Trace minimization method via penalty for linear response eigenvalue problems

Convergence analysis of Lanczos-type methods for the linear response eigenvalue problem

A note on the homotopy method for linear algebraic eigenvalue problems

Distributed control of large-scale dynamical systems poses a new challenge to the field of control driven by the technological advances of modern communication networks. A particular challenge is the distributed design of such control systems. Here, a distributed iterative controller synthesis method for continuous time linear systems using a gradient descent method is presented. One of the main contributions is the determination of the step size according to a distributed Barzilai-Borwein (BB) method. As the control objective, we treat the finite horizon linear quadratic cost functional. The gradient approach uses communication only with direct neighbors and is based on the forward simulation of the system states and the backwards simulation of adjoint states. The effectiveness of the approach is shown by means of numerical simulations.

In the field of operator algebra, the concept of antieigenvalue theory was first introduced by Karl Gustafson with a special emphasis on accretive operators. In a wide range of scientific applications, antieigenvalue naturally occurs. Due to the inclusion of nonlinear Euler equations in the antieigenvalue theory, computing antieigenvalues is a difficult task as compared to that of computing eigenvalues of the operator. In the current paper, we consider linear two-parameter eigenvalue problems (LTEP) and will discuss the abstract algebraic setting of the problem as proposed by Atkinson. We analyze the generalized antieigenvalue pair and their corresponding generalized antieigenvectors for LTEP using the consequences of the Cauchy Schwarz inequality. Some generalized antieigenvalue bounds will also be derived. Generalized antieigenvalues and their corresponding generalized antieigenvectors will be calculated solving their relevant optimization problem. For numerical computations, three examples will be provided. Real symmetric matrices are used in the first case, while real diagonal matrices are used in the second case and finally arbitrary matrices are considered.

A technique is developed and presented which improves the accuracy of the predictions for the eigenvalues of the usual first order perturbation scheme for the generalized symmetric linear algebraic eigenvalue problem. The improved method very simply replaces the usual B‐norms appearing in the denominator of the expressions for the first order changes in the eigenvalues by BP‐norms where BP is the perturbed B matrix. With increasing sizes for the perturbations the improved technique produces results whose accuracy approaches those of the usual second order predictions. Numerical studies are presented to support the analysis.

Consider algorithms to solve eigenvalue problems for partial differential equations describing the bending of a von Karman elastic plate. Here we explore numerical techniques based on a variational principle, Newton's iterations and numerical continuation. The variational approach uses the Galerkin spectral method. First we study the linearized problem. Second, eigenfunctions of the nonlinear equations describing post-buckling behaviour of the von Karman plate are calculated. The plate is supposed to be totally clamped and compressed along its four sides. The basis functions in the variational procedure are trigonometric functions. They are chosen to match the boundary conditions. Effective computational techniques allow us to detect bifurcation points and to trace branches of solutions. Numerical examples demonstrate the efficiency of the methods. The proposed algorithms are applicable to similar problems involving elliptic differential equations.

Lamb waves are mechanical waves guided in a plate with stress-free boundaries. In practice, the plate is often in contact with a fluid. The propagation of guided waves may change substantially in this case due to interaction with the adjacent medium, e.g., acoustic radiation may lead to leakage of energy into the fluid domain. The resulting plate modes are called leaky Lamb waves, and their attenuation due to leakage is an important parameter for the design of ultrasonic devices that exploit such waves. Modern methods to calculate dispersion characteristics of guided waves, i.e., the wavenumber-frequency relationship, solve the corresponding discretized eigenvalue problem. This kind of model for leaky Lamb waves with the analytically exact fluid interaction leads to a nonlinear eigenvalue problem, which is difficult to solve. We present a change in variables that yields an equivalent polynomial eigenvalue problem. The latter can be rewritten as a linear eigenvalue problem in state space. As a result, conventional numerical eigenvalue solvers can be used to robustly determine the sought wavenumbers. In contrast to traditional root-finding of the characteristic equations, the presented method guarantees to find all solutions and naturally handles complex wavenumbers. Additionally, no initial guesses for the wavenumbers are required.

A theoretical study of the resonances of an elastic plate in a compressible flow in a two-dimensional duct is presented. Due to the fluid–structure coupling, a quadratic eigenvalue problem is involved, in which the resonance frequencies k solve the equations λ(k) = k2, where λ is the eigenvalue of a self-adjoint operator of the form A + kB. In a previous paper, we have proved that a linear eigenvalue problem is recovered if the plate is rigid or the fluid at rest. We focus here on the general problem for which elasticity and flow are jointly present and derive a lower bound for the number of resonances. The expression of this bound, based on the solution of two linear eigenvalue problems, points out that the coupling between elasticity and flow generally reduces the number of resonances. This estimate is validated numerically.

To solve quadratic eigenvalue problems (QEPs), especially the gyroscopic systems, two methods are proposed: an iterative direct detection method (DDM) of the complex eigenvalues of the original QEP, and a split-linearization method (SLM) for finding the solvent matrix, which results to a standard linear eigenvalue problem (LEP) being solved to compute all eigenvalues by the symmetry extension. Reducing the dimension to one-half, the LEP is recast in a simpler QEP involving the square of the solvent. We set up two new merit functions which are minimized to detect the complex eigenvalues from the original QEP and a simpler QEP. For each eigen-parameter the merit function consists of the Euclidean norm of each derived eigen-equation, whose vector variable is solved from a derived inhomogeneous linear system. Then, the golden section search algorithm is employed to minimize the merit functions and locate the complex eigenvalue as a local minimal point. The results are compared with that computed by the cyclic-reduction-based solvent (CRS) method.

Engineering problems governed by interval differential equations under static and dynamic conditions led to interval system of linear equations and interval eigenvalue problems. Chapter 8 dealt with solving interval system of linear equations using interval analysis and in this chapter, we have focused on solving eigenvalue problems having interval parameters.

To generalize the Fisher Discriminant Analysis (FDA) algorithm to the case of discriminant functions belonging to a nonlinear, finite dimensional function space F (Nonlinear FDA or NFDA), it is sufficient to expand the input data by computing the output of a basis of F when applied to it [1,2,3,4]. The solution to NFDA can then be found like in the linear case by solving a generalized eigenvalue problem on the between- and within-classes covariance matrices (see e.g. [5]). The goal of NFDA is to find linear projections of the expanded data (i.e., nonlinear transformations of the original data) that minimize the variance within a class and maximize the variance between different classes. Such a representation is of course ideal to perform classification. The application of NFDA to pattern recognition is particularly appealing, because for a given input signal and a fixed function space it has no parameters and it is easy to implement and apply. Moreover, given C classes only C - 1 projections are relevant [5]. As a consequence, the feature space is very small and the algorithm has low memory requirements and high speed during recognition. Here we apply NFDA to a handwritten digit recognition problem using the MNIST database, a standard and freely available set of 70,000 handwritten digits (28 × 28 pixels large), divided into a training set (60,000 digits) and a test set (10,000 digits). Several established pattern recognition methods have been applied to this database by Le Cun et al. [6]. Their paper provides a standard reference work to benchmark new algorithms. We perform NFDA on spaces of polynomials of a given degree d, whose corresponding basis functions include all monomials up to order d in all input variables. It is clear that the problem quickly becomes intractable because of the high memory requirements. For this reason, the input dimensionality is first reduced by principal component analysis. On the preprocessed data we then apply NFDA by expanding the training patterns in the polynomial space and solving the linear FDA eigenvalue problem. As mentioned above, since we have 10 classes we only need to compute the first 9 eigenvectors. Since the within-class variance is minimized, the patterns belonging to different classes tend to cluster in the feature space when projected on the eigenvectors. For this reason we classify the digits with a simple method such as Gaussian classifiers. We perform simulations with polynomials of degree 2 to 5. With polynomials of degree 2 the explosion in the dimensionality of the expanded space with increasing number of input dimensions is relatively restricted, so that it is possible to use up to 140 dimensions. With higher order polynomials one has to rely on a smaller number of input dimensions, but since the function space gets larger and includes new nonlinearities, one obtains a remarkable improvement in performance. The best performance is achieved with polynomials of degree 3 and 35 input dimensions, with an error rate of 1.5% on test data. This error rate is comparable to but does not outperform that of the most elaborate algorithms (Table 1). The performance of NFDA is however remarkable considering the simplicity of the method and the fact that it has no a priori knowledge on the problem, in contrast for example to the LeNet-5 algorithm [6] which has been designed specifically for handwritten character recognition. In addition, for recognition, NFDA has to store and compute only 9 functions and has thus small memory requirements and a high recognition speed. It is also possible to formulate NFDA using the kernel trick, in which case one can in principle use function spaces of infinite dimensionality [1,2,3,4]. However, the limiting factor in that formulation is the number of training patterns, which makes it not realistic for this application. The performance of NFDA could be further improved using for example a more problem-specific preprocessing of the patterns (e.g., by increasing the size of the training set with new patterns generated by artificial distortion of the original one), boosting techniques, or mixture of experts with other algorithms [5,6].

Because of its vital role of the trust-region subproblem (TRS) in various applications, for example, in optimization and in ill-posed problems, there are several factorization-free algorithms for solving the large-scale sparse TRS. The truncated Lanczos approach proposed by N. I. M. Gould, S. Lucidi, M. Roma, and P. L. Toint [SIAM J. Optim., 1999, 9: 504–525] is a natural extension of the classical Lanczos method for the symmetric linear system and eigenvalue problem and, indeed follows the classical Rayleigh-Ritz procedure for eigenvalue computations. It consists of 1) projecting the original TRS to the Krylov subspaces to yield smaller size TRS’s and then 2) solving the resulted TRS’s to get the approximates of the original TRS. This paper presents a posterior error bounds for both the global optimal value and the optimal solution between the original TRS and their projected counterparts. Our error bounds mainly rely on the factors from the Lanczos process as well as the data of the original TRS and, could be helpful in designing certain stopping criteria for the truncated Lanczos approach.

The QR factorisation is a cornerstone of numerical linear algebra, essential for solving overdetermined linear systems, eigenvalue problems, and various scientific computing tasks. However, computing it for ill-conditioned tall-and-skinny (TS) matrices on large-scale distributed-memory systems, particularly those with multiple GPUs, presents significant challenges in balancing numerical stability, high performance, and efficient communication. Traditional Householder-based QR methods provide numerical stability but perform poorly on TS matrices due to their reliance on memory-bound kernels. This paper introduces a novel algorithm for computing the QR factorisation of ill-conditioned TS matrices based on CholeskyQR methods. Although CholeskyQR is fast, it typically fails due to severe loss of orthogonality for ill-conditioned inputs. To solve this, our new algorithm, mCQRGSI+, combines the speed of CholeskyQR with stabilising techniques from the Gram–Schmidt process. It is specifically optimised for distributed multi-GPU systems, using adaptive strategies to balance computation and communication. Our analysis shows the method achieves accuracy comparable to Householder QR, even for extremely ill-conditioned matrices (condition numbers up to 1016). Scaling experiments demonstrate speedups of up to 12× over ScaLAPACK and 16× over SLATE’s CholeskyQR2. This work delivers a method that is both robust and highly parallel, advancing the state-of-the-art for this challenging class of problems.

We study the regularity in weighted Sobolev spaces of Schrödinger-type eigenvalue problems, and we analyze their approximation via a discontinuous Galerkin (dG) [Formula: see text] finite element method. In particular, we show that, for a class of singular potentials, the eigenfunctions of the operator belong to analytic-type non-homogeneous weighted Sobolev spaces. Using this result, we prove that an isotropically graded [Formula: see text] dG method is spectrally accurate, and that the numerical approximation converges with exponential rate to the exact solution. Numerical tests in two and three dimensions confirm the theoretical results and provide an insight into the behavior of the method for varying discretization parameters.

The purpose of this paper is to analyze a mixed method for the linear elasticity eigenvalue problem, which approximates numerically the stress, displacement, and rotation, by piecewise (<i>k</i> + 1), <i>k</i>, and (<i>k</i> + 1) th degree polynomial functions (<i>k</i> ≥ 1), respectively. The numerical eigenfunction of stress is symmetric. By the discrete <i>H</i><sup style="">1</sup>-stability of numerical displacement, we prove an <i>O </i>(<i>h^k</i>⁺²) approximation to the <i>L</i><sup style="">2</sup>-orthogonal projection of the eigenspace of exact displacement for the eigenvalue problem with a proper regularity assumption. Thus via postprocessing, we obtain a better approximation to the eigenspace of exact displacement for the eigenproblem than conventional methods. We also prove that numerical approximation to the eigenfunction of stress is locking free with respect to the Poisson ratio. We introduce a hybridization to reduce the mixed method to a condensed eigenproblem and prove an <i>O </i>(<i>h</i>²) initial approximation (independent of the inverse of the elasticity operator) of the eigenvalue for the nonlinear eigenproblem by using the discrete <i>H</i>¹-stability of numerical displacement, while only an <i>O </i>(<i>h</i>) approximation can be obtained if we use the traditional inf-sup condition. Finally, we report some numerical experiments. © 2023 The Author(s).