Homotopy algorithm for symmetric eigenvalue problems

Summary The homotopy method can be used to solve eigenvalue-eigenvector problems. The purpose of this paper is to report the numerical experience of the homotopy method of computing eigenpairs for real symmetric tridiagonal matrices together with a couple of new theoretical results. In practice, it is rerely of any interest to compute all the eigenvalues. The homotopy method, having the order preserving property, can provide any specific eigenvalue without calculating any other eigenvalues. Besides this advantage, we note that the homotopy algorithm is to a large degree a parallel algorithm. Numerical experimentation shows that the homotopy method can be very efficient especially for graded matrices.Research was supported in part by NSF under Grant DMS-8701349     

Asymptotic expansions for the Sturm-Liouville problem by homotopy perturbation method

The asymptotic formulae for the eigenvalues and eigenfunctions of Sturm-Liouville problem with the Dirichlet boundary conditions when the potential is square integrable on [0, 1] are obtained by using homotopy perturbation method.     

Bounds for eigenvalues of second-order elliptic differential operators

Summary We derive bounds for the firstN eigenvalues of a linear second-order elliptic differential operator on a bounded domain, subject to mixed boundary conditions. The results are achieved by a combination of (a generalized version of) Kato's estimates and a homotopy algorithm.     

A new application of the homotopy analysis method: Solving the Sturm–Liouville problems

In this paper, the homotopy analysis method (HAM) is applied to numerically approximate the eigenvalues of the second and fourth-order Sturm–Liouville problems. These eigenvalues are calculated by starting the HAM algorithm with one initial guess. In this paper, it can be observed that the auxiliary parameter ℏ, which controls the convergence of the HAM approximate series solutions, also can be used in predicting and calculating multiple solutions. This is a basic and more important qualitative difference in analysis between HAM and other methods.     

A new modification of He’s homotopy perturbation method for rapid convergence of nonlinear undamped oscillators

In this paper we present a new efficient modification of the homotopy perturbation method with x ³ force nonlinear undamped oscillators for the first time that will accurate and facilitate the calculations. The He’s homotopy perturbation method is modified by adding a term to linear operator depends on the equation and boundary conditions. We find that this modified homotopy perturbation method works very well for the wide range of time and boundary conditions for nonlinear oscillator. Only two or three iteration leads to high accuracy of the solutions. We then conduct a comparative study between the new modification and the homotopy perturbation method for strongly nonlinear oscillators. Numerical illustrations are investigated to show the accurate of the techniques. The new modified method accelerates the rapid convergence of the solution, reduces the error solution and increases the validity range. The new modification introduces a promising tool for many nonlinear problems.     

Solving generalized equations via homotopies

Global Newton methods for computing solutions of nonlinear systems of equations have recently received a great deal of attention. By using the theory of generalized equations, a homotopy method is proposed to solve problems arising in complementarity and mathematical programming, as well as in variational inequalities. We introduce the concepts of generalized homotopies and regular values, characterize the solution sets of such generalized homotopies and prove, under boundary conditions similar to Smale’s [10], the existence of a homotopy path which contains an odd number of solutions to the problem. We related our homotopy path to the Newton method for generalized equations developed by Josephy [3]. An interpretation of our results for the nonlinear programming problem will be given.     

A study of the eigenvalue sensitivity by homotopy and perturbation methods

In this work, the sensitivity to material characteristics of eigenvalues is studied. From an initial structure, some defects of material or/and geometry are introduced. A method is proposed to solve the new eigenvalue problem from the initial one without using classical techniques. This method is based on the association of a homotopy transformation and the perturbation method.     

On homotopy varieties

Given an algebraic theory T, a homotopy T-algebra is a simplicial set where all equations from T hold up to homotopy. All homotopy T-algebras form a homotopy variety. We will give a characterization of homotopy varieties analogous to the characterization of varieties.     

Numerical solution of singularly perturbed fifth order two point boundary value problem

We will consider Adomain decomposition method and the homotopy method to solve a fifth order singularly perturbed BVP arising in viscoelastic flows. The success and pitfalls of the methods will be investigated. Numerical testing will be provided to show the efficiency of the methods proposed. Comparison with the work of others will also be done.     

Exact completion of path categories and algebraic set theory: Part I: Exact completion of path categories

We introduce the notion of a “category with path objects”, as a slight strengthening of Kenneth Brown's classical notion of a “category of fibrant objects”. We develop the basic properties of such a category and its associated homotopy category. Subsequently, we show how the exact completion of this homotopy category can be obtained as the homotopy category associated to a larger category with path objects, obtained by freely adjoining certain homotopy quotients. In a second part of this paper, we will present an application to models of constructive set theory. Although our work is partly motivated by recent developments in homotopy type theory, this paper is written purely in the language of homotopy theory and category theory, and we do not presuppose any familiarity with type theory on the side of the reader.     

A reflection on the implicitly restarted Arnoldi method for computing eigenvalues near a vertical line

In this article, we will study the link between a method for computing eigenvalues closest to the imaginary axis and the implicitly restarted Arnoldi method. The extension to computing eigenvalues closest to a vertical line is straightforward, by incorporating a shift. Without loss of generality we will restrict ourselves here to computing eigenvalues closest to the imaginary axis.In a recent publication, Meerbergen and Spence discussed a new approach for detecting purely imaginary eigenvalues corresponding to Hopf bifurcations, which is of interest for the stability of dynamical systems. The novel method is based on inverse iteration (inverse power method) applied on a Lyapunov-like eigenvalue problem. To reduce the computational overhead significantly a projection was added.This method can also be used for computing eigenvalues of a matrix pencil near a vertical line in the complex plane. We will prove in this paper that the combination of inverse iteration with the projection step is equivalent to Sorensen’s implicitly restarted Arnoldi method utilizing well-chosen shifts.     

Analysis of parameterized quadratic eigenvalue problems in computational acoustics with homotopic deviation theory

This paper analyzes a family of parameterized quadratic eigenvalue problems from acoustics in the framework of homotopic deviation theory. Our specific application is the acoustic wave equation (in 1D and 2D) where the boundary conditions are partly pressure release (homogeneous Dirichlet) and partly impedance, with a complex impedance parameter ζ. The admittance t = 1/ζ is the classical homotopy parameter. In particular, we study the spectrum when t → ∞. We show that in the limit part of the eigenvalues remain bounded and converge to the so‐called kernel points. We also show that there exist the so‐called critical points that correspond to frequencies for which no finite value of the admittance can cause a resonance. Finally, the physical interpretation that the impedance condition is transformed into a pressure release condition when |t| → ∞ enables us to give the kernel points in closed form as eigenvalues of the discrete Dirichlet problem. Copyright © 2006 John Wiley & Sons, Ltd.     

Über einige formale Eigenschaften von Faserungen und h-Faserungen

The aim of this paper is to study some formal properties of fibrations (=continuous maps which have the covering homotopy property), cofibrations (=maps which have the homotopy extension property), h-fibrations ([2],6.4) and h-cofibrations([2],2.2). We introduce the notion of a generalized homotopy system in a category (2). This notion will be selfdual. To show that fibrations and cofibrations, h-fibrations and h-cofibrations are dual notions, no use will be made of adjoint functors. Our approach admits the transition from a categoryC to other categories, e.g. to the categoryC _B of objects over a given object B ofC.     

解约束非凸规划问题的同伦方法的收敛性定理

本文在利用组合内点同伦方法求解约束非凸规划问题时,得到了一些新的收敛性定理.证明了同伦映射为正则映射的条件下,选取合适的同伦方程,用此同伦方法得到的K-K-T点一定是问题局部最优解.     

Fixed points of compact approachable multimaps in topological vector spaces

Applying the homotopy extension theorem for compact approachable vector fields with star-shaped values, we prove the existence of fixed points and eigenvalues of compact approachable multimaps in topological vector spaces. In particular, we give the Birkhoff-Kellogg Theorem for compact approachable multimaps with star-shaped values in normal spaces.     

Perturbation theorems for positive eigenvalues of countably condensing maps

Using a homotopy extension theorem, we give a perturbation theorem for countably condensing maps in a more general setting. Moreover, we prove perturbation theorems for positive eigenvalues of countably condensing maps in locally convex topological vector spaces which include the case of condensing maps due to Jerofsky [1].     

Minimal compact global attractor for a damped semilinear wave equation

Abstract

The asymptotic behavior of eigenvalues of an elliptic operator with a divergence form is discussed. The coefficients of the operator are discontinuous through a boundary of a subdomain and degenerate to zero on the subdomain when a parameter tends to zero. We will prove that the eigenvalues approach eigenvalues of the Laplacian on the subdomain or on the complement. We will obtain precise asymptotic behavior of their convergence.     

The Budget Constrained Multi-Product Newsboy Problem with Reactive Production: A Problem from Entrepreneurial Network Construction

This paper develops an extended newsboy model and presents a formulation for this model. This new model has solved the budget contained multi-product newsboy problem with the reactive production. This model can be used to describe the status of entrepreneurial network construction. We use the Lagrange multiplier procedure to deal with our problem, but it is too complicated to get the exact solution. So we introduce the homotopy method to deal with it. We give the flow chart to describe how to get the solution via the homotopy method. We also illustrate our model in both the classical procedure and the homotopy method. Comparing the two methods, we can see that the homotopy method is more exact and efficient.     

Comparison of two reliable analytical methods based on the solutions of fractional coupled Klein–Gordon–Zakharov equations in plasma physics

In this paper, homotopy perturbation transform method and modified homotopy analysis method have been applied to obtain the approximate solutions of the time fractional coupled Klein–Gordon–Zakharov equations. We consider fractional coupled Klein–Gordon–Zakharov equation with appropriate initial values using homotopy perturbation transform method and modified homotopy analysis method. Here we obtain the solution of fractional coupled Klein–Gordon–Zakharov equation, which is obtained by replacing the time derivatives with a fractional derivatives of order α ∈ (1, 2], β ∈ (1, 2]. Through error analysis and numerical simulation, we have compared approximate solutions obtained by two present methods homotopy perturbation transform method and modified homotopy analysis method. The fractional derivatives here are described in Caputo sense.     

Spectral homotopy analysis method and its convergence for solving a class of nonlinear optimal control problems

A combination of the hybrid spectral collocation technique and the homotopy analysis method is used to construct an iteration algorithm for solving a class of nonlinear optimal control problems (NOCPs). In fact, the nonlinear two-point boundary value problem (TPBVP), derived from the Pontryagin’s Maximum Principle (PMP), is solved by spectral homotopy analysis method (SHAM). For the first time, we present here a convergence proof for SHAM. We treat in detail Legendre collocation and Chebyshev collocation. It is indicated that Legendre collocation gives the same numerical results with Chebyshev collocation. Comparisons are made between SHAM, Matlab bvp4c generated results and results from literature such as homotopy perturbation method (HPM), optimal homotopy perturbation method (OHPM) and differential transformations.