Eigenvalue Clusters Research Articles

Composite convergence bounds based on Chebyshev polynomials and finite precision conjugate gradient computations

The conjugate gradient method (CG) for solving linear systems of algebraic equations represents a highly nonlinear finite process. Since the original paper of Hestenes and Stiefel published in 1952, it has been linked with the Gauss-Christoffel quadrature approximation of Riemann-Stieltjes distribution functions determined by the data, i.e., with a simplified form of the Stieltjes moment problem. This link, developed further by Vorobyev, Brezinski, Golub, Meurant and others, indicates that a general description of the CG rate of convergence using an asymptotic convergence factor has principal limitations. Moreover, CG is computationally based on short recurrences. In finite precision arithmetic its behaviour is therefore affected by a possible loss of orthogonality among the computed direction vectors. Consequently, any consideration concerning the CG rate of convergence relevant to practical computations must include analysis of effects of rounding errors. Through the example of composite convergence bounds based on Chebyshev polynomials, this paper argues that the facts mentioned above should become a part of common considerations on the CG rate of convergence. It also explains that the spectrum composed of small number of well separated tight clusters of eigenvalues does not necessarily imply a fast convergence of CG or other Krylov subspace methods.

Abstract 171: CRP, IL-18, and BNP are Associated with Regional Brain Atrophy: Results from the Dallas Heart Study

Purpose: Multiple biomarkers have been associated with total brain atrophy. However, little is known about their relationship to segmental atrophy in a large, multi-ethnic, population-based sample. Materials and Methods: 3D-MPRAGE brain images obtained at 3T from 2082 participants of the Dallas Heart Study (DHS) 2 were analyzed with Freesurfer and outlier analysis was performed. Divisive eigenvalue clustering of 89 brain segments yielded 24 groups with linked atrophy patterns. Plasma C-reactive protein (CRP), IL-18, homocsysteine and B-type natriuretic peptide (BNP) obtained 7 years prior during DHS 1 were available for 1343, 840, 1333 and 1331 participants, respectively. Multivariate linear regression analysis with adjustments for age, ethnicity, and gender were used to demonstrate associations between biomarkers and atrophy clusters. Results: Nine atrophy clusters were associated with CRP, three atrophy clusters were associated with IL-18, and six atrophy clusters were associated with BNP (Table 1). Homocysteine did not have any significant correlations. Conclusions: The markers studied had associations with distinct patterns of segmental atrophy indicating they may have unique interactions in different brain regions. This suggests that distinct inflammatory and other pathways may be at work in specific regions of the brain and that their localized effects may be obscured by approaches evaluating solely total brain volumes. Table 1:

Asymptotic Density of Eigenvalue Clusters for the Perturbed Landau Hamiltonian

We consider the Landau Hamiltonian (i.e. the 2D Schrödinger operator with constant magnetic field) perturbed by an electric potential V which decays sufficiently fast at infinity. The spectrum of the perturbed Hamiltonian consists of clusters of eigenvalues which accumulate to the Landau levels. Applying a suitable version of the anti-Wick quantization, we investigate the asymptotic distribution of the eigenvalues within a given cluster as the number of the cluster tends to infinity. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the Radon transform of the perturbation potential V.

Convergence of Multipower Defect-Correction for spectral computations of integral operators

In this paper we propose the Multipower Defect-Correction method, a generalization of the double iteration, to compute a cluster of eigenvalues and the associated invariant subspace from discretized integral operators. It consists of an inner/outer iteration where, inside a defect-correction iteration, p power iteration steps are performed. The approximate inverse used in the defect correction is built with an approximation to the reduced resolvent operator of a coarse discretization of the integral operator. The proposed method computes eigenpairs approximations by refining initial approximations obtained from a coarser dimensional problem. It is therefore meant for large dimensional problems. Furthermore, the kernel of the integral operator may be weakly singular. We provide a proof for the convergence of this Multipower Defect-Correction method. A numerical example illustrating the theory and the behavior of the method is also presented.

An invariant subspace‐based approach to the random eigenvalue problem of systems with clustered spectrum

SUMMARYThe repeated or closely spaced eigenvalues and corresponding eigenvectors of a matrix are usually very sensitive to a perturbation of the matrix, which makes capturing the behavior of these eigenpairs very difficult. Similar difficulty is encountered in solving the random eigenvalue problem when a matrix with random elements has a set of clustered eigenvalues in its mean. In addition, the methods to solve the random eigenvalue problem often differ in characterizing the problem, which leads to different interpretations of the solution. Thus, the solutions obtained from different methods become mathematically incomparable. These two issues, the difficulty of solving and the non‐unique characterization, are addressed here. A different approach is used where instead of tracking a few individual eigenpairs, the corresponding invariant subspace is tracked. The spectral stochastic finite element method is used for analysis, where the polynomial chaos expansion is used to represent the random eigenvalues and eigenvectors. However, the main concept of tracking the invariant subspace remains mostly independent of any such representation. The approach is successfully implemented in response prediction of a system with repeated natural frequencies. It is found that tracking only an invariant subspace could be sufficient to build a modal‐based reduced‐order model of the system. Copyright © 2012 John Wiley & Sons, Ltd.

Band invariants for perturbations of the harmonic oscillator

We study the direct and inverse spectral problems for semiclassical operators of the form S=S0+ℏ2V, where S0=12(−ℏ2ΔRn+|x|2) is the harmonic oscillator and V:Rn→R is a tempered smooth function. We show that the spectrum of S forms eigenvalue clusters as ℏ tends to zero, and compute the first two associated “band invariants”. We derive several inverse spectral results for V, under various assumptions. In particular we prove that, in two dimensions, generic analytic potentials that are even with respect to each variable are spectrally determined (up to a rotation).

On the response of large-amplitude internal waves to upstream disturbances

AbstractLarge-amplitude internal solitary waves generate shear flows that intensify from the wings of the waves to their maxima. Upstream perturbations of the hydrostatic equilibrium in the form of wave packets along the path of wave propagation are expected to trigger shear instability and ultimately generate Kelvin–Helmholtz roll-ups. In contrast, as shown here with accurate simulations of incompressible stratified Euler equations, large internal waves can act as suppressors of perturbations. The precise understanding of the mechanisms leading to different outcomes, including whether instability is excited, is the focus of this work. Under the action of shear flows, small-amplitude wave packets undergo stretching and filamentation, which lead to significant absorption of perturbation energy into the background shear. It is found that this typical behaviour is present in the self-induced shear by internal waves, regardless of whether the shear is stable or unstable, and can leave a quieter state in the wave’s wake for a wide range of perturbation parameters. In the unstable case, even once perturbations are selected to excite the instability, our results show that this absorption can act to reduce growth in the strong-shear region, effectively making roll-up development observable only downstream of the wave crest. Our approach is both analytical and numerical; a model valid for relatively thin pycnoclines and suitable for local spectral analysis is devised and used. Energy diagnostics on the simulations are implemented to validate the numerics and illustrate the energy exchanges between background wave flow and its shear. A link between the absorption mechanism and the clustering of local eigenvalues along the wave is proposed. This promotes an energetic coupling among neutral modes stronger than what may be expected to occur in slowly varying flows, and gives rise to multi-modal transient dynamics of the kind often referred to as non-normality effects. For those cases in which the wave-induced shear meets the conditions for local instability, it is found that the growth of disturbances is selective with respect to the sign of the mode excited upstream. Elements of this phenomenon are interpreted by asymptotic analysis for spatial growth in time-independent slowly varying media.

Least squares h-p spectral element method for elliptic eigenvalue problems

In this paper we discuss how to solve elliptic eigenvalue problems on polygonal domains using least squares methods. Exponential rate of convergence has been obtained for approximate eigenvalues as well as the eigenfunctions. The result is proved when the boundary of underlying domain is smooth and also when the boundary contains corners. The computational results confirm the error estimates. The results proved in the paper are valid for multiple or clustered eigenvalues also. The method works for non-self adjoint problems too.

Semiclassical Szegö limit of resonance clusters for the hydrogen atom Stark Hamiltonian

We study the weighted averages of resonance clusters for the hydrogen atom with a Stark electric field in the weak field limit. We prove a semiclassical Szego-type theorem for resonance clusters showing that the limiting distribution of the resonance shifts concentrates on the classical energy surface corresponding to a rescaled eigenvalue of the hydrogen atom Hamiltonian. This result extends Szego-type results on eigenvalue clusters to resonance clusters. There are two new features in this work: first, the study of resonance clusters requires the use of non-self-adjoint operators, and second, the Stark pertur- bation is unbounded so control of the perturbation is achieved using localization properties of coherent states corresponding to hydrogen atom eigenvalues.

Linear Precoding for MIMO Systems with Low-Complexity Receivers

This paper considers large multiple-input multiple-output (MIMO) communication systems with linear precoding and linear minimum mean-squared error (LMMSE) equalization based on the iterative conjugate gradient (CG) algorithm. Convergence of the CG algorithm is fast when the eigenvalues of the received signal's covariance matrix are clustered, suggesting that mean-squared error and receiver complexity can be managed with judicious precoder design. In order to accelerate convergence of an iterative CG receiver, we incorporate constraints on two measures of eigenvalue clustering into the precoder design. Closed-form solutions to the optimal precoders are derived using majorization theory and convex optimization techniques. We show that if there are constraints on receiver complexity, the proposed precoders can improve performance for large MIMO systems operating over slowly time-varying fading channels.

A Preconditioning Framework for Sequences of Diagonally Modified Linear Systems Arising in Optimization

We propose a framework for building preconditioners for sequences of linear systems of the form $(A+\Delta_k) x_k=b_k$, where $A$ is symmetric positive semidefinite and $\Delta_k$ is diagonal positive semidefinite. Such sequences arise in several optimization methods, e.g., in affine-scaling methods for bound-constrained convex quadratic programming and bound-constrained linear least squares, as well as in trust-region and overestimation methods for convex unconstrained optimization problems and nonlinear least squares. For all the matrices of a sequence, the preconditioners are obtained by updating any preconditioner for $A$ available in the $LDL^T$ form. The preconditioners in the framework satisfy the natural requirement of being effective on slowly varying sequences; furthermore, under an additional property they are also able to cluster eigenvalues of the preconditioned matrix when some entries of $\Delta_k$ are sufficiently large. We present two low-cost preconditioners sharing the above-mentioned p...

A note on generating finer-grain parallelism in a representation tree

SUMMARY The representation tree lies at the heart of the algorithm of Multiple Relatively Robust Representations for computing orthogonal eigenvectors of a symmetric tridiagonal matrix without Gram–Schmidt. A representation tree describes the incremental shift relations between relatively robust representations of eigenvalue clusters of an unreduced tridiagonal matrix, which are needed to strongly separate close eigenvalues in the relative sense. At the bottom of the representation tree, each leaf defines a relatively isolated eigenvalue to high relative accuracy. The shape of the representation tree plays a pivotal role for complexity and available parallelism: a deeper tree consisting of multiple levels of nodes involves tasks associated to more work (i.e., eigenvalue refinement to resolve eigenvalue clusters) and less parallelism (i.e., a longer critical path as well as potential data movement and synchronization). An embarrassingly parallel, ideal tree on the other hand consists of a root and leaves only. As highly parallel hybrid graphics processing unit/multicore platforms with large memory now become available as commodity platforms, exploiting parallelism in traditional algorithms becomes key to modernizing the components of standard software libraries such as LAPACK. This paper focuses on LAPACK's Multiple Relatively Robust Representations algorithm and investigates the critical case where a representation tree contains a long sequential chain of large (fat) nodes that hamper parallelism. This key problem needs to be addressed as it concerns all sorts of computing environments, distributed computing, symmetric multiprocessor, as well as hybrid graphics processing unit/multicore architectures. We present an improved representation tree that often offers a significantly shorter critical path and finer computational granularity of smaller tasks that are easier to schedule. In a study of selected synthetic and application matrices, we show that an average 75% reduction in the length of the critical path and 82% reduction in task granularity can be achieved. Copyright © 2011 John Wiley & Sons, Ltd.

Reduced Order Models Using Generalized Eigenanalysis

This article discusses the use of generalized eigenanalysis to extract reduced order models from the linearization of structural and aeroelastic problems written in differential-algebraic form. These problems may arise from multi-body analysis and in general from mixed approaches, where a high degree of generality and modelling flexibility are sought. A method based on a shift technique is proposed, that allows to exploit the regularity of the matrix pencil resulting from the linearization of differential-algebraic problems. Alternatively, the generalized Schur decomposition, or QZ decomposition, is directly used to select a cluster of eigenvalues related to the dynamic states. The two approaches are used to reduce the model to ordinary differential in state-space form. The two methods are applied to simple numerical problems, highlighting their robustness and versatility compared to other techniques. They are also applied to numerical models of a high-altitude long endurance aircraft obtained using a free general-purpose multi-body solver and a dedicated mixed variational solver.

Detecting Localization in an Invariant Subspace

A normalized eigenvector, or, more interestingly, an invariant subspace, is localized if its significant entries are defined by just part(s) of the matrix and negligible elsewhere. This paper presents two new procedures to detect such localization in eigenvectors of a symmetric tridiagonal matrix. The procedures are intended for use before the actual eigenvector computation. If localization is found, one may reduce costs by computing the vectors just from the relevant matrix regions. Practical eigensolvers from numerical libraries such as LAPACK and ScaLAPACK already inspect a given tridiagonal T for off-diagonal entries that are of small magnitude relative to the matrix norm. These so-called splitting points indicate that T breaks into smaller blocks, each one defining a subset of eigenvalues and localized eigenvectors. However, localization can occur even when none of the off-diagonals is particularly small. Our study investigates this more complicated phenomenon in the context of invariant subspaces belonging to isolated eigenvalue clusters.

Analysis of the Laplacian and spectral operators on the Vicsek set

We study the spectral decomposition of the Laplacian on a family of fractals $\mathcal{VS}_n$ that includes the Vicsek set for $n=2$, extending earlier research on the Sierpinski Gasket. We implement an algorithm [23] for spectral decimation of eigenfunctions of the Laplacian, and explicitly compute these eigenfunctions and some of their properties. We give an algorithm for computing inner products of eigenfunctions. We explicitly compute solutions to the heat equation and wave equation for Neumann boundary conditions. We study gaps in the ratios of eigenvalues and eigenvalue clusters. We give an explicit formula for the Green's function on $\mathcal{VS}_n$. Finally, we explain how the spectrum of the Laplacian on $\mathcal{VS}_n$ converges as $n \to \infty$ to the spectrum of the Laplacian on two crossed lines (the limit of the sets $\mathcal{VS}_n$.)

Preconditioning the modified conjugate gradient method

In this paper, the convergence analysis of the conventional conjugate Gradient method was reviewed. And the convergence analysis of the modified conjugate Gradient method was analysed with our extension on preconditioning the algorithm. Convergence of the algorithm is a function of the condition number of M-1A. Again, this work broadens our understanding that the modified CGM yields the exacts result after n-iterations, and further proves that the CGM algorithm is quicker if there are duplicated eigenvalues. Given infinite floating point precision, the number of iterations required to compute an exact solution is at most the number of distinct eigenvalues. It was discovered that the modified CGM algorithm converges more quickly when eigenvalues are clustered together than when they are irregularly distributed between a given interval. The effectiveness of a preconditioner is determined by the condition number of the matrix and occasionally by its clustering of eigenvalues. For large scale application, CGM should always be used with a pre-conditioner to improve convergence.KEYWORDS: Convergence, Conjugate Gradient, eigenvalue, preconditioning.

Preconditioned conjugate gradient algorithms for nonconvex problems with box constraints

The paper describes new conjugate gradient algorithms for large scale nonconvex problems with box constraints. In order to speed up convergence the algorithms employ scaling matrices which transform the space of original variables into the space in which Hessian matrices of the problem’s functionals have more clustered eigenvalues. This is done by applying limited memory BFGS updating matrices. Once the scaling matrix is calculated, the next few conjugate gradient iterations are performed in the transformed space. The box constraints are treated efficiently by the projection. We also present a limited memory quasi-Newton method which is a special version of our general algorithm. The presented algorithms have strong global convergence properties, in particular they identify constraints active at a solution in a finite number of iterations. We believe that they are competitive to the L-BFGS-B method and present some numerical results which support our claim.

Deflated block Krylov subspace methods for large scale eigenvalue problems

We discuss a class of deflated block Krylov subspace methods for solving large scale matrix eigenvalue problems. The efficiency of an Arnoldi-type method is examined in computing partial or closely clustered eigenvalues of large matrices. As an improvement, we also propose a refined variant of the Arnoldi-type method. Comparisons show that the refined variant can further improve the Arnoldi-type method and both methods exhibit very regular convergence behavior.

SVD-tail: a new linear-sampling reconstruction method for inverse scattering problems

A new efficient numerical procedure (SVD-tail) is proposed for the reconstruction of the shape and volume of unknown objects from measurements of their radiation in the far field. At variance with previously published linear-sampling methods where the solution is constructed as a regularized solution of the far-field equations using a variant of the Tikhonov–Morozov type, the new algorithm uses a new eigenspace recovery technique which exploits the combined presence of error in the operator and of eigenvalue clusters. Its performance on a battery of examples and its comparison with existing methods are shown to be promising.

A Linear-Time Complex-Valued Eigenvalue Solver for Full-Wave Analysis of Large-Scale On-Chip Interconnect Structures

This paper proposes a linear-time complex-valued eigenvalue solver for solving large-scale on-chip interconnect problems. The fast eigenvalue solution is achieved by eigenvalue clustering, fast system reduction with negligible computational cost, and fast linear-time solution of the reduced system. Numerical and experimental results are presented to demonstrate the accuracy and efficiency of the proposed method.