Rayleigh Quotient Minimization Research Articles

Robust Rayleigh quotient minimization and generalized eigenvalue problems

We study the problem of minimizing the non-linear trace quotient trace(V^TG(V)V)/trace(V^TH(V)V) over the Stiefel manifold of all n x p matrices with orthonormal columns. Hereby we assume G(V) and H(V) to be symmetric and positive definite for all V. In this way we generalize the robust Rayleigh quotient optimization introduced by Bai et al. in the article "Robust rayleigh quotient minimization and nonlinear eigenvalue problems". We show a possible way to minimize the nonlinear trace quotient by joining and generalizing different known techniques, e.g. the SCF-iteration, and examine it by testing a small example.

A Maximum Likelihood Method for Joint DOA and Polarization Estimation Based on Manifold Separation

The use of the polarization diversity of a target signal at a polarization-sensitive antenna array can enhance the target detection and tracking capabilities of a radar. In this article, the manifold separation steering vector modeling technique is used to develop a maximum likelihood method for joint direction of arrival (DOA) and polarization estimation. Manifold separation can incorporate antenna array nonideal characteristics (e.g., cross polarization, mutual coupling) into the estimation algorithm using array calibration measurements. In the proposed technique, the estimation problem is formulated as a generalized Rayleigh quotient minimization problem that is transformed into a determinant minimization problem. Both the azimuth and elevation angles are estimated using the fast Fourier transform. Unlike the existing manifold separation based polarimetric element space (PES) multiple signal classification method and the PES Capon method, the proposed method can obtain DOA and polarization estimates based on very small-size primary data samples, even with a single sample, which makes the proposed method more suitable for nonstationary target polarization. The performance of the proposed method is demonstrated through simulations. The Cramer-Rao lower bound for joint DOA and polarization is also used for comparison with empirical errors.

Rayleigh quotient minimization method for symmetric eigenvalue problems

In this paper, we present a new method, which is referred to as the Rayleigh quotient minimization method, for computing one extreme eigenpair of symmetric matrices. This method converges globally and attains cubic convergence rate locally. In addition, inexact implementations and its numerical stability of the Rayleigh quotient minimization method are explored. Finally, we use numerical experiments to demonstrate the convergence properties and show the competitiveness of the new method for solving symmetric eigenvalue problems.

Rayleigh quotient minimization for absolutely one-homogeneous functionals

In this paper we examine the problem of minimizing generalized Rayleigh quotients of the form , where both J and H are absolutely one-homogeneous functionals. This can be viewed as minimizing J where the solution is constrained to be on a generalized sphere with , where H is any norm or semi-norm. The solution admits a nonlinear eigenvalue problem, based on the subgradients of J and H. We examine several flows which minimize the ratio. This is done both by time-continuous flow formulations and by discrete iterations. We focus on a certain flow, which is easier to analyze theoretically, following the theory of Brezis on flows with maximal monotone operators. A comprehensive theory is established, including convergence of the flow. We then turn into a more specific case of minimizing graph total variation on the L1 sphere, which approximates the Cheeger-cut problem. Experimental results show the applicability of such algorithms for clustering and classification of images.

Comparative study of methods for integrated model identification and state of charge estimation of lithium-ion battery

Model-based observers appeal to both research and industry utilization due to the high accuracy and robustness. To further improve the robustness to dynamic work conditions and battery ageing, the online model identification is integrated to the state estimation, giving rise to the co-estimation methods. This paper systematically compares three types of co-estimation methods for the online state of charge of lithium-ion battery. This first method is dual extended Kalman filter which uses two parallel filters for co-estimation. The second method is a typical data-model fusion method which uses recursive least squares for model identification and extended Kalman filter for state estimation. Meanwhile, a noise compensating method based on recursive total least squares and Rayleigh quotient minimization is exploited for online model identification, which is further designed in conjunction with the extended Kalman filter to estimate the state of charge. Simulation and experimental studies are carried out to compare the performances of three methods in terms of the accuracy, convergence property, and noise immunity. The computing cost and tuning effort are further discussed to give insights to the application prospective of different methods.

Online Model Identification and State-of-Charge Estimate for Lithium-Ion Battery With a Recursive Total Least Squares-Based Observer

The state-of-charge (SOC) observer with online model adaption generally has high accuracy and robustness. However, the unexpected sensing of noises is shown to cause the biased identification of model parameters. To address this problem, a novel technique which integrates a recursive total least squares (RTLS) with an SOC observer is proposed to enhance the online model identification and SOC estimate. An efficient method is exploited to solve the Rayleigh quotient minimization which lays the basis of the RTLS. The number of multiplies, divides, and square roots is elaborated to show the low computational complexity of the developed RTLS. Simulation and experimental results show that the proposed RTLS-based observer attenuates the model identification bias caused by noise corruption effectively, and, thereby, provides a more reliable estimation of SOC. The proposed method is further compared with several available methods to highlight its superiority in terms of accuracy and the robustness to noise corruption.

Generalized RQ Minimization With Applications in Array Transmit Beamforming

This letter deals with generalized Rayleigh quotient (RQ) minimization with possibly singular matrices in the quadratic forms. An extended theorem providing closed-form analytical solutions for different cases is derived. It is shown that the theorem is an extension of the typical RQ minimization. Next, its applications in array transmit beamforming under the criterion of minimum integrated sidelobe level are illustrated. Finally, theoretical analyses and numerical comparisons are provided to validate both the theorem and the beamformers.

Colocated MIMO radar waveform design for transmit beampattern formation

In this paper, colocated MIMO radar waveform design is considered by minimizing the integrated side-lobe level to obtain beam patterns with lower side-lobe levels than competing methods. First, a quadratic programming problem is formulated to design beam patterns by using the criteria for a minimal integrated side-lobe level. A theorem is derived that provides a closed-form analytical optimal solution that appears to be an extension of the Rayleigh quotient minimization for a possibly singular matrix in quadratic form. Such singularities are shown to occur in the problem of interest, but proofs for the optimum solution in these singular matrix cases could not be found in the literature. Next, an additional constraint is added to obtain beam patterns with desired 3 dB beamwidths, resulting in a nonconvex quadratically constrained quadratic program which is NP-hard. A semidefinite program and a Gaussian randomized semidefinite relaxation are used to determine feasible solutions arbitrarily close to the solution to the original problem. Theoretical and numerical analyses illustrate the impacts of changing the number of transmitters and orthogonal waveforms employed in the designs. Numerical comparisons are conducted to evaluate the proposed design approaches.

Optimization algorithms on the Grassmann manifold with application to matrix eigenvalue problems

This article deals with the Grassmann manifold as a submanifold of the matrix Euclidean space, that is, as the set of all orthogonal projection matrices of constant rank, and sets up several optimization algorithms in terms of such matrices. Interest will center on the steepest descent and Newton’s methods together with applications to matrix eigenvalue problems. It is shown that Newton’s equation in the proposed Newton’s method applied to the Rayleigh quotient minimization problem takes the form of a Lyapunov equation, for which an existing efficient algorithm can be applied, and thereby the present Newton’s method works efficiently. It is also shown that in case of degenerate eigenvalues the optimal solutions form a submanifold diffeomorphic to a Grassmann manifold of lower dimension. Furthermore, to generate globally converging sequences, this article provides a hybrid method composed of the steepest descent and Newton’s methods on the Grassmann manifold together with convergence analysis.

A Riemannian symmetric rank-one trust-region method

The well-known symmetric rank-one trust-region method--where the Hessian approximation is generated by the symmetric rank-one update--is generalized to the problem of minimizing a real-valued function over a $$d$$ d -dimensional Riemannian manifold. The generalization relies on basic differential-geometric concepts, such as tangent spaces, Riemannian metrics, and the Riemannian gradient, as well as on the more recent notions of (first-order) retraction and vector transport. The new method, called RTR-SR1, is shown to converge globally and $$d+1$$ d + 1 -step q-superlinearly to stationary points of the objective function. A limited-memory version, referred to as LRTR-SR1, is also introduced. In this context, novel efficient strategies are presented to construct a vector transport on a submanifold of a Euclidean space. Numerical experiments--Rayleigh quotient minimization on the sphere and a joint diagonalization problem on the Stiefel manifold--illustrate the value of the new methods.

Novel passive localization algorithm based on weighted restricted total least square

A novel multi-observer passive localization algorithm based on the weighted restricted total least square (WRTLS) is proposed to solve the bearings-only localization problem in the presence of observer position errors. Firstly, the unknown matrix perturbation information is utilized to form the WRTLS problem. Then, the corresponding constrained optimization problem is transformed into an unconstrained one, which is a generalized Rayleigh quotient minimization problem. Thus, the solution can be got through the generalized eigenvalue decomposition and requires no initial state guess process. Simulation results indicate that the proposed algorithm can approach the Cramer-Rao lower bound (CRLB), and the localization solution is asymptotically unbiased.

Novel passive localization algorithm based on double side matrix-restricted total least squares

In order to solve the bearings-only passive localization problem in the presence of erroneous observer position, a novel algorithm based on double side matrix-restricted total least squares (DSMRTLS) is proposed. First, the aforementioned passive localization problem is transferred to the DSMRTLS problem by deriving a multiplicative structure for both the observation matrix and the observation vector. Second, the corresponding optimization problem of the DSMRTLS problem without constraint is derived, which can be approximated as the generalized Rayleigh quotient minimization problem. Then, the localization solution which is globally optimal and asymptotically unbiased can be got by generalized eigenvalue decomposition. Simulation results verify the rationality of the approximation and the good performance of the proposed algorithm compared with several typical algorithms.

On the Stiffness of the Tangent Modulus Tensor in Elastoplasticity

In finite element analysis of pressure vessels undergoing elastoplastic deformation, low stiffness of the tangent modulus tensor will engender low stiffness in the tangent stiffness matrix, posing a risk of computational difficulties such as poor convergence. The current investigation presents the explicit tangent modulus tensor in an elastoplastic model based on a Von Mises yield surface with isotropic work hardening, and the associated flow rule. The stiffness of the tangent modulus tensor is assessed by deriving explicit expressions for its minimum eigenvalue using both tensor diagonalization and Rayleigh quotient minimization. The derived expressions are validated computationally. Using the minimum eigenvalue, the stiffness is found to depend on the current path in stress space. The results of the current investigation suggest a way of following a stress path, which bypasses low stiffness, while attaining the prescribed load.

Efficient parallel solution to large‐size sparse eigenproblems with block FSAI preconditioning

SUMMARYThe choice of the preconditioner is a key factor to accelerate the convergence of eigensolvers for large‐size sparse eigenproblems. Although incomplete factorizations with partial fill‐in prove generally effective in sequential computations, the efficient preconditioning of parallel eigensolvers is still an open issue. The present paper describes the use of block factorized sparse approximate inverse (BFSAI) preconditioning for the parallel solution of large‐size symmetric positive definite eigenproblems with both a simultaneous Rayleigh quotient minimization and the Jacobi–Davidson algorithm. BFSAI coupled with a block diagonal incomplete decomposition proves a robust and efficient parallel preconditioner in a number of test cases arising from the finite element discretization of 3D fluid‐dynamical and mechanical engineering applications, outperforming FSAI even by a factor of 8 and exhibiting a satisfactory scalability. Copyright © 2011 John Wiley & Sons, Ltd.

Solving symmetric eigenvalue problem via genetic algorithms: Serial versus parallel implementation

Abstract We propose a new method for computing more than one eigenvalue of small and medium sized real symmetric matrices simultaneously by minimizing a suitably defined average Rayleigh Quotient ( ρ a v ) for the targeted group of eigenvalues by Genetic Algorithm. The proposed method is tested on two benchmark matrices of varying dimensions. Performance statistics is presented both as functions of dimensions of the matrices and as functions of the number of eigenvalues being sought simultaneously. A comparison is made with sequential search for multiple eigenvalues by minimization of Rayleigh Quotient with successive projections of unwanted eigenvectors. Parallel implementation of the algorithm shows an edge over its serial counterpart when larger number of eigenvalues are sought simultaneously and the dimensions of the matrices are higher.

Block tensor conjugate gradient-type method for Rayleigh quotient minimization in two-dimensional case

A method for solving a partial algebraic eigenvalues problem is constructed. It exploits tensor structure of eigenvectors in two-dimensional case. For a symmetric matrix represented in tensor format, the method finds low-rank approximations to the eigenvectors corresponding to the smallest eigenvalues. For sparse matrices, execution time and required memory for the proposed method are proportional to the square root of miscellaneous overall number of unknowns, whereas this dependence is usually linear. To maintain tensor structure of vectors at each iteration step, low-rank approximations are performed, which introduces errors into the original method. Nevertheless, the new method was proved to converge. Convergence rate estimates are obtained for various tensor modifications of the abstract one-step method. It is shown how the convergence of a multistep method can be derived from the convergence of the corresponding one-step method. Several modifications of the method with an low-rank approximation techniques were implemented on the basis of the block conjugate gradient method. Their performance is compared on numerical examples.

New eigensolvers for large-scale nanoscience simulations

We present results for applications to nanosystems of state-of-the-art iterative eigensolvers based on conjugate gradients and variants of Davidson in the context of semi-empirical plane wave electronic structure calculations. We are concerned with the computation of electronic and optical properties of nanosystems using the Energy SCAN method to compute interior eigenstates around the band gap that determine their properties. Numerically, this interior Hermitian eigenvalue problem poses several challenges, with respect to both accuracy and efficiency. All the iterative eigensolvers are seeking the minimal eigenvalues of the folded operator with reference shift in the band-gap. The tested methods include standard conjugate-gradient (CG)-based Rayleigh quotient minimization, Locally optimal block-preconditioned CG (LOBPCG) and two variants of the (Jacobi-)Davidson method: JDQMR and GD+1. Our experimental results indicated that Davidson-type methods are often the fastest.

A Rayleigh quotient minimization algorithm based on algebraic multigrid

AbstractThis paper presents a new algebraic extension of the Rayleigh quotient multigrid (RQMG) minimization algorithm to compute the smallest eigenpairs of a symmetric positive definite pencil (A, M). Earlier versions of RQMG minimize the Rayleigh quotient over a hierarchy of geometric grids. We replace the geometric mesh information with the algebraic information defined by an algebraic multigrid preconditioner. At each level, we minimize the Rayleigh quotient with a block preconditioned algorithm. Numerical experiments illustrate the efficiency of this new algorithm to compute several eigenpairs. Copyright © 2007 John Wiley & Sons, Ltd.

Diagonalization of a real-symmetric Hamiltonian by genetic algorithm: A recipe based on minimization of Rayleigh quotient

A genetic algorithm-based recipe involving minimization of the Rayleigh quotient is proposed for the sequential extraction of eigenvalues and eigenvectors of a real symmetric matrix with and without basis optimization. Important features of the method are analysed, and possible directions of development suggested

PARTIAL SOLUTION OF EIGENPROBLEMS BY A DECOMPOSITION METHOD—I. THEORY

This paper demonstrates the feasibility of using a recently developed technique, the ‘r-variable algorithm’ (see references under Ng), to solve the partial eigenproblem. The method involves decomposing the Rayleigh quotient minimization problem into smaller subproblems, where each subproblem is comprised of the solution of a system of linear equations using backward and forward substitutions. The striking features of the method lie in its simplicity and the minimal storage requirements. Examples are included to illustrate the accuracy of the approach in solving symmetric eigenproblems.