Iterative Projection Method Research Articles

Projection iterative methods for extended general variational inequalities

In this paper, we introduce and study a new class of variational inequalities involving three operators, which is called the extended general variational inequality. Using the projection technique, we show that the extended general variational inequalities are equivalent to the fixed point and the extended general Wiener-Hopf equations. This equivalent formulation is used to suggest and analyze a number of projection iterative methods for solving the extended general variational inequalities. We also consider the convergence of these new methods under some suitable conditions. Since the extended general variational inequalities include general variational inequalities and related optimization problems as special cases, results proved in this paper continue to hold for these problems.

An oblique projection iterative method to compute Drazin inverse and group inverse

In this paper, an oblique projection iterative method is presented to compute matrix equation AXA = A of a square matrix A with ind ( A ) = 1 . By this iterative method, when taken the initial matrix X 0 = A , the group inverse A g can be obtained in absence of the roundoff errors. If we use this iterative method to the matrix equation A k XA k = A k , a group inverse ( A k ) g of matrix A k is got, then we use the formulae A d = A k - 1 ( A k ) g , the Drazin inverse A d can be obtained.

Spatial Weighting and Iterative Projection Methods for EOFs

Abstract Often there is a need to consider spatial weighting in methods for finding spatial patterns in climate data. The focus of this paper is on techniques that maximize variance, such as empirical orthogonal functions (EOFs). A weighting matrix is introduced into a generalized framework for dealing with spatial weighting. One basic principal in the design of the weighting matrix is that the resulting spatial patterns are independent of the grid used to represent the data. A weighting matrix can also be used for other purposes, such as to compensate for the neglect of unrepresented subgrid-scale variance or, in the form of a prewhitening filter, to maximize the signal-to-noise ratio of EOFs. The new methodology is applicable to other types of climate pattern analysis, such as extended EOF analysis and maximum covariance analysis. The increasing availability of large datasets of three-dimensional gridded variables (e.g., reanalysis products and model output) raises special issues for data-reduction methods such as EOFs. Fast, memory-efficient methods are required in order to extract leading EOFs from such large datasets. This study proposes one such approach based on a simple iteration of successive projections of the data onto time series and spatial maps. It is also demonstrated that spatial weighting can be combined with the iterative methods. Throughout the paper, multivariate statistics notation is used, simplifying implementation as matrix commands in high-level computing languages.

Sidelobe Reduction Through Element Phase Control in Uniform Subarrayed Array Antennas

When dealing with the synthesis subarrayed array antennas, the phase control necessary for beam-steering purposes can be also exploited to reduce the undesired secondary lobes caused by the periodic spatial distribution of the amplitude excitations on the aperture when contiguous and identical subarrays are used. In order to determine the phase values of the array elements for a fixed subarray amplitude weighting, the iterative projection method (IPM) is adopted. Some representative results are shown to assess the effectiveness of the method.

Construction of solutions of integro-differential equations with restrictions and control by projection-iterative method

We substantiate the application of the projection-iterative method to the solution of a boundary-value problem for integro-differential equations with restrictions and control.

Approximation methods for a class of discrete Wiener-Hopf equations

In this paper, we consider approximation methods for operator equations of the form \[Au + Bu = f,\] where \(A\) is a discrete Wiener-Hopf operator on \(l_p\) (\(1 \leq p \lt \infty\)) which symbol has roots on the unit circle with arbitrary multiplicities (not necessary integers). Conditions on perturbation \(B\) and \(f\) are given in order to guarantee the applicability of projection-iterative methods. Effective error estimates, and simultaneously, decaying properties for solutions are obtained in terms of some smooth spaces.

On the Kaczmarz iterative method and its generalizations

Under study is the family of iterative projection methods that stems from the work of Kaczmarz in 1937. We propose some alternating block algorithms of Kaczmarz type in Krylov subspaces with a relaxation parameter and acceleration in Krylov spaces. The results are presented and discussed of numerical experiments for some model problems.

Methods for solving a singular integral equation with Cauchy kernel on the real line

We study exact and approximate methods for solving a singular integral equation with Cauchy kernel on the real line. On the basis of the theory of positive operators, we prove an existence and uniqueness theorem for this equation in the space of Lebesgue square integrable functions. This theorem is then used to give a theoretical justification of general projection and projection-iteration methods as well as an iteration method for solving this equation.

RESTARTING PROJECTION METHODS FOR RATIONAL EIGENPROBLEMS ARISING IN FLUID‐SOLID VIBRATIONS

For nonlinear eigenvalue problems T(λ)x = 0 satisfying a minmax characterization of its eigenvalues iterative projection methods combined with safeguarded iteration are suitable for computing all eigenvalues in a given interval. Such methods hit their limitations if a large number of eigenvalues is required. In this paper we discuss restart procedures which are able to cope with this problem, and we evaluate them for a rational eigenvalue problem governing vibrations of a fluid‐solid structure.

Extended general variational inequalities

In this work, we introduce and consider a new class of general variational inequalities involving three nonlinear operators, which is called the extended general variational inequalities. Noor [M. Aslam Noor, Projection iterative methods for extended general variational inequalities, J. Appl. Math. Comput. (2008) (in press)] has shown that the minimum of nonconvex functions can be characterized via these variational inequalities. Using a projection technique, we establish the equivalence between the extended general variational inequalities and the general nonlinear projection equation. This equivalent formulation is used to discuss the existence of a solution of the extended general variational inequalities. Several special cases are also discussed.

Smoothness-constrained projection method for particle analysis based on forward light scattering

The iterative projection method, originally proposed by Kaczmarz and Cimmino, was recently applied to particle size analysis with forward light scattering. Modification was made to improve the convergent procedure. However, there are still limitations. It is found that the solutions are oscillatory when the method is applied to a set of underdetermined linear equations without any additional constraints. A new version of the projection method, combined with the constraints of nonnegativity and smoothness, is proposed and is studied in the presence of experimental noise. Compared with the early versions of the projection method, this one shows fast convergence, is more stable against random noise, and is insensitive to particle size distributions.

GLOBAL CONVERGENCE OF RTLSQEP: A SOLVER OF REGULARIZED TOTAL LEAST SQUARES PROBLEMS VIA QUADRATIC EIGENPROBLEMS

The total least squares (TLS) method is a successful approach for linear problems if both the matrix and the right hand side are contaminated by some noise. In a recent paper Sima, Van Huffel and Golub suggested an iterative method for solving regularized TLS problems, where in each iteration step a quadratic eigenproblem has to be solved. In this paper we prove its global convergence, and we present an efficient implementation using an iterative projection method with thick updates.

The Metric Nearness Problem

Metric nearness refers to the problem of optimally restoring metric properties to distance measurements that happen to be nonmetric due to measurement errors or otherwise. Metric data can be important in various settings, for example, in clustering, classification, metric-based indexing, query processing, and graph theoretic approximation algorithms. This paper formulates and solves the metric nearness problem: Given a set of pairwise dissimilarities, find a “nearest” set of distances that satisfy the properties of a metric—principally the triangle inequality. For solving this problem, the paper develops efficient triangle fixing algorithms that are based on an iterative projection method. An intriguing aspect of the metric nearness problem is that a special case turns out to be equivalent to the all pairs shortest paths problem. The paper exploits this equivalence and develops a new algorithm for the latter problem using a primal-dual method. Applications to graph clustering are provided as an illustration. We include experiments that demonstrate the computational superiority of triangle fixing over general purpose convex programming software. Finally, we conclude by suggesting various useful extensions and generalizations to metric nearness.

Robust expansion of subspaces in iterative projection methods for large eigenvalue problems

AbstractThe Jacobi–Davidson method is known to converge at least quadratically if the correction equation is solved exactly, and it is common experience that the fast convergence is maintained if the correction equation is solved only approximately. Here we derive the Jacobi–Davidson method in a way that explains this robust behavior. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Reducing sparse nonlinear eigenproblems by automated multi-level substructuring

The automated multi-level substructuring method (AMLS) was recently suggested as an alternative to iterative projection methods for computing eigenpairs of huge matrix eigenproblems in the context of structural engineering. Taking advantage of a substructuring on several levels the method constructs a projected problem of much smaller dimension which still yields satisfactory accuracy over a wide frequency range of interest. In this paper we generalise the AMLS method to certain classes of nonlinear eigenvalue problems which can be partitioned into an essential linear and positive definite pencil and a small residual. The efficiency of the method is demonstrated by numerical examples modeling damped vibrations of a structure with nonproportional damping, a gyroscopic eigenproblem, and a rational eigenproblem governing free vibrations of a fluid–solid structure.

Stationary Schrödinger equations governing electronic states of quantum dots in the presence of spin-orbit splitting

In this work we derive a pair of nonlinear eigenvalue problems corresponding to the one-band effective Hamiltonian accounting for the spin-orbit interaction governing the electronic states of a quantum dot. We show that the pair of nonlinear problems allows for the minmax characterization of its eigenvalues under certain conditions which are satisfied for our example of a cylindrical quantum dot and the common InAs/GaAs heterojunction. Exploiting the minmax property we devise an efficient iterative projection method simultaneously handling the pair of nonlinear problems and thereby saving about 25% of the computation time as compared to the Nonlinear Arnoldi method applied to each of the problems separately.

A Jacobi–Davidson method for nonlinear and nonsymmetric eigenproblems

For the nonlinear eigenvalue problem T( λ) x = 0 we consider a Jacobi–Davidson type iterative projection method for computing a few eigenvalues close to a given parameter. We discuss the numerical solution of the projected eigenvalue problems in particular for nonsymmetric systems. We present methods how to prevent the algorithm from converging to the same eigenpair repeatedly. To verify the Jacobi–Davidson method it is applied to a rational eigenvalue problem governing damped vibrations of a structure and to a damped gyroscopic eigenvalue problem.

Even systems of functional differential equations with restrictions and methods for their solution

We propose an approach to the investigation of even systems of functional differential equations with restrictions and control. According to this approach, the investigation of the consistency of the considered problem is reduced to the investigation of the solvability of a system of integral equations. We substantiate the application of the iteration and projection-iterative methods to this problem.

A version of the projection-iterative method for systems of linear differential equations with delay of neutral type and restrictions

We establish consistency conditions for systems of linear differential equations with constant delay of neutral type and restrictions. The applicability of the projection-iterative method to these problems is justified.

Iterative projection methods for computing relevant energy states of a quantum dot

A computational technique for computing relevant energy levels and corresponding wave functions of an electron confined by a 3D quantum dot embedded in a semiconductor matrix are studied. Assuming an energy and position dependent electron effective mass approximation this problem is governed by a rational eigenvalue problem. We discuss the application of iterative projection method of Arnoldi and Jacobi-Davidson type. Projected problems of small dimension are solved efficiently by safeguarded iteration.