COMPUTING EIGENVECTORS OF NORMAL MATRICES WITH SIMPLE INVERSE ITERATION

It is well-known that if we have an approximate eigenvalue A of a normal matrix A of order n, a good approximation to the corresponding eigenvector u can be computed by one inverse iteration provided the position, say kmax, of the largest component of u is known. In this paper we give a detailed theoretical analysis to show relations between the eigenvector u and vector Xk, k = 1, … ,n, obtained by simple inverse iteration, i.e., the solution to the system (A - I)x = ek with ek the kth column of the identity matrix I. We prove that under some weak conditions, the index kmax is of some optimal properties related to the smallest residual and smallest approximation error to u in spectral norm and Probenius norm. We also prove that the normalized absolute vector v = |u|/||u|| of u can be approximated by the normalized vector of (||x1||2, … ||xn||2)T- We also give some upper bounds of |u(k)| for those 'optimal' indexes such as Fernando's heuristic for kmax without any assumptions. A stable double orthogonal factorization method and a simpler but may less stable approach are proposed for locating the largest component of u.

AN ADAPTIVE INVERSE ITERATION FEM FOR THE INHOMOGENEOUS DIELECTRIC WAVEGUIDES

We introduce an adaptive finite element method for computing electromagnetic guided waves in a closed, inhomogeneous, pillared three-dimensional waveguide at a given frequency based on the inverse iteration method. The problem is formulated as a generalized eigenvalue problems. By modifying the exact inverse iteration algorithm for the eigenvalue problem, we design a new adaptive inverse iteration finite element algorithm. Adaptive finite element methods based on a posteriori error estimate are known to be successful in resolving singularities of eigenfunctions which deteriorate the finite element convergence. We construct a posteriori error estimator for the electromagnetic guided waves problem. Numerical results are reported to illustrate the quasi-optimal performance of our adaptive inverse iteration finite element method.

On Computer Implementation for Comparison of Inverse Numerical Schemes for Non-Linear Equations

In this research article,we interrogate two new modifications in inverse Weierstrass iterative method for estimating all roots of non-linear equation simultaneously.These modifications enables us to accelerate the convergence order of inverse Weierstrass method from 2 to 3.Convergence analysis proves that the orders of convergence of the two newly constructed inverse methods are 3.Using computer algebra system Mathematica,we find the lower bound of the convergence order and verify it theoretically.Dynamical planes of the inverse simultaneous methods and classical iterative methods are generated using MATLAB(R2011b),to present the global convergence properties of inverse simultaneous iterative methods as compared to classical methods.Some non-linear models are taken from Physics,Chemistry and engineering to demonstrate the performance and efficiency of the newly constructed methods.Computational CPU time,and residual graphs of the methods are provided to present the dominance behavior of our newly constructed methods as compared to existing inverse and classical simultaneous iterative methods in the literature.

Inverse kinematics of a heavy duty manipulator with 6-DOF based on dual quaternion

An iterative method is introduced successfully to solve the inverse kinematics of a 6-DOF manipulator of a tunnel drilling rig based on dual quaternion, which is difficult to get the solution by Denavit-Hartenberg(D-H) based methods. By the intuitive expression of dual quaternion to the orientation of rigid body, the coordinate frames assigned to each joint are established all in the same orientation, which does not need to use the D-H procedure. The compact and simple form of kinematic equations, consisting of position equations and orientation equations, is also the consequence of dual quaternion calculations. The iterative process is basically of two steps which are related to solving the position equations and orientation equations correspondingly. First, assume an initial value of the iterative variable; then, the position equations can be solved because of the reduced number of unknown variables in the position equations and the orientation equations can be solved by applying the solution from the position equations, which obtains an updated value for the iterative variable; finally, repeat the procedure by using the updated iterative variable to the position equations till the prescribed accuracy is obtained. The method proposed has a clear geometric meaning, and the algorithm is simple and direct. Simulation for 100 poses of the end frame shows that the average running time of inverse kinematics calculation for each demanded pose of end-effector is 7.2 ms on an ordinary laptop, which is good enough for practical use. The iteration counts 2-4 cycles generally, which is a quick convergence. The method proposed here has been successfully used in the project of automating a hydraulic rig.

Sensitivity and inverse analysis methods for parameter intervals

This paper proposes a sensitivity analysis method for engineering parameters using interval analyses.This method substantially extends the application of interval analysis method.In this scheme,parameter intervals and decision-making target intervals are determined using the interval analysis method.As an example,an inverse analysis method for uncertainty is presented.The intervals of unknown parameters can be obtained by sampling measured data.Even for limited measured data,robust results can also be obtained with the inverse analysis method,which can be intuitively evaluated by the uncertainty expressed in terms of an interval.For complex nonlinear problems,an iteratively optimized inverse analysis model is proposed.In a given set of loose parameter intervals,all the unknown parameter intervals that satisfy the measured information can be obtained by an iteratively optimized inverse analysis model.The influences of measured precisions and the number of parameters on the results of the inverse analysis are evaluated.Finally,the uniqueness of the interval inverse analysis method is discussed.

Homotopy Method for Inverse Design of the Bulbous Bow of A Container Ship

The homotopy method is utilized in the present inverse hull design problem to minimize the wave-making coefficient of a 1300 TEU container ship with a bulbous bow. Moreover, in order to improve the computational efficiency of the algorithm, a properly smooth function is employed to update the homotopy parameter during iteration. Numerical results show that the homotopy method has been successfully applied in the inverse design of the ship hull. This method has an advantage of high performance on convergence and it is credible and valuable for engineering practice.

Acoustic impedance inversion of zero-offset VSP data

Highly precise acoustic impedance inversion is a key technology for pre-drilling prediction by VSP data. In this paper, based on the facts that VSP data has high resolution, high signal to noise ratio, and the downgoing and upgoing waves can be accurately separated, we propose a method of predicting the impedance below the borehole in front of the bit using VSP data. First, the method of nonlinear iterative inversion is adopted to invert for impedance using the VSP corridor stack. Then, by modifying the damping factor in the iteration and using the preconditioned conjugate gradient method to solve the equations, the stability and convergence of the inversion results can be enhanced. The results of theoretical models and actual data demonstrate that the method is effective for pre-drilling prediction using VSP data.

Development and application of iterative facies-constrained seismic inversion

To improve the accuracy of inversion results,geological facies distributions are considered as additional constraints in the inversion process.However,the geological facies itself also has its own uncertainty.In this paper,the initial sedimentary facies maps are obtained by integrated geological analysis from well data,seismic attributes,and deterministic inversion results.Then the fi rst iteration of facies-constrained seismic inversion is performed.According to that result and other data such as geological information,the facies distribution can be updated using cluster analysis.The next round of facies-constrained inversion can then be performed.This process will be repeated until the facies inconsistency or error before and after the inversion is minimized.It forms a new iterative facies-constrained seismic inversion technique.Compared with conventional facies-constrained seismic inversion,the proposed method not only can reduces the non-uniqueness of seismic inversion results but also can improves its resolution.As a consequence,the sedimentary facies will be more consistent with the geology.A practical application demonstrated that the superposition relationship of sand bodies could be better delineated based on this new seismic inversion technique.The result highly increases the understanding of reservoir connectivity and its accuracy,which can be used to guide further development.

Application of optical diffraction method in designing phase plates

Continuous phase plate（CPP）,which has a function of beam shaping in laser systems,is one kind of important diffractive optics.Based on the Fourier transform of the Gerchberg-Saxton（G-S） algorithm for designing CPP,we proposed an optical diffraction method according to the real system conditions.A thin lens can complete the Fourier transform of the input signal and the inverse propagation of light can be implemented in a program.Using both of the two functions can realize the iteration process to calculate the near-field distribution of light and the far-field repeatedly,which is similar to the G-S algorithm.The results show that using the optical diffraction method can design a CPP for a complicated laser system,and make the CPP have abilities of beam shaping and phase compensation for the phase aberration of the system.The method can improve the adaptation of the phase plate in systems with phase aberrations.

Magnetotelluric data inversion with seismic data constraint

In the paper we present a new method to invert the interior structure in the basement or ancient hidden hill by using magnetotelluric (MT) data with seismic data constraint. We first obtain the thickness and resistivity of each layer above the basement or buried hill by the inversion of seismic and log data and create a geoelectrical model for the layers above the basement or hidden hill. Then with the reference to the inversion of 1D MT data, a geoelectrical model for the layers below the basement or hidden hill is created. On the basis of the above initial model, we present an effective and practical forward method, i.e., a model-matched approach to conduct forward inversion arithmetic. Finally, by the method of conjugate gradient iteration, a forward and backward iterative calculation is made. Taking No. 618 profile of Shengli Oil Field as an example, we have found out that the tectonic information that is unreflective in the seismic data below the basement is better reflected in the inversion result.

A Method of Disc Inclination Correction Based on the Inversion Model of Rotation Law

Under the traditional dynamic model,the conventional method for solving the rotation angle of a rigid body is to use the fixed-axis rotation law of the rigid body,but the known rotation shaft position must be used as a prerequisite.In practical work,for the rotation of a rigid body under multiple forces,solving the shaft is often a difficult problem.In this paper,we consider the rigid body of the disc is subjected to the force of uneven magnitude from multiple angles,the position of the rotating shaft is obtained by iterative inversion through the rigid body rotation law and the dichotomy method.After the position of the shaft is determined,we establish a differential equation model based on the law of rigid body rotation,the rotation angle of the rigid body thus being solved based on this model.Furthermore,an optimization algorithm such as genetic algorithm is used to search for a correction scheme to return the rigid body to equilibrium at any given deflection angle.The model and method are based on computer to explore the law of rotation,the practical application of them play an important role in studying the concentric drum movement and the balance of handling furniture.

Iterative inverse beamforming algorithm and its application in multiple targets detection of passive sonar

With the increasing of detection ability of passive sonar,the weak signal detection problem in multiple interferences becomes more and more important.In the time/bearing record（TBR） display of sonar detection,when there exist traces of multiple interferences,the identification of weak signal is difficult or impossible.The adaptive noise cancellation technique provides the theoretical basis for suppressing strong interferences.But the solution for finding the steady-state optimum filter matrix is quite difficult due to the real time calculation of inverse matrix of input data correlation matrix.The iterative inverse beamforming（IBF） algorithm for solving the optimum filter vector,which is expressed by inverse matrix of the ocean environment data,is derived in this paper,by which,the optimum filter can be eventually expressed as a sum of series simple matrices of constructed from sensor data.Based on the algorithm proposed in this paper,some examples of at sea experiment are provided.The strong interferences are cancelled and the weak signal is emerged,even it didn＇t appear in the conventional beamforming（CBF） processing.

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 Cramér-Rao lower bound accuracy. Simulation results show the validity of the theoretical derivation and superiority of the new algorithm.

Global algorithms for maximal eigenpair

This paper is a continuation of our previous work [Front. Math. China, 2016, 11（6）： 1379-1418] where an efficient algorithm for computing the maximal eigenpalr was introduced first for tridiagonal matrices and then extended to the irreducible matrices with nonnegative off-diagonal elements. This paper introduces mainly two global algorithms for computing the maximal eigenpair in a rather general setup, including even a class of real （with some negative off-diagonal elements） or complex matrices.

Improved global algorithms for maximal eigenpair

This paper is a continuation of our previous paper[Front.Math.China,2017,12(5):10231043]where global algorithms for computing the maximal cigcnpair were introduced in a rather general setup.The efficiency of the global algorithms is improved in this paper in terms of a good use of power iteration and two quasi-symmetric techniques.Finally,the new algorithms are applied to Hua’s economic optimization model.

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.

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.

Active Source Tomography in Northwestern Xinjiang,China:Implication for Mineral Distribution

The main aim of this work is to understand the distribution of minerals by obtaining a shallow velocity structure around the Karatungk（喀拉通克） region.Data were acquired in 2009 by a denser array in deploying a transportable seismometer with 4.5 Hz vertical geophone.All the P-wave arrival times are picked automatically with Akaike information criterion,and then checked man-machine interactively by short-receiver geometry.The database for local active-source tomographic in-version involves 4 241 P-wave arrival time readings from 96 shots and three quarry blasts.Checker-board tests aimed at checking the reliability of the obtained velocity models are presented.The result-ing Vp distribution slices show a complicated 3-D structure beneath this area and offer a better under-standing of three well-defined mineral deposits.Near the surface we observe a series of zones with slightly high-velocity which probably reflect potential deposits.Based on features of metallic ores we attempt to delimit their distributions and stretched directions.

Bi-iterative least squares algorithms for blind channel identification and equalization with second-order statistics

We present an adaptive algorithm for blind identification and equalization of single-input multiple-output （SIMO） FIR channels with second-order statistics. We first reformulate the blind channel identification problem into a low-rank matrix approximation solution based on the QR decomposition of the received data matrix. Then, a fast recursive algorithm is developed based on the bi-iterative least squares （Bi-LS） subspace tracking method. The new algorithm requires only a computational complexity of O（md2） at each iteration, or even as low as O（md） if only equalization is necessary, where m is the dimension of the received data vector （or the row rank of channel matrix） and d is the dimension of the signal subspace （or the column rank of channel matrix）. To overcome the shortcoming of the back substitution, an inverse QR iteration algorithm for subspace tracking and channel equalization is also developed. The inverse QR iteration algorithm is well suited for the parallel implementation in the systolic array. Simulation results are presented to illustrate the effectiveness of the proposed algorithms for the channel identification and equalization.