Power Iteration Method Research Articles

Krylov Subspace Iterations for Deterministic k-Eigenvalue Calculations

The Implicitly Restarted Arnoldi Method (IRAM), a Krylov subspace iterative method, applied to k-eigenvalue calculations for criticality problems in deterministic transport codes is discussed. A computationally efficient alternative to the power iteration method that is typically used for such problems, the IRAM not only finds the largest eigenvalue but also several additional higher order eigenvectors with little extra computational cost. Implementation requires only modest changes to existing power iteration coding present in an SN transport program. Numerical results are presented for three-dimensional SN transport on unstructured tetrahedral meshes to compare the IRAM results with those computed using the traditional, unaccelerated power iteration method. The results indicate that the IRAM can be an efficient and powerful technique, especially for problems with dominance ratios approaching unity.

一回の実験データに基づく制御器パラメータチューニングの新しいアプローチ―Ｆｉｃｔｉｔｉｏｕｓ Ｒｅｆｅｒｅｎｃｅ Ｉｔｅｒａｔｉｖｅ Ｔｕｎｉｎｇの提案

In this paper, we propose one of the powerful iterative tuning methods for the parameters of the controller of a closed loop system. The method we provide here requires only one-shot experiment and then an iterative non-linear optimization for obtaining the optimum parameters of the controller can be performed off-line by using the fictitious reference signal computed by the first experimental data (from this reason, we refer to our method as “fictitious reference iterative tuning”, which is abbreviated to “FRIT”). The main concept and the algorithm of FRIT are provided as our main results of this paper. Moreover, we give a control experiment of the cart-system in order to show the validity of FRIT.

Computing the Higher k-Eigenfunctions by Monte Carlo Power Iteration: A Conjecture

Most Monte Carlo transport codes estimate the fundamental k-eigenfunction by means of a power iteration method. A modified power iteration method appears to generate the higher eigenfunctions for some Monte Carlo transport problems. This technical note describes the method as well as some plausibility arguments about why the method works. At this time, no formal proof exists to show that the method converges to the desired eigenfunction.

On Two Variants of an Algebraic Wavelet Preconditioner

A recursive method of constructing preconditioning matrices for the nonsymmetric stiffness matrix in a wavelet basis is proposed for solving a class of integral and differential equations. It is based on a level-by-level application of the wavelet scales decoupling the different wavelet levels in a matrix form just as in the well-known nonstandard form. The result is a powerful iterative method with built-in preconditioning leading to two specific algebraic multilevel iteration algorithms: one with an exact Schur preconditioning and the other with an approximate Schur preconditioning. Numerical examples are presented to illustrate the efficiency of the new algorithms.

Structural/functional assignment of unknown bacteriophage T4 proteins by iterative database searches

Among the total of 274 orfs within bacteriophage T4, only half have been reasonably well characterized, and the functions of the rest have remained obscure. In order to predict the molecular functions of the orfs, a position-specific iterated (PSI)-BLAST search of bacteriophage T4 against the sequence database of known 3D structures was carried out. PSI-BLAST is one of the most powerful iterative sequence search methods using multiple sequence alignment, with the ability to detect many more proteins with distant homology than standard pairwise methods. The 3D structures of proteins are considered to be better preserved than the sequences, and the detected distantly homologous proteins are likely to possess highly similar 3D structures. Thirteen orfs of phage T4, whose homologues were not detected by standard pairwise methods, were found to have significantly homologous counterparts by this method. The plausibility of the results was confirmed by checking whether important residues at substrate/ligand-binding sites were conserved. Among them, two orfs, vs.1 and e.1, which are similar to Escherichia coli lytic enzyme and MutT protein, respectively, had not been studied previously. Also, gp rIIA, a rapid lysis protein, whose gene structure had been intensively studied during the development of molecular biology in the 1950s and yet whose molecular function remains unknown, has an N-terminal domain that is significantly similar to the N-terminal region of the heat shock protein Hsp90.

Efficient Computational Model for Inductive Plasma Flows

A new approach to the numerical modeling of inductive plasmas is presented. The governing magnetohydrodynamic equations are discretized in a second-order accurate finite volume manner. A straightforward pressure-stabilized flow field solver is introduced as an alternative to the staggered-mesh solvers used in traditional algorithms. It is argued that the widely used integral boundary formulation for the electric field is computationally expensive and cannot be incorporated into an efficient iterative solution procedure. This approach is abandoned in favor of a 'far field' formulation of the electric field, such that powerful iterative methods can be applied to speed up the calculation. The discretized equations are solved through a damped Picard method and an approximate and full Newton method. Efficient linear algebra methods are used to solve the linear systems arising from the iterative methods. An appropriate linearization of the strongly positive Joule heating source term is important for the convergence at the linear level. The new model is tested on an LTE argon inductive plasma computation. The proposed approximate Newton method is found to converge substantially better than the Picard method. The full Newton method appears promising, although its successful use on fine meshes still requires improvements to the linear solver used.

The implicit restarted Arnoldi method, an efficient alternative to solve the neutron diffusion equation

To calculate the neutronic steady state of a nuclear power reactor core and its subcritical modes, it is necessary to solve a partial eigenvalue problem. In this paper, an implicit restarted Arnoldi method is presented as an advantageous alternative to classical methods as the Power Iteration method and the Subspace Iteration method. The efficiency of these methods, has been compared calculating the dominant Lambda modes of several configurations of the Three Mile Island reactor core.

Preconditioning of Indefinite and Almost Singular Finite Element Elliptic Equations

This paper deals with two ways of solving discretized finite element elliptic equations with indefinite and almost singular matrices. Such problems typically arise when applying the shifted inverse power iteration (SIPI) method to the generalized eigenvalue problem $A{\bf u}=\lambda\,B{\bf u}$ where A is defined from some discretized equation by a finite element self-adjoint coercive second-order elliptic operator and B is a mass-matrix operator. Both methods explore two-by-two block partitioning of the given matrices. One of the main matrix blocks corresponds to a coarse space (or, equivalently, to coarse-grid degrees of freedom) and contains the main singularity of the original problem. The other major block, possibly indefinite, is not as close to singular as the original matrix and is inverted by a preconditioned minimal residual (MINRES) method in the first method. The second method, which we use for comparison, exploits preconditioned MINRES iterations for the reduced problem obtained by eliminating the block unknowns that correspond to the coarse discretization space. Here each iteration involves solving a coarse-grid problem. Numerical examples are presented that demonstrate the various aspects of both approaches for solving generalized eigenvalue problems applied to second-order finite element elliptic equations.

Numerical Analysis of Asymmetrical-Core Two-Mode Fiber Interferometers Subjected to Bending

A finite element formulation for solving the scalar wave equation is described. The proposed formulation leads to an eigenvalue problem defined in terms of positive definite matrices. Therefore, convergence of the two first eigenvectors/eigenvalues to modes LP01 and LP1, and their correspondent propagation constants is guaranteed. A computer code was written to implement the finite element formulation. The software assembles the matrix eigenvalue problem and solves it either with the sub-space iteration method or the power iteration method. The software also computes the coordinates of the centroid of the modal field distribution of each mode. The finite element model can be used to assess the effects of bending the optical fiber about an arbitrary axis on the guided light. The numerical analysis accounts for the mechanical deformation of the fiber cross section and the variation of the refractive index profile due to all strain components present in pure bending. A comparison of the analyses of light propagation in the fiber under unstrained and strained conditions enables one to estimate the bend sensitivity of a two-mode fiber interferometer. Numerical results confirm experimental observations that a small asymmetry of the core index profile may lead to a significant bend sensitivity of a two-mode fiber interferometer. The present model was used to assess the influence of several parameters on the bend sensitivity of an asymmetrical core two-mode fiber interferometer. The excellent agreement between the numerical and experimental results indicates that this numerical model is a reliable tool to analyze the effect of bending strains in twomode fiber interferometers.

The representation of numbers as sums of unlike powers. II

n= Zx,+ i=1 The principal tool in the proof is the Hardy-Littlewood circle method, incorporating results and techniques of a powerful new iterative method developed by Vaughan and Wooley ([Va3], [Va4], [VW1], [Wol], [VW2]). In section 2, an algorithm developed in [Fo] for optimizing the parameters in mixed power mean value theorems is generalized and analyzed. Section 3 details a more sophisticated method of generating mixed power mean value theorems, by a limited adaptation of the iterative method itself. The form of these estimates offers many advantages over those of section 2, and provides the key to the elimination of the 16th power from (1.1). These mean value theorems are then applied to the proof of Theorem 1 in section 4. The tools developed here are applicable to a wide range of mixed power representation problems, and we briefly illustrate in section 5 the application to the problem of determining the number of terms required to represent all large n, when the lowest power used is a kth power instead of a square. Throughout, n is a large natural number whose representation as a sum of mixed powers is at issue, and E is an arbitrarily small positive real number. Constants implied by the Landau Oand Vinogradov <<-symbols may depend on E or k. For a real number x, write e(x) for e27iX, [x] for the greatest integer not exceeding x and llxll for the distance from x to the nearest integer. Unless otherwise specified, lowercase Latin letters denote natural numbers and Greek letters denote real numbers. Let A(P, R) denote the set of natural numbers not exceeding P with no

Computer Mechanics. Design Study of Flying Heads Having Unsymmetric Lubricating Surface for Minimum Spacing Variation over Disk Surface. Derivation of Iterative Solution Method for Inverse Problem Using Unbalanced Force and Moments.

Flying head sliders with constant spacing during head motion over a disk surface are designed. For this purpose, a powerful iterative method to solve the inverse problem is derived, wherein the slider operating attitude is obtained from a given load and a pivoting point. At first, static and dynamic Reynolds equations are solved to give not only the load and the pressure center, but also the stiffnesses of the lubricating film. Then unbalanced force and moments are divided by stiffnesses to provide new attitude parameters to be used in the next iteration. A newly designed slider having inclined grooved rails (IGR slider) is found to feature a minimum spacing variation of 3.9%, which is comparable to 3.3% obtained for the currently designed transverse pressure contour slider (TPC slider). The IGR slider has excellent dynamic characteristics of large damping even for a high-frequency region, while damping of the TPC slider decreases considerably to a negative value with increasing frequency.

Semivectorial polarised finite difference method for optical waveguides with arbitrary index profiles

A simple accurate method, which automatically takes full account of the discontinuities in the normal electric field components across any arbitrary distribution of internal dielectric interfaces, is presented for the determination of polarised solutions of the Helmholtz wave equation. The application of the shifted inverse power iteration method to the resulting matrices, enables any required propagation eigenvalue (not necessarily the fundamanetal mode) to be determined, together with its corresponding electric field profile. It is found that the results, which are presented for various semiconductor rib waveguide structures, compare favourably with published vector finite element and scalar results.

An Efficient Fully Implicit Simulator

Abstract This paper describes Program for Oil Reservoir Simulation (pores), an efficient general-purpose black-oil simulator now in production use for modeling North Sea fields. The fully implicit finite-difference equations are solved for each timestep with a Newton-Raphson procedure. The resulting large sets of linear equations are usually solved simultaneously by a new and powerful iterative method that uses a preconditioned conjugate gradient algorithm with an enforced column-sum condition to accelerate convergence. A sequential solution option is available, and direct matrix inversion methods also are provided. Gas condensate problems are handled by a variable switching technique. Four examples are presented to illustrate the power and efficiency of the program.

Electromagnetic induction in the Earth in electrical contact with the oceans

Summary. Integral equations are formulated to solve the problem of electromagnetic induction in the Earth in the presence of thin finitely conducting oceans. Unlike earlier models, the oceans are assumed to be in electrical contact with the mantle and leakage of oceanic electric current is therefore possible. A powerful iterative method is developed to solve the very large system of equations and is applied successfully to a model problem. A feature of the integral equations is that they lead to a good first approximation. When fed into the iterative scheme this enhances the rate of convergence. Variable mesh size also appears possible.

A highly stable current or voltage source

A powerful iterative method of current stabilization is described which uses FETs in a chain. Zero temperature coefficient for small temperature excursions may be obtained at a certain current value characteristic of the individual transistor. By means of one or two fixed resistors very stable voltages of almost any value may be produced. Measurements on a three FET device show it to be a promising replacement for a standard cell.