Inverse Iteration Research Articles

Nonintrusive parametric solutions in structural dynamics

A nonintrusive reduced order method able to solve a parametric modal analysis is proposed in this work. The main objective is being able to efficiently identify how a variation of user-defined parameters affects the dynamic response of the structure in terms of fundamental natural frequencies and corresponding mode shapes. A parametric version of the inverse power method (IPM) is presented by using the proper generalised decomposition (PGD) rationale. The proposed approach utilises the so-called encapsulated PGD toolbox and includes a new algorithm for computing the square root of a parametric object. With only one offline computation, the proposed PGD-IPM approach provides an analytical parametric expression of the smallest (in magnitude) eigenvalue (or natural frequency) and corresponding eigenvector (mode shape), which contains all the possible solutions for every combination of the parameters within pre-defined ranges. A Lagrange multiplier deflation technique is introduced in order to compute subsequent eigenpairs, which is also valid to overcome the stiffness matrix singularity in the case of a free-free structure. The proposed approach is nonintrusive and it is therefore possible to be integrated with commercial finite element (FE) packages. Two numerical examples are shown to underline the properties of the technique. The first example includes one material and one geometric parameter. The second example shows a more realistic industrial example, where the nonintrusivity of the approach is demonstrated by employing a commercial FE package for assembling the FE matrices. Finally, a multi-objective optimisation study is performed proving that the developed method could significantly assist the decision-making during the preliminary phase of a new design process.

Extending Zhong-Williams scheme to solve repeated-root wave modes

The eigenvalue scheme proposed by Zhong and Williams in 1995 is a well-conditioned and efficient approach to solve the wave propagation characteristics of periodic structures. However, the original scheme is valid only for single-root wave modes and cannot be directly employed to solve repeated-root wave modes. This limits the application of Zhong-Williams scheme when the cross-section of the structure has certain kind of symmetry (such as cylinders and square-section beams). In this paper we extend Zhong-Williams scheme and make it applicable for structures with repeated-root wave modes. The improvements are two-fold. First, the infinite values caused by repeated roots while solving the generalized symplectic eigen-problem are eliminated by a more general expression. Second, the ill-conditioning caused by repeated roots while recovering the wave modes is resolved by a novel approach based on the inverse power method. A square-section beam is used to verify the proposed method on the propagating waves. Forced response and energy flow analysis of a cylindrical shell and a stiffened cylindrical shell are conducted using wave modes, to verify the proposed method on the overall wave basis. Results show that the proposed improvements of Zhong-Williams scheme are obligatory for the analysis of structures with repeated-root wave modes.

Extended Wittrick–Williams algorithm for eigenvalue solution of stochastic dynamic stiffness method

This paper proposes an efficient and reliable eigenvalue solution technique for analytical stochastic dynamic stiffness (SDS) formulations of beam built-up structures with parametric uncertainties. The SDS formulations are developed based on frequency-dependent shape functions in conjunction with both random-variable and random-field structural parameters. The overall numerical framework is aimed towards representing the broadband dynamics of structures using very few degrees of freedom. This paper proposes a novel approach combining the Wittrick–Williams(WW) algorithm, the Newton iteration method and numerical perturbation method to extract eigensolutions from SDS formulations. First, the eigenvalues and eigenvectors of the deterministic DS formulations are computed by the WW algorithm and the corresponding mode finding technique, which are used as the initial solution. Then, a numerical perturbation technique based on the inverse iteration and homotopy method is proposed to update the eigenvectors and eigenvalues. The robustness and efficiency of the proposed method are guaranteed through several technique arrangements. Through numerical examples, the proposed method is demonstrated to be robust within the whole frequency range. This method provides an efficient and reliable tool for stochastic analysis of eigenvalue problems relevant to free vibration and buckling analysis of built-up structures.

Pseudo-orthogonality for graph 1-Laplacian eigenvectors and applications to higher Cheeger constants and data clustering

The data clustering problem consists in dividing a data set into prescribed groups of homogeneous data. This is an NP-hard problem that can be relaxed in the spectral graph theory, where the optimal cuts of a graph are related to the eigenvalues of graph 1-Laplacian. In this paper, we first give new notations to describe the paths, among critical eigenvectors of the graph 1-Laplacian, realizing sets with prescribed genus. We introduce the pseudo-orthogonality to characterize m3(G), a special eigenvalue for the graph 1-Laplacian. Furthermore, we use it to give an upper bound for the third graph Cheeger constant h3(G), that is, h3(G) ⩽ m3(G). This is a first step for proving that the k-th Cheeger constant is the minimum of the 1-Laplacian Raylegh quotient among vectors that are pseudo-orthogonal to the vectors realizing the previous k − 1 Cheeger constants. Eventually, we apply these results to give a method and a numerical algorithm to compute m3 (G), based on a generalized inverse power method.

Inexact Inverse Subspace Iteration with Preconditioning Applied to Quadratic Matrix Polynomials

Abstract An inexact variant of inverse subspace iteration is used to find a small invariant pair of a large quadratic matrix polynomial. It is shown that linear convergence is preserved provided the inner iteration is performed with increasing accuracy. A preconditioned block GMRES solver is employed as inner iteration. The preconditioner uses the strategy of “tuning” which prevents the inner iteration from increasing and therefore results in a substantial saving in costs. The accuracy of the computed invariant pair can be improved by the addition of a post-processing step involving very few iterations of Newton’s method. The effectiveness of the proposed approach is demonstrated by numerical experiments.

High surface accuracy and pretension design for mesh antennas based on dynamic relaxation method

Cable truss structures are widely used in large satellite antennas with ultrahigh electromagnetic performance. The electromagnetic performance is directly determined by the mesh reflector's surface accuracy, which triggers research on the pretension design. In this paper, a novel pretension design method for both high surface accuracy and uniform tension distribution is proposed based on the dynamic relaxation (DR) method. In this approach, members of the cable network are regarded as string elements with 3 degrees of freedom to resist the axial tensile force, and members of the supporting truss are regarded as Euler-Bernoulli beams with 6 degrees of freedom to resist the axial and transverse forces and the bending and torsion moments. The DR method is adopted to find the static equilibrium of the whole mesh antenna, including both the cable mesh and the supporting truss. Thus, the deformation of the cables and rods and their interactive effects are fully captured. Then, to find an expected high-precision mesh surface with a uniform tension distribution, the inverse iteration algorithm (IIA) is adopted to adjust the original cable lengths of the cable network in each DR calculation cycle until the convergence criterion is met. Finally, this approach is effectively applied to the pretension design of symmetric and asymmetric mesh antennas, and the results indicate that the method is effective in obtaining both high surface accuracy and uniform tension distribution with high computational efficiency.

Approximation of the first eigenpair of the p(x)-Laplacian using WEB-spline based mesh-free method

Our objective in this article is to develop weighted extended B-spline (WEB-spline) approximation of the first eigenpair of p(x)-Laplacian problem. One of the reason for choosing WEB-spline based mesh-free method for solving p(x)-Laplacian problem is that, it has additional advantages compared to usual finite element method. For example, it does not require any mesh generation procedure and hence eliminates the difficult, time consuming preprocessing step in finite element method. Also, it combines the computational advantages of B-splines and geometric flexibility of standard mesh-based elements. To compute the first eigenpair of p(x)-Laplacian problem, we examine the problem numerically by applying an inverse power method with three different minimization techniques and we compare the results obtained. Finally, we study the asymptotic behaviour of the eigenvalue problem of p(x)-Laplacian and validate the results comparing with those given in the literature.

Computing tensor Z-eigenvalues via shifted inverse power method

The positive definiteness of an even degree homogeneous polynomial plays an important role in the stability study of nonlinear autonomous systems via Lyapunov’s direct method in automatic control, the detection of P or P0 tensor in tensor complementarity problems and spectral hypergraph theory, and more. Owing to the positive definiteness of an even degree homogeneous polynomial is equivalent to that of an even order symmetric tensor. In this paper, we propose a shifted inverse power method for computing tensor Z-eigenpairs, which can be viewed as a generalization of the inverse power method for matrices case. We also formulate it as a fixed point iteration form, and reveal that the relationship between the fixed points and the Z-eigenvectors of symmetric tensors. The advantages of the proposed method are simple operations and readily comprehensible convergence analysis. An efficient initialization strategy is also developed, which makes the proposed method converges to a better solution compared to not using the initialization strategy. Finally, we present applications of the proposed method in nonlinear autonomous systems, the detection of P or P0 tensor and symmetric tensor Z-eigenproblems, some numerical results are reported to illustrate the effectiveness of the proposed method.

A generalization of inverse power method for computing eigenpairs of symmetric tensors

In this paper, we propose generalized inverse power methods with variable shifts for finding the smallest H-/Z-eigenvalue and associated H-/Z-eigenvector of symmetric tensors. The methods are guaranteed to always converge to a H-/Z-eigenpair. Furthermore, for an even order nonsingular symmetric -tensor, the proposed method with any positive initial point always converges to the smallest H-eigenvalue. Numerical results are reported to illustrate that the proposed methods often can find the smallest H-/Z-eigenvalue instead of other H-/Z-eigenvalues of symmetric tensors. Moreover, we can always get the smallest H-eigenvalue for an even order nonsingular symmetric -tensor.

Smoothed-Adaptive Perturbed Inverse Iteration for Elliptic Eigenvalue Problems

Abstract We present a perturbed subspace iteration algorithm to approximate the lowermost eigenvalue cluster of an elliptic eigenvalue problem. As a prototype, we consider the Laplace eigenvalue problem posed in a polygonal domain. The algorithm is motivated by the analysis of inexact (perturbed) inverse iteration algorithms in numerical linear algebra. We couple the perturbed inverse iteration approach with mesh refinement strategy based on residual estimators. We demonstrate our approach on model problems in two and three dimensions.

The Homogeneous B 1 Model as Polynomial Eigenvalue Problem

The homogeneous version of the B 1 leakage model is a non-linear eigenvalue problem which is generally solved iteratively by a root-finding algorithm, combined to the supplementary eigenvalue problem of the multiplication factor. This problem is widely used for ordinary cross section preparation in reactor analysis. Our work approximates this problem with a polynomial eigenvalue problem, which can be easily transformed into an ordinary linear generalized eigenproblem of size equal to the initial one multiplied by the polynomial degree used for the approximation of a transcendental function. This procedure avoids recurring to numerical root-finding methods supported by extra eigenvalue problems. The solution of the fundamental buckling with increasing approximation order is compared to the reference value obtained by inverse iterations.

A locking‐free shifted inverse iteration based on multigrid discretization for the elastic eigenvalue problem

We propose a locking‐free shifted inverse iteration based on multigrid discretization and prove the error estimates of eigenvalues and eigenfunctions. Besides, we also present a multigrid correction scheme based on the multilevel correction. Numerical experiments coincide in our theoretical analysis and validate that our scheme is highly efficient.

A general adaptive finite element eigen-algorithm stemming from Wittrick-Williams algorithm

Stemming from Wittrick-Williams algorithm for transcendental eigenproblems, a general adaptive finite element (FE) procedure is proposed for general linear eigenproblems. The pivot of the proposed procedure is the counting method based on Sturm sequences, a simplified version of the Wittrick-Williams algorithm for linear matrix eigenproblems, based on which a general eigen-algorithm is developed by integrating a number of effective approaches such as a two-phased divide-and-conquer scheme, a guided-and-guarded technique for eigenvalue seeking, the inverse and subspace iterations for mode finding, etc. The adopted error estimator in the adaptive FE procedure is the recently developed one which is based on the element energy projection (EEP) technique for pointwise error estimation and hence is able to control errors by maximum norm with local mesh refinement. Representative numerical examples of plates and shells are given to show the validity, reliability and efficiency of the proposed procedure.

ADAPTIVE SOLUTION OF THE NEUTRON DIFFUSION EQUATION WITH HETEROGENEOUS COEFFICIENTS USING THE MIXED FINITE ELEMENT METHOD ON STRUCTURED MESHES

The neutron transport equation can be used to model the physics of the nuclear reactor core. Its solution depends on several variables and requires a lot of high precision computations. One can simplify this model to obtain the SPN equation for a generalized eigenvalue problem. In order to solve this eigenvalue problem, we usually use the inverse power iteration by solving a source problem at each iteration. Classically, this problem can be recast in a mixed variational form, and then discretized by using the Raviart-Thomas-Nédélec Finite Element. In this article, we focus on the steady-state diffusion equation with heterogeneous coefficients discretized on Cartesian meshes. In this situation, it is expected that the solution has low regularity. Therefore, it is necessary to refine at the singular regions to get better accuracy. The Adaptive Mesh Refinement (AMR) is one of the most effective ways to do that and to improve the computational time. The main ingredient for the refinement techniques is the use of a posteriori error estimates, which gives a rigorous upper bound of the error between the exact and numerical solution. This indicator allows to refine the mesh in the regions where the error is large. In this work, some mesh refinement strategies are proposed on the Cartesian mesh for the source problem. Moreover, we numerically investigate an algorithm which combines the AMR process with the inverse power iteration to handle the generalized eigenvalue problem.

Diffusion recursive total least square algorithm over adaptive networks and performance analysis

In most existing distributed estimation algorithms, many scholars adopted the assumption that the output signal of the system is noisy and the input signal is sufficiently accurate. However, in real applications, the filter input vector usually contains noise. In such situations, it has been proved that the adaptive filtering algorithm using total least squares (TLS) method has shown better performance than classical least squares (LS) method. So in this paper, we propose a diffusion recursive total least squares (DRTLS) algorithm by using the single inverse power iterations in the distributed network. Then, we analyze the mean and mean square behaviors of the proposed DRTLS algorithm. In addition, to reduce the computational complexity of the DRTLS algorithm,an improved algorithm called DCD-DRTLS is also proposed by using the dichotomous coordinate descent (DCD) iterations. At last, simulation results illustrate the superiority of the proposed algorithms and the validity of the theoretical analysis results.

Novel strategies for modal-based structural material identification

In this work, we present modal-based methods for model calibration in structural dynamics, and address several key challenges in the solution of gradient-based optimization problems with eigenvalues and eigenvectors, including the solution of singular Helmholtz problems encountered in sensitivity calculations, non-differentiable objective functions caused by mode swapping during optimization, and cases with repeated eigenvalues. Unlike previous literature that relied on direct solution of the eigenvector adjoint equations, we present a parallel iterative domain decomposition strategy (Adjoint Computation via Modal Superposition with Truncation Augmentation) for the solution of the singular Helmholtz problems. For problems with repeated eigenvalues we present a novel Mode Separation via Projection algorithm, and in order to address mode swapping between inverse iterations we present a novel Injective mode ordering metric. We present the implementation of these methods in a massively parallel finite element framework with the ability to use measured modal data to extract unknown structural model parameters from large complex problems. A series of increasingly complex numerical examples are presented that demonstrate the implementation and performance of the methods in a massively parallel finite element framework [7,5], using gradient-based optimization techniques in the Rapid Optimization Library (ROL) [21].

Fourier spectral method on sparse grids for computing ground state of many-particle fractional Schrödinger equations

ABSTRACT In this paper, we consider the Fourier spectral method on the sparse grids for computing the ground state of the many-particle fractional Schrödinger equations. The appropriate sparse grids for many-particle fractional Schrödinger equations are given, and the estimation for the number of grid points is obtained. Then, the iterative scheme of the inverse power method is presented to compute the ground state. In the numerical experiments, we consider two kinds of fractional Schrödinger equations, i.e. the fractional Schrödinger equation with harmonic potential and the electronic fractional Schrödinger equation. For the fractional Schrödinger equation with harmonic potential, when the scaled Planck constant ℏ is equal to 1, the sparse grid method has the obvious advantages in the high-resolution approximation both for the integer-order problem and the fractional-order problem, but when ℏ is equal to 1/4, the advantage of sparse grid method exists only for the fractional-order problem. We also show that for the electronic fractional Schrödinger equation, the sparse grid method is much better than the full grid method when the number of electrons is relatively large.

PDE eigenvalue iterations with applications in two-dimensional photonic crystals

We consider PDE eigenvalue problems as they occur in two-dimensional photonic crystal modeling. If the permittivity of the material is frequency-dependent, then the eigenvalue problem becomes nonlinear. In the lossless case, linearization techniques allow an equivalent reformulation as an extended but linear and Hermitian eigenvalue problem, which satisfies a Gårding inequality. For this, known iterative schemes for the matrix case such as the inverse power or the Arnoldi method are extended to the infinite-dimensional case. We prove convergence of the inverse power method on operator level and consider its combination with adaptive mesh refinement, leading to substantial computational speed-ups. For more general photonic crystals, which are described by the Drude–Lorentz model, we propose the direct application of a Newton-type iteration. Assuming somea prioriknowledge on the eigenpair of interest, we prove local quadratic convergence of the method. Finally, numerical experiments confirm the theoretical findings of the paper.

Идентификация параметров дискретных стохастических систем методом обратных итераций

Получены условия глобальной сходимости алгоритмов, основанных на обратных итерациях в переменной метрике, в задаче идентификации параметров дискретной стохастической системы с возмущениями в невязке уравнения и наблюдениях процессов. Доказана сходимость оценок параметров к истинному значению при увеличении объема выборки наблюдений истинного процесса. Приведены примеры расчетов The article addresses the problem of identifying parameters of discrete stochastic systems with perturbations in the residual of the equation and observation of variables. The identification functional in the problem has a complex nature of isosurfaces, which is why universal minimization algorithms based on estimates of the first and second derivatives have a small radius of convergence. It is proposed to employ efficient computational identification algorithms with inverse iterations in a variable metric for solving the convergence problem for two classes of systems with simple correspondence between matrix elements and parameters of equivalent systems without state variables. These algorithms are used for systems without state variables due to the large radius and high convergence rate since the 1970s. At first, a theorem on the conditions for convergence of inverse iterations from almost any initial approximation to a small neighborhood of the global minimum of the identification functional was proved. Secondly, a theorem on the convergence of the points of the global minimum of the identification functional to the desired true value with an increase in the sample size of observations is taken into account. Assumption of a zero first and restricted second moments of stochastic disturbances in the residual of the equation and observation of variables was made. The convergence of inverse iterations is shown numerically in a model example with significant values of disturbances. The result of the article is new theorems on the conditions of global convergence of computational algorithms with inverse iterations in the problem with mixed disturbances and the justification of possibility of using these algorithms to identify the parameters for discrete stochastic systems of two classes with a simple correspondence between matrix elements and parameters of equivalent systems without state variables

Numerical Solution of One-Speed 1-Dimensional Diffusion Equation based on Finite Difference and Source Iteration

Numerical methods are in favor to solve the impenetrable diffusion equations for neutron transport in contrast to using the analytical approach. This work applied a finite difference scheme and inverse power iteration to solve the one-group diffusion problem numerically for a one-dimensional slab containing fissile materials and heavy water. Both numerical and analytical solutions were obtained for comparison. MATLAB was employed to perform iterations and the minimum mesh size search. The effect of the initial guess on the accuracy and computational efficiency was investigated. Arguably, there is an appreciable acceleration for convergence by more intuitive initial estimations, as evidenced by the comparative results in the iteration counts decreasing from 799 to 8 by means of a random sampling method in place of the conventional initial guess method based on uniform flux when searching for the minimum size of the mesh necessary to meet the prescriptive accuracy.