Iterative Projection Method Research Articles

A projection iterative algorithm boundary element method for the Signorini problem

We propose a projection iterative algorithm based on a fixed point equation for solving a certain class of Signorini problem. The satisfaction of the Signorini boundary conditions is verified in a projection iterative manner, and at each iterative step, an elliptic mixed boundary value problem is solved by a boundary element method which is suitable for any domain. We prove the convergence of the algorithm by the property of projection. The advantage of this algorithm is that it is easy to be implemented and converge quickly. Some numerical results show the accuracy and effectiveness of the algorithm.

Projection iterative method for solving general variational inequalities

In this paper, we suggest and analyze a new projection iterative method for solving general variational inequalities by using a new step size. We also prove the global convergence of the proposed method under some suitable conditions. Some preliminary numerical experiments are included to illustrate the advantage and efficiency of the proposed method.

THREE-STEP PROJECTION METHOD FOR GENERAL VARIATIONAL INEQUALITIES

In this paper, we suggest and analyze a new three-step iterative projection method for solving general variational inequalities in conjunction with a descent direction. We prove that the new method is globally convergent under suitable mild conditions. An example is given to illustrate the advantage and efficiency of the proposed method.

Analysis And Efficient Solution Of Stationary Schrödinger Equation Governing Electronic States Of Quantum Dots And Rings In Magnetic Field

AbstractIn this work the one-band effective Hamiltonian governing the electronic states of a quantum dot/ring in a homogenous magnetic field is used to derive a pair/quadruple of nonlinear eigenvalue problems corresponding to different spin orientations and in case of rotational symmetry additionally to quantum number -L•i. We show, that each of those pair/quadruple of nonlinear problems allows for the min-max characterization of its eigenvalues under certain conditions, which are satisfied for our examples and the common InAs/GaAs heterojunction. Exploiting the minmax property we devise efficient iterative projection methods simultaneously handling the pair/quadruple of nonlinear problems and thereby saving up to 40% of the computational time as compared to the nonlinear Arnoldi method applied to each of the problems separately.

An Iterative Shrinking Projection Method for Solving Fixed Point Problems of Closed and ϕ‐Quasi‐Strict Pseudocontractions along with Generalized Mixed Equilibrium Problems in Banach Spaces

We provide some new type of mappings associated with pseudocontractions by introducing some actual examples in smooth and strictly convex Banach spaces. Moreover, we also find the significant inequality related to the mappings mentioned in the paper and the mappings defined from generalized mixed equilibrium problems on Banach spaces. We propose an iterative shrinking projection method for finding a common solution of generalized mixed equilibrium problems and fixed point problems of closed and ϕ‐quasi‐strict pseudo‐contractions. Our results hold in reflexive, strictly convex, and smooth Banach spaces with the property (K). The results of this paper improve and extend the corresponding results of Zhou and Gao (2010) and many others.

Some Aspects of Extended General Variational Inequalities

Noor (“Extended general variational inequalities,” 2009, “Auxiliary principle technique for extended general variational inequalities,” 2008, “Sensitivity analysis of extended general variational inequalities,” 2009, “Projection iterative methods for extended general variational inequalities,” 2010) introduced and studied a new class of variational inequalities, which is called the extended general variational inequality involving three different operators. This class of variational inequalities includes several classes of variational inequalities and optimization problems. The main motivation of this paper is to review some aspects of these variational inequalities including the iterative methods and sensitivity analysis. We expect that this paper may stimulate future research in this field along with novel applications.

Hyperspectral Unmixing Using Statistics of Q Function

Proposed technique of hyperspectral unmixing is apparent to implement and compute the results in a very fast and efficient manner. To reducing the computational complexity and to estimation of hyperspectral data we adopted a statistical method of median absolute deviation about median. Number of end-members is enumerating by self iterative subspace projection method which depends on Pearson correlation. The mixing matrix is inferred by using Q function projections. A set of tests with real hyperspectral data evaluates the performance and illustrates the effectiveness of the proposed method. For the evaluation of proposed method, the results are compared with the results of vertex component analysis. The experimental results show the effectiveness of proposed method on hyperspectral unmixing. targets Alunite, Buddingtonite, Calcite, Kaolinite, and Muscovite are detected well and have high spectral similarities. Hyperspectral remote sensing is used in a large array of real life applications e.g. Surveillance, Mineralogy, Physics, and Agriculture. The complete work is prepared by using MATLAB.

Projection iterative methods for solving some systems of general nonconvex variational inequalities

In this article, we introduce and consider a new system of general nonconvex variational inequalities involving four different operators. We use the projection operator technique to establish the equivalence between the system of general nonconvex variational inequalities and the fixed points problem. This alternative equivalent formulation is used to suggest and analyse some new explicit iterative methods for this system of nonconvex variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Since this new system includes the system of nonconvex variational inequalities, variational inequalities and related optimization problems as special cases, results obtained in this article continue to hold for these problems. Our results can be viewed as a refinement and an improvement of the previously known results for variational inequalities.

Multiband analysis and synthesis of spectro-temporal modulations of Fourier spectrogram

The two-dimensional spectro-temporal modulation filtering concept of the auditory model [T. Chi, P. Ru, and S. A. Shamma, J. Acoust. Soc. Am. 118(2), 887-906 (2005)] is implemented on the Fourier spectrogram. The Fourier magnitude spectrogram is analyzed in terms of its joint spectro-temporal modulations, which embed the temporal dynamics and spectral structures. Instead of iterative projection methods, the overlap-and-add method is adopted to invert modified Fourier spectrograms back to sounds. The proposed framework not only provides a similar spectro-temporal analytical process for sounds as the auditory model but also produces synthesized sounds with better quality in a timely manner, which makes proposed framework feasible to human speech recognition (HSR) applications as well.

Pathway analysis of super-exchange electronic couplings in electron transfer reactions using a multi-configuration self-consistent field method

We present a novel pathway analysis of super-exchange electronic couplings in electron transfer reactions using localized molecular orbitals from multi-configuration self-consistent field (MCSCF) calculations. In our analysis, the electronic coupling and the tunneling pathways can be calculated in terms of the configuration interaction (CI) Hamiltonian matrix obtained from the localized MCSCF wave function. Making use of the occupation restricted multiple active spaces (ORMAS) method can effectively produce the donor, acceptor, and intermediate configuration state functions (CSFs) and CIs among these CSFs. In order to express the electronic coupling as a sum of individual tunneling pathways contributions, we employed two perturbative methods: Löwdin projection-iteration method and higher-order super-exchange method. We applied them to anion couplings of butane-1,4-diyl and pentane-1,5-diyl. The results were (1) the electronic couplings calculated from the two perturbative methods were in reasonable agreement with those from a non-perturbative method (one-half value of the energy difference between the ground and first excited states), (2) the main tunneling pathways consisted of a small number of lower-order super-exchange pathways where bonding, anti-bonding, or extra-valence-shell orbitals were used once or twice, and (3) the interference among a huge number of higher-order super-exchange pathways significantly contributed to the overall electronic coupling, whereas each of them contributed only fractionally. Our method can adequately take into account both effects of non-dynamical electron correlation and orbital relaxation. Comparing with the analyses based on the Koopmans' theorem (ignoring both effects) and the ORMAS-CIs from frozen localized reference orbitals (ignoring the effect of orbital relaxation), we discuss these effects.

Sparsity driven metal part reconstruction for artifact removal in dental CT.

Metal artifact removal (MAR) is one of the most important issues in x-ray CT reconstruction. Various methods have been suggested for metal artifact removal, among which projection modification and iterative methods are most popular. While those methods mainly focus on removing background artifacts, for some applications such as dental CT the correct reconstruction of metallic inserts is also important. For this application, we formulate the MAR problem as a sparse recovery problem since metallic inserts usually occupy very little volume within a field of view. One of the main advantages of this approach is to overcome the inconsistency of sinograms from metal artifacts by imposing a geometric constraint, "sparsity". As a side product of this formulation, a significant reduction of the sample views is feasible for metal part reconstruction without sacrificing quality, thanks to the compressed sensing theory, which minimizes the additional computational overhead. Numerical results confirm that metallic inserts can be accurately reconstructed with a significant reduction of computation time.

Construction of approximate solutions of systems of differential equations with parameters and restrictions

We propose a new approach to the investigation of boundary-value problems for systems of differential equations with parameters and restrictions. This approach uses ideas of both iterative and projection methods. We develop and substantiate a new modification of the projection-iterative method for the construction of approximate solutions of boundary-value problems that has significant advantages over the existing versions of the method.

Iterative Projection Methods for Large-Scale Nonlinear Eigenvalue Problems

In this presentation we review iterative projection methods for sparse nonlinear eigenvalue problems which have proven to be very efficient. Here the eigenvalue problem is projected to a subspace V of small dimension which yields approximate eigenpairs. If an error tolerance is not met then the search space V is expanded in an iterative way with the aim that some of the eigenvalues of the reduced matrix become good approximations to some of the wanted eigenvalues of the given large matrix. Methods of this type are the nonlinear Arnoldi method, the Jacobi–Davidson method, and the rational Krylov method.

Stein implicit Runge–Kutta methods with high stage order for large-scale ordinary differential equations

The Runge–Kutta method is one of the most popular implicit methods for the solution of stiff ordinary differential equations. For large problems, the main drawback of such methods is the cost required at each integration step for computing the solution of a nonlinear system of equations. In this paper, we propose to reduce the cost of the computation by transforming the linear systems arising in the application of Newton's method to Stein matrix equations. We propose an iterative projection method onto block Krylov subspaces for solving numerically such Stein matrix equations. Numerical examples are given to illustrate the performance of our proposed method.

Low-dose x-ray phase-contrast and absorption CT using equally sloped tomography

Tomographic reconstruction from undersampled and noisy projections is often desirable in transmission CT modalities for purposes of low-dose tomography and fast acquisition imaging. However under such conditions, due to the violation of the Nyquist sampling criteria and the presence of noise, reconstructions with acceptable accuracy may not be possible. Recent experiments in transmission electron tomography and coherent diffraction microscopy have shown that the technique of equally sloped tomography (EST), an exact tomographic method utilizing an oversampling iterative Fourier-based reconstruction, provides more accurate image reconstructions when the number of projections is significantly undersampled relative to filtered back projection and algebraic iterative methods. Here we extend this technique by developing new reconstruction algorithms which allow for the incorporation of advanced mathematical regularization constraints, such as the nonlocal means total variational model, in a manner that is consistent with experimental projections. We then evaluate the resulting image quality of the developed algorithm through simulations and experiments at the European Synchrotron Radiation Facility on image quality phantoms using the x-ray absorption and phase contrast CT modalities. Both our simulation and experimental results have indicated that the method can reduce the number of projections by 60–75% in parallel beam modalities, while achieving comparable or better image quality than the conventional reconstructions. As large-scale and compact synchrotron radiation facilities are currently under rapid development worldwide, the implementation of low-dose x-ray absorption and phase-contrast CT can find broad applications in biology and medicine using these advanced x-ray sources.

Successive projection iterative method for solving matrix equation [formula omitted

In this paper, we present a new iterative method (successive projection iterative method) to solve matrix equation A X = B , where A is a symmetric positive definite (SPD) matrix. Based on this method an algorithm is proposed and proved to be convergent. In addition, analysis of the algorithm and numerical experiments are also given to show the efficiency of the method.

Stochastic iterative projection methods for large linear systems

We suggest a randomized version of the projection methods belonging to the class of a “row-action” methods which work well both for systems with quadratic nonsingular matrices and for overdetermined systems. These methods belong to a type known as Projection on Convex Sets methods. Here we present a method beyond the conventional Markov chain based Neumann–Ulam scheme. The main idea is in a random choice of blocks of rows in the projection method so that in average, the convergence is improved compared to the conventional periodic choice of the rows. We suggest an acceleration of the row projection method by using the Johnson–Lindenstrauss (J–L) theorem to find, among the randomly chosen rows, in a sense an optimal row. We extend this randomized method for solving linear systems coupled with systems of linear inequalities. Applied to finite-difference approximations of boundary value problems, the method appears to be an extremely efficient Random Walk algorithm whose convergence is exponential, and the cost does not depend on the dimension of the matrix. In addition, the algorithm calculates the solution in all grid points, and is easily parallelizable.

Stability Analysis of the Quasidiffusion Method on Periodic Heterogeneous 1D Transport Problems

We study the convergence of the quasidiffusion (QD) method on one-dimensional spatially periodic heterogeneous problems. The QD method is a nonlinear projection-iterative method. A Fourier analysis of the linearized QD equations is performed. The convergence rates of the QD method in the vicinity of the solution are obtained. We also analyze the Second Moment (SM) method, which can be interpreted as a linear version of the QD method. The presented analysis gives a new insight on the convergence behavior of the QD method in a discretized form and reveals the differences in the convergence of the QD and SM methods. Numerical results are presented to confirm theoretical predictions.

Projection-Iterative Methods for a Class of Difference Equations

We develop a theoretical framework for projection-iterative methods to solve operator equations of the form Au + Bu = f, where A is a Toeplitz operator in a Banach space $$l_{p}({\mathbb{N}}) \quad (1 \leq p < \infty)$$ , B is considered as a perturbation (of general form) of A, and f is a given element in this space. The methods are adopted for application to general situations, in particular, to the equations in which A need not be a Fredholm operator. The idea to involve iteration procedures and the technique which we apply allow to obtain conditions on perturbations for convergence and effective error estimates in terms of some weighted spaces (without any restrictions on the norms for perturbations). Based on established evaluations we derive further information about decaying properties of the solutions. The obtained results are illustrated by considering concrete classes of equations as, for instance, equations corresponding to Jacobi type operators.

Methods for the solution of boundary-value problems for weakly nonlinear integro-differential equations with parameters and restrictions

We establish conditions for the existence of solutions of boundary-value problems for weakly nonlinear integro-differential equations with parameters and restrictions. We also substantiate the applicability of iterative and projection-iterative methods for the solution of these problems.