order to obtain a solution. These simplifications lead to the true solution of an approximate problem. Our approach is to find a good approximate solution to the true problem. We approximate the optimal decision in each period with a low dimensional parameterization, thus reformulating the problem as a non-linear, simulation-based optimization in the parameter space. The dimension of the reformulated optimization problem becomes linear in the number of periods. The approach is extendable to other problems where similar solution characteristics are known.