Constrained Eigenvalue Problem Research Articles

A computational framework for homogenization and multiscale stability analyses of nonlinear periodic materials

AbstractThis article presents a consistent computational framework for multiscale first‐order finite strain homogenization and stability analyses of rate‐independent solids with periodic microstructures. The homogenization formulation is built on a priori discretized microstructure, and algorithms for computing the matrix representations of the homogenized stresses and tangent moduli are consistently derived. The homogenization results lose their validity at the onset of first bifurcation, which can be computed from multiscale stability analysis. The multiscale instabilities include: (a) microscale structural instability calculated by Bloch wave analysis; and (b) macroscale material instability calculated by rank‐1 convexity checks on the homogenized tangent moduli. Implementation details of the Bloch wave analysis are provided, including the selection of wave vector space and the retrieval of real‐valued buckling mode from complex‐valued Bloch wave. Three methods are detailed for solving the resulted constrained eigenvalue problem—two condensation methods and a null‐space projection method. Both the homogenization and stability analyses are verified using numerical examples including hyperelastic and elastoplastic periodic materials. Various microscale buckling phenomena are demonstrated. Aligned with theoretical results, the numerical results show that the microscopic long wavelength buckling can be equivalently detected by the loss of rank‐1 convexity of the homogenized tangent moduli.

Constrained Eigenvalue Minimization of Incomplete Pairwise Comparison Matrices by Nelder-Mead Algorithm

Pairwise comparison matrices play a prominent role in multiple-criteria decision-making, particularly in the analytic hierarchy process (AHP). Another form of preference modeling, called an incomplete pairwise comparison matrix, is considered when one or more elements are missing. In this paper, an algorithm is proposed for the optimal completion of an incomplete matrix. Our intention is to numerically minimize a maximum eigenvalue function, which is difficult to write explicitly in terms of variables, subject to interval constraints. Numerical simulations are carried out in order to examine the performance of the algorithm. The results of our simulations show that the proposed algorithm has the ability to solve the minimization of the constrained eigenvalue problem. We provided illustrative examples to show the simplex procedures obtained by the proposed algorithm, and how well it fills in the given incomplete matrices.

Linear Constrained Rayleigh Quotient Optimization: Theory and Algorithms

We consider the following constrained Rayleigh quotient optimization problem (CRQopt) $$ \min_{x\in \mathbb{R}^n} x^{T}Ax\,\,\mbox{subject to}\,\, x^{T}x=1\,\mbox{and}\,C^{T}x=b, $$ where $A$ is an $n\times n$ real symmetric matrix and $C$ is an $n\times m$ real matrix. Usually, $m\ll n$. The problem is also known as the constrained eigenvalue problem in the literature because it becomes an eigenvalue problem if the linear constraint $C^{T}x=b$ is removed. We start by equivalently transforming CRQopt into an optimization problem, called LGopt, of minimizing the Lagrangian multiplier of CRQopt, and then an problem, called QEPmin, of finding the smallest eigenvalue of a quadratic eigenvalue problem. Although such equivalences has been discussed in the literature, it appears to be the first time that these equivalences are rigorously justified. Then we propose to numerically solve LGopt and QEPmin by the Krylov subspace projection method via the Lanczos process. The basic idea, as the Lanczos method for the symmetric eigenvalue problem, is to first reduce LGopt and QEPmin by projecting them onto Krylov subspaces to yield problems of the same types but of much smaller sizes, and then solve the reduced problems by some direct methods, which is either a secular equation solver (in the case of LGopt) or an eigensolver (in the case of QEPmin). The resulting algorithm is called the Lanczos algorithm. We perform convergence analysis for the proposed method and obtain error bounds. The sharpness of the error bound is demonstrated by artificial examples, although in applications the method often converges much faster than the bounds suggest. Finally, we apply the Lanczos algorithm to semi-supervised learning in the context of constrained clustering.

Robustness Can Be Cheap: A Highly Efficient Approach to Discover Outliers under High Outlier Ratios

Efficient detection of outliers from massive data with a high outlier ratio is challenging but not explicitly discussed yet. In such a case, existing methods either suffer from poor robustness or require expensive computations. This paper proposes a Low-rank based Efficient Outlier Detection (LEOD) framework to achieve favorable robustness against high outlier ratios with much cheaper computations. Specifically, it is worth highlighting the following aspects of LEOD: (1) Our framework exploits the low-rank structure embedded in the similarity matrix and considers inliers/outliers equally based on this low-rank structure, which facilitates us to encourage satisfying robustness with low computational cost later; (2) A novel re-weighting algorithm is derived as a new general solution to the constrained eigenvalue problem, which is a major bottleneck for the optimization process. Instead of the high space and time complexity (O((2n)2)/O((2n)3)) required by the classic solution, our algorithm enjoys O(n) space complexity and a faster optimization speed in the experiments; (3) A new alternative formulation is proposed for further acceleration of the solution process, where a cheap closed-form solution can be obtained. Experiments show that LEOD achieves strong robustness under an outlier ratio from 20% to 60%, while it is at most 100 times more memory efficient and 1000 times faster than its previous counterpart that attains comparable performance. The codes of LEOD are publicly available at https://github.com/demonzyj56/LEOD.

Optimal duration-bandwidth localized antisymmetric biorthogonal wavelet filters

We present a design of a new class of compactly supported antisymmetric biorthogonal wavelet filter banks which have the analysis as well as the synthesis filters of even-length. Here, the analysis and the synthesis filters are designed to have minimum joint duration-bandwidth localization (JDBL). The design of filters has been formulated as a direct time-domain linearly constrained eigenvalue problem that does not involve any parametrization and iterations. The optimal analysis and synthesis filters have been obtained as the eigenvectors of the positive definite matrices. The closed form analytic expression for the objective function has been presented. The perfect reconstruction and regularity conditions have been incorporated in the design by employing time-domain matrix characterization. The method can control duration and bandwidth localizations of the analysis and synthesis filters, independently. A few design examples have been presented and compared with previous works. The performance of the optimal filter banks designed by employing the proposed method has been evaluated in image coding and signal denoising applications.

Buckling analysis of structures under combined loading with acceleration forces

The structures of concern in this study are subject to two types of forces: dead loads from the acceleration imposed on the structures as well as the installed operation machines and the additional adjustable forces. We wish to determine the critical values of the adjustable forces when buckling of the structures occurs. The mathematical statement of such a problem gives rise to a constrained eigenvalue problem (CEVP) in which the dominant eigenvalue is subject to an equality constraint. A numerical algorithm for solving the CEVP is proposed in which an iterative method is employed to identify an interval embracing the target eigenvalue. The algorithm is applied to four engineering application examples finding the critical loads of a fixed-free beam subject to its own body force, two plane structures and one wide-flange beam using shell elements when acceleration force is present. The accuracy is demonstrated using the first example whose classical solution exists. The significance of the equality constraint in the EVP is shown by comparing the solutions without the constraint on the eigenvalue. Effectiveness and accuracy of the numerical algorithm are presented.

Mixed methods for viscoelastodynamics and topology optimization

A truly-mixed approach for the analysis of viscoelastic structures and continua is presented. An additive decomposition of the stress state into a viscoelastic part and a purely elastic one is introduced along with an Hellinger-Reissner variational principle wherein the stress represents the main variable of the formulation whereas the kinematic descriptor (that in the case at hand is the velocity field) acts as Lagrange multiplier. The resulting problem is a Differential Algebraic Equation (DAE) because of the need to introduce static Lagrange multipliers to comply with the Cauchy boundary condition on the stress. The associated eigenvalue problem is known in the literature as constrained eigenvalue problem and poses several difficulties for its solution that are addressed in the paper. The second part of the paper proposes a topology optimization approach for the rationale design of viscoelastic structures and continua. Details concerning density interpolation, compliance problems and eigenvalue-based objectives are given. Worked numerical examples are presented concerning both the dynamic analysis of viscoelastic structures and their topology optimization.

Analysis of plasmon resonances in metallic nanostructures in proximity to dielectric objects with application to heat-assisted magnetic recording

A novel approach to the calculation of plasmon resonance in metallic nanoparticle located nearby a dielectric object is presented. The plasmon resonance problem for such structure is formulated as a constrained eigenvalue problem for specific coupled boundary integral equations. By solving this eigenvalue problem, the resonance frequencies (wavelengths) of the metallic nanoparticle as well as the corresponding plasmon modes are computed. In this paper, two examples of application are considered and a good agreement between the computational results and analytical solution as well as with available experimental and numerical data is demonstrated.

A Multiplicity Result for a Class of Elliptic Problems on a Compact Riemannian Manifold

The existence of multiple solutions for a constrained eigenvalue problem on a compact Riemannian manifold without boundary is proved. In the framework of variational analysis, we use an Arcoya---Carmona type three critical points result. As an application an Emden---Fowler type equation is given.

Curvilinear vector finite difference approach to the computation of waveguide modes

We describe here a Vector Finite Difference approach to the evaluation of waveguide eigenvalues and modes for rectangular, circular and elliptical waveguides. The FD is applied using a 2D cartesian, polar and elliptical grid in the waveguide section. A suitable Taylor expansion of the vector mode function allows to take exactly into account the boundary condition. To prevent the raising of spurious modes, our FD approximation results in a constrained eigenvalue problem, that we solve using a decomposition method. This approach has been evaluated comparing our results to the analytical modes of rectangular and circula rwaveguide, and to known data for the elliptic case.

A smoothing Broyden-like method for polyhedral cone constrained eigenvalue problem

For the polyhedral cone constrained eigenvalue problem over apolyhedral cone, based on its nonsmooth transformed version and asmoothing technique, we propose a modified smoothing Broyden-likemethod and establish its convergence under suitable conditions. Thegiven computational experiments show the efficiency of the proposedmethod.

A constrained eigenvalue problem and nodal and modal control of vibrating systems

A constrained non-homogeneous linear eigenvalue problem is introduced. The application given to the problem is of finding the frequency and amplitudes of exciting forces that impose constraints on the configurations of vibratory modes. The scope of the problem is wider. It is shown that the problem may be transformed to a singular unsymmetric generalized eigenvalue problem. Depending on the given data the problem may have finite, infinite or empty spectrum. The solvability of the problem is analysed. Examples demonstrate the application and the various results.

An eigenvalue problem for elastic cracks in free space

We study in this paper an eigenvalue problem for the linear elasticity equations in three-dimensional space. The problem is defined in the whole space cut by a planar crack. The eigenvalue appears in a linear condition relating the traction to the jump in displacements across the crack. We prove for such problems that an eigenspace containing eigenfunctions, which do not average to zero over the crack is in general not simple. Then we prove for a more constrained eigenvalue problem, where the direction of slip over the crack is imposed, that the first eigenspace is in that case simple. Copyright © 2009 John Wiley & Sons, Ltd.

Quantum cosmological Friedman models with a massive Yang–Mills field

We prove the existence of a spectral resolution of the Wheeler–DeWitt equation when the matter field is provided by a massive Yang–Mills field. The resolution is achieved by first solving the free eigenvalue problem for the gravitational field and then the constrained eigenvalue problem for the Yang–Mills field. In the latter case, the mass of the Yang–Mills field assumes the role of the eigenvalue.

Preconditioning constrained eigenvalue problems

The purpose of our paper is to introduce a robust preconditioning scheme for the numerical solution of the leftmost eigenvalues and corresponding eigenvectors of a constrained eigenvalue problem. This constrained eigenvalue problem is congruent to a nonsymmetric eigenvalue problem with nontrivial Jordan blocks associated with infinite eigenvalues. The proposed preconditioning scheme is relevant to the application of Krylov subspace methods and preconditioned eigensolvers. The two key results are a semi-orthogonal decomposition and a transformation process that implicitly combines a preconditioning step followed by abstract projection onto the subspace associated with the finite eigenvalues. Numerical results demonstrate the effectiveness of the preconditioning scheme.

On the natural vibrations of linear structures with constraints

The undamped natural vibrations of a constrained linear structure are given by the solutions to a generalized eigenvalue problem derived from the equations of motion for the constrained system involving Lagrangian multipliers. The eigenvalue problem derived is defined by the mass matrix of the unconstrained structure and a non-symmetric and singular stiffness matrix for the constrained system. The character of the solution of the eigenvalue problem of the constrained system is stated and proved in a Theorem. Applications of the constrained eigenvalue problem to some simple structures are demonstrated. Finally a condition for the calculation of the damped natural vibrations for the constrained structure in terms of the undamped mode shapes is formulated.

A Theory of the Optimal Policy of Oil Recovery by Secondary Displacement Processes

We consider the stability of the displacement of a viscous fluid in a porous medium by a less viscous fluid containing a solute. For the case in which the total amount of injected solute is fixed, we formulate a theory for the optimal injection policy, i.e., the policy which minimizes the growth constant of the Saffman–Taylor–Chouke instability. The result is a nonlinear constrained eigenvalue problem with the viscosity ratio, $\alpha $, of the two fluids and the total solute, N, as parameters. We develop perturbation solutions for large and small N and numerical solutions for all N. Several features of the optimal policy which have been noted or demonstrated empirically by previous workers are proven, and new features of the solution are discussed.