Inverse Iteration Research Articles

Power iteration and inverse power iteration for eigenvalue complementarity problem

SummaryIn this paper, an inverse complementarity power iteration method (ICPIM) for solving eigenvalue complementarity problems (EiCPs) is proposed. Previously, the complementarity power iteration method (CPIM) for solving EiCPs was designed based on the projection onto the convex cone K. In the new algorithm, a strongly monotone linear complementarity problem over the convex cone K is needed to be solved at each iteration. It is shown that, for the symmetric EiCPs, the CPIM can be interpreted as the well‐known conditional gradient method, which requires only linear optimization steps over a well‐suited domain. Moreover, the ICPIM is closely related to the successive quadratic programming (SQP) via renormalization of iterates. The global convergence of these two algorithms is established by defining two nonnegative merit functions with zero global minimum on the solution set of the symmetric EiCP. Finally, some numerical simulations are included to evaluate the efficiency of the proposed algorithms.

Lightweight Topology Optimization with Buckling and Frequency Constraints Using the Independent Continuous Mapping Method

This research focuses on the lightweight topology optimization method for structures under the premise of meeting the requirements of stability and vibration characteristics. A new topology optimization model with the constraints of natural frequencies and critical buckling loads and the objective of minimizing the structural volume is established and solved based on the independent continuous mapping method. The eigenvalue equations and composite exponential filter functions are applied to convert the optimization formulation into a continuous, solvable mathematical programming model. In the process of topology optimization, suitable initial values of the filter functions are chosen to avoid local modes, and the dynamic frequency gap constraints are added in the optimal model to prevent mode switches. Furthermore, for the optimal structures with grey elements obtained by the continuous optimization model, the bisection–inverse iteration is applied to search the optimal discrete structures. Finally, a detailed scheme is given for the buckling and frequency topology optimization problem. Numerical examples illustrate that the modelling method of minimizing the economic index with given performance requirements is practical and feasible for multi-performance topology optimization problems.

Asymptotic convergence of spectral inverse iterations for stochastic eigenvalue problems

We consider and analyze applying a spectral inverse iteration algorithm and its subspace iteration variant for computing eigenpairs of an elliptic operator with random coefficients. With these iterative algorithms the solution is sought from a finite dimensional space formed as the tensor product of the approximation space for the underlying stochastic function space, and the approximation space for the underlying spatial function space. Sparse polynomial approximation is employed to obtain the first one, while classical finite elements are employed to obtain the latter. An error analysis is presented for the asymptotic convergence of the spectral inverse iteration to the smallest eigenvalue and the associated eigenvector of the problem. A series of detailed numerical experiments supports the conclusions of this analysis.

Local convergence analysis of inverse iteration algorithm for computing the H-spectral radius of a nonnegative weakly irreducible tensor

In this paper, we present an inverse iteration algorithm, to find the H-spectral radius and the associated positive eigenvector of a nonnegative weakly irreducible tensor, which always preserve the positivity of approximate eigenvectors. The local quadratical convergence of the proposed algorithm is established based on a basic result of the Newton’s method for solving nonlinear equations. Some numerical examples illustrate the efficiency of the proposed algorithm.

A quantum eigensolver for symmetric tridiagonal matrices

The eigenproblem of a symmetric tridiagonal matrix widely appears in physical applications and scientific computing. The classical solvers are the symmetric QR iteration, divide-and-conquer, bisection and inverse iteration, etc. For a symmetric tridiagonal matrix of dimension N, the cost of these methods scales at least as $$O(N^2)$$ . In this work, we present an efficient quantum algorithm for solving this problem through quantum simulation of resonant transitions. The cost of the algorithm can scale as $$poly(\log \,N)$$ , provided a good initial guess on the eigenstates of the system is obtained. By coupling a probe qubit to a quantum register that represents a system, resonance dynamics can be observed on the probe qubit when its frequency matches a transition frequency in the system. Energy eigenvalues of the system can be obtained by locating resonant transition frequencies between the probe qubit and transitions from a reference state to eigenstates of the system through observing dynamics of the probe qubit for different frequencies. And the system can be evolved to one of its eigenstate with known eigenvalue by inducing an appropriate resonant transition between the probe qubit and the system.

An initial acoustic impedance modeling method based on seismic attributes

It is common that an initial impedance model should be created first, while performing acoustic impedance inversion. The traditional modeling is horizontally interpolated using inverse distance power method for multiple wells, and vertically controlled by seismic horizons. This method was relatively simple, and could not reflect the seismic facies changes in complex geological conditions, inversion accuracy was not high.

Low-Rank Solution Methods for Stochastic Eigenvalue Problems

We study efficient solution methods for stochastic eigenvalue problems arising from discretization of self-adjoint partial differential equations with random data. With the stochastic Galerkin approach, the solutions are represented as generalized polynomial chaos expansions. A low-rank variant of the inverse subspace iteration algorithm is presented for computing one or several minimal eigenvalues and corresponding eigenvectors of parameter-dependent matrices. In the algorithm, the iterates are approximated by low-rank matrices, which leads to significant cost savings. The algorithm is tested on two benchmark problems, a stochastic diffusion problem with some poorly separated eigenvalues, and an operator derived from a discrete stochastic Stokes problem whose minimal eigenvalue is related to the inf-sup stability constant. Numerical experiments show that the low-rank algorithm produces accurate solutions compared to the Monte Carlo method, and it uses much less computational time than the original algorithm without low-rank approximation.

Binding energies of quantum dipole in plane

We propose a numerical algorithm based on a discrete variable representation and shifted inverse iterations and apply it to for the analysis of the bound states of edge dislocation modelled by a quantum dipole in a plane. The good agreement with results of recent papers of Amore et al [J. Phys. B 45, 235004 (2012)] was obtained. The error estimates of the previous results of low-lying states energies of other authors were not known due to limitations of the variational approaches and this paper fills this gap presenting calculated low-lying bound states energies by non-variational technique. The probability densities of low-lying states were calculated.

Localized Computation of Eigenstates of Random Schrödinger Operators

This paper concerns the numerical approximation of low-energy eigenstates of the linear random Schrödinger operator. Under oscillatory high-amplitude potentials with a sufficient degree of disorder it is known that these eigenstates localize in the sense of an exponential decay of their moduli. We propose a reliable numerical scheme which provides localized approximations of such localized states. The method is based on a preconditioned inverse iteration including an optimal multigrid solver which spreads information only locally. The practical performance of the approach is illustrated in various numerical experiments in two and three space dimensions and also for a nonlinear random Schrödinger operator.

Shape Design of Cable-net for Parabolic Cylindrical Deployable Antenna

A method of cable-net shape design based on the equilibrium matrix method is proposed for a new parabolic cylindrical deployable antenna structure with fewer modules. And the inverse iteration method is adopted to find the shape of cable-truss structure with considering the truss deformation induced by cable tension. Firstly, the ideal geometrical configuration of the locally symmetric support cable is designed for the given truss. Then, the pretension distribution of the cable is solved by the equilibrium matrix method under the circumstance of the unchanged topology of cable-net structure, position of nodes and boundary condition. In addition, the inverse iteration method is adopted to find the shape of cable-truss structure. Finally, the validity of the method is verified by simulation analysis.

On convergence of the inverse iteration algorithm for modified Prony method

On convergence of the inverse iteration algorithm for modified Prony method

Blind multichannel identification based on Kalman filter and eigenvalue decomposition

A noise-robust approach for blind multichannel identification is proposed on the basis of Kalman filter and eigenvalue decomposition. It is proved that the state vector composed of the multichannel impulse responses is nothing but the eigenvector corresponding to the maximum eigenvalue of the filtered state-error correlation matrix. This eigenvector can be computed iteratively with the so-called ‘power method’ to reduce the complexity of the algorithm. Furthermore, it is found that the computation of the inverse of the filtered state-error correlation matrix is much easier than itself, the wanted state vector can be computed from this inverse matrix with the so-called ‘inverse power method’. Therefore, two algorithms are proposed on the basis of the eigenvalue decomposition of the filtered state-error correlation matrix and its inverse matrix, respectively. In addition, for reducing the computing complexity of the proposed algorithms, matrix factorization such as QR-, LU- and Cholesky-factorizations are exploited to accelerate the computation of the algorithms. Simulations show that the proposed algorithms perform well over a wide range of the signal-to-noise ratio of the multichannel signals.

Alternating iterative methods for solving tensor equations with applications

Recently, the alternating direction method of multipliers (ADMM) and its variations have gained great popularity in large-scale optimization problems. This paper is concerned with the solution of the tensor equation $\mathscr{A}\textbf {x}^{m-1}=\textbf {b}$ in which $\mathscr{A}$ is an m th-order and n-dimensional real tensor and b is an n-dimensional real vector. By introducing certain auxiliary variables, we transform equivalently this tensor equation into a consensus constrained optimization problem, and then propose an ADMM type method for it. It turns out that each limit point of the sequences generated by this method satisfies the Karush-Kuhn-Tucker conditions. Moreover, from the perspective of computational complexity, the proposed method may suffer from the curse-of-dimensionality if the size of the tensor equation is large, and thus we further present a modified version (as a variant of the former) turning to the tensor-train decomposition of the tensor $\mathscr{A}$ , which is free from the curse. As applications, we establish the associated inverse iteration methods for solving tensor eigenvalue problems. The performed numerical examples illustrate that our methods are feasible and efficient.

Approximation theorem for principle eigenvalue of discrete p-Laplacian

For the principle eigenvalue of discrete weighted p-Laplacian on the set of nonnegative integers, the convergence of an approximation procedure and the inverse iteration is proved. Meanwhile, in the proof of the convergence, the monotonicity of an approximation sequence is also checked. To illustrate these results, some examples are presented.

An eigenvector‐based iterative procedure for the free‐interface component modal synthesis method

SummaryA novel eigenvector‐based iteration procedure is developed for the free‐interface component modal synthesis (CMS) method. To derive the iteration formula, Kron's substructuring is employed to distribute the computations of subspace iteration, and the free‐interface component modes are chosen as the initial guess. Then, the modal transformation matrix of the first‐order approximated Kron's CMS method is proved to be the free‐interface component modes with one step of Kron's inverse iteration. The proposed CMS method has the advantages of both free‐interface CMS approximation and subspace iteration: on one hand, the CMS approximation provides a high‐quality initial guess and distributes the computational cost of the subspace iteration; on the other hand, the subspace iteration provides a more efficient way for truncation compensation and is compatible with using deflation, shifting, and restarting for further enhancements on the efficiency. Numerical examples show that the efficiency of the proposed method is higher than that of the conventional simultaneous iterative Kron's CMS method, especially for obtaining a large number of high‐precision modes. Moreover, the proposed method is as efficient as the global Lanczos method for first‐time analysis without parallelization while retaining the advantages of CMS methods for reanalysis tasks, parallelization, etc.

Structural total least squares algorithm for locating multiple disjoint sources based on AOA/TOA/FOA in the presence of system error

Single-station passive localization technology avoids the complex time synchronization and information exchange between multiple observatories, and is increasingly important in electronic warfare. Based on a single moving station localization system, a new method with high localization precision and numerical stability is proposed when the measurements from multiple disjoint sources are subject to the same station position and velocity displacement. According to the available measurements including the angle-of-arrival (AOA), time-of-arrival (TOA), and frequency-of-arrival (FOA), the corresponding pseudo linear equations are deduced. Based on this, a structural total least squares (STLS) optimization model is developed and the inverse iteration algorithm is used to obtain the stationary target location. The localization performance of the STLS localization algorithm is derived, and it is strictly proved that the theoretical performance of the STLS method is consistent with that of the constrained total least squares method under first-order error analysis, both of which can achieve the Cramer-Rao lower bound accuracy. Simulation results show the validity of the theoretical derivation and superiority of the new algorithm.

On the power method for quaternion right eigenvalue problem

In this paper, we study the power method of the right eigenvalue problem of a quaternion matrix A . If A is Hermitian, we propose the power method that is a direct generalization of that of complex Hermitian matrix. When A is non-Hermitian, by applying the properties of quaternion right eigenvalues, we propose the power method for computing the standard right eigenvalue with the maximum norm and the associated eigenvector. We also briefly discuss the inverse power method and shift inverse power method for the both cases. The real structure-preserving algorithm of the power method in the two cases are also proposed, and numerical examples are provided to illustrate the efficiency of the proposed power method and inverse power method.

Phase retrieval from local measurements: Improved robustness via eigenvector-based angular synchronization

Abstract We improve a phase retrieval approach that uses correlation-based measurements with compactly supported measurement masks [30] . Our approach admits deterministic measurement constructions together with a robust, fast recovery algorithm that consists of solving a system of linear equations in a lifted space, followed by finding an eigenvector (e.g., via an inverse power iteration). Theoretical reconstruction error guarantees from [30] are improved as a result for the new and more robust reconstruction approach proposed herein. Numerical experiments demonstrate robustness and computational efficiency that compete with other approaches on large problems. Along the way, we show that this approach also trivially extends to phase retrieval problems based on windowed Fourier measurements.

Inverse iteration algorithm for neutron depth profiling

This work is based on the method of Monte Carlo simulation and probabilistic inversion for neutron depth profiling. The depth and concentration distribution of boron in a silicon matrix material is calculated. According to NDP experimental configuration at CARR, the energy spectra of a standard sample (SRM2137) are measured and simulated using MCNP and Geant4 software, and the concentration-depth profile of boron in SRM2137 is obtained by adopting an inverse iteration method through MATLAB software. It is shown that the inverse iteration calculation for NDP is feasible.

Efficient algorithm for principal eigenpair of discrete p-Laplacian

This paper is a continuation of the author’s previous papers [Front. Math. China, 2016, 11(6): 1379–1418; 2017, 12(5): 1023–1043], where the linear case was studied. A shifted inverse iteration algorithm is introduced, as an acceleration of the inverse iteration which is often used in the non-linear context (the p-Laplacian operators for instance). Even though the algorithm is formally similar to the Rayleigh quotient iteration which is well-known in the linear situation, but they are essentially different. The point is that the standard Rayleigh quotient cannot be used as a shift in the non-linear setup. We have to employ a different quantity which has been obtained only recently. As a surprised gift, the explicit formulas for the algorithm restricted to the linear case (p = 2) is obtained, which improves the author’s approximating procedure for the leading eigenvalues in different context, appeared in a group of publications. The paper begins with p-Laplacian, and is closed by the non-linear operators corresponding to the well-known Hardy-type inequalities.