Non-iterative Methods Research Articles

Coupled-cluster methods correct through sixth order

Several coupled-cluster approximations to the full CCSDTQ model, correct through sixth order, are presented and illustrated by numerical comparions to full CI. Both iterative and non-iterative methods built upon underlying CCSDT solutions are considered. Unlike CCSDTQ, which is an ∼ n 10 computational procedure, the computational dependence of all approximations is no worse than ∼ n 9. The latter follows use of the 2 n rule of perturbation theory. The average errors compared to full CI for different methods are 0.1 to 0.3 mhartree, compared to 0.05 mhartree for CCSDTQ.

Full data utilization in PVI using the 3D radon transform

An algorithm is described for three-dimensional (3D) image reconstruction in positron volume imaging (PVI) using the inversion of the 3D radon transform (RT) for a truncated cylindrical detector geometry. A discrete version of the 3D RT is formed in a 3D histogram from the event-line coordinates of the detected events. The histogram entries represent the plane integrals of the activity in the field of view. Conventional inversion of the X-ray transform (XT) excludes oblique events in non-iterative reconstruction methods. By employing a spatially varying angular acceptance of events into the histogrammed plane integrals of the 3D RT, it is possible to include most of the detected oblique events in the reconstruction of the image using the standard 3D RT inversion formula. The oblique events are added to the histogram with a partial weight compared to those in complete (XT) projections. This single-pass reconstruction image has better statistical noise properties than images formed by RT inversion from complete XT projections, but only for some detector geometries is it significantly better.

Numerical evaluation of iterative and noniterative methods for the solution of the nonlinear Richards equation

Numerical evaluation of iterative and noniterative methods for the solution of the nonlinear Richards equation

On geostatistical formulations of the groundwater flow inverse problem

The problem of estimating log-transmissivity when it is assumed to be a random field can be faced using indirect iterative methods or using the so-called geostatistical approach. A close examination of the two formulations suggests that basic assumptions are similar. In fact, if the iterative method is stopped after one iteration of a Gauss-Newton method, starting from the mean log- T, we show that both estimates and estimation covariance matrices obtained with the two approaches are identical. We then argue that the main difference between the two formulations arises from the way linearization is performed, which is global in non-iterative methods and only local in iterative methods. As a result, the latter are less constrained by linearity than the former and lead to better estimates and to more consistent estimation covariance matrices. This is illustrated with two synthetic examples for several degrees of spatial variability.

Numerical evaluation of iterative and noniterative methods for the solution of the nonlinear Richards equation

Several noniterative procedures for solving the nonlinear Richards equation are introduced and compared to the conventional Newton and Picard iteration methods. Noniterative strategies for the numerical solution of transient, nonlinear equations arise from explicit or linear time discretizations, or they can be obtained by linearizing an implicit differencing scheme. We present two first order accurate linearization methods, a second order accurate two‐level “implicit factored” scheme, and a second order accurate three‐level “Lees” method. The accuracy and computational efficiency of these four schemes and of the Newton and Picard methods are evaluated for a series of test problems simulating one‐dimensional flow processes in unsaturated porous media. The results indicate that first order accurate schemes are inefficient compared to second order accurate methods; that second order accurate noniterative schemes can be quite competitive with the iterative Newton and Picard methods; and that the Newton scheme is no less efficient than the Picard method, and for strongly nonlinear problems can outperform the Picard scheme. The two second order accurate noniterative schemes appear to be attractive alternatives to the iterative methods, although there are concerns regarding the stability behavior of the three‐level scheme which need to be resolved. We conclude that of the four noniterative strategies presented, the implicit factored scheme is the most promising, and we suggest improved formulations of the method.

Non-iterative 2D rational approximation

The rational approximation of two-dimensional (2D) digital filters in the spatial domain is examined. Non-iterative methods are presented for finding 2D recursive digital filters to approximate, in some sense, a desired 2D pulse response over a finite area. The methods require only linear calculations and they are quite straight forward. Two examples are given to illustrate the utility of the presented techniques.

Locating and identifying compact ferrous objects

A solution to the problem of magnetostatic location and identification of compact ferrous objects of arbitrary shape is presented. It is shown that, in practice, the inversion of the magnetostatic dipole field or field gradient is a necessary first step toward determining object location and identity. Several iterative and noniterative methods of determinating the dipole moment and location from field or gradient measurements are described and compared. It is shown that, given the dipole-moment estimates, it is possible to determine the identity of the dipole source in practical situations by pattern recognition. A unique prototype total field magnetometer which is capable of explicitly and accurately locating and identifying axially symmetric compact ferrous objects is described. It has performed well in preliminary tests using spheres and spheroids.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

因子分析における統計的推測:最近の発展

Several methods of statistical inference in factor analysis are first reviewed under the normality assumption as well as the case in which no assumption about the population ditribution is made. Normal theory inference, whether it is asymptotically efficient, is shown to be robust against a wide class of distributions which may be encountered in many practical fields. For small or medium sample sizes, simple procedures of estimation, such as simple least-squares and noniterative methods, are recommended, whereas the method of maximum likelihood or other asymptotically efficient ways should be utilized for large samples.

Multi-reference coupled-cluster methods for ionization potentials with partial inclusion of triple excitations

An extension of multi-reference coupled-cluster (MRCC) methods to include some effects of triple excitations for the direct calculation of ionization potentials is presented. A series of non-iterative methods are proposed, derived in analogy to the successful non-iterative inclusions of triple excitations for ground states. Results of two of these methods, MRCCSD+T(3) and MRCCSD+T*(3) are presented for N 2, H 2CO, and CH 2( 1A 1). We also consider how ionization potentials vary with alternative ways of determining the correlated ground state.

Coupled-cluster methods that include connected quadruple excitations, T4: CCSDTQ-1 and Q(CCSDT)

Several coupled-cluster methods that include the connected T 4 contribution have been formulated and implemented. All are correct through the fifth-order energy. CCSDTQ-1 reproduces the full CI results for BH, FH and H 2O to a mean absolute error of 0.36 mhartree, compared to 0.93 mhartree for CCSDT. Based on CCSDTQ-1, some noniterative methods for the inclusion of quadruples have also been considered. One of them, denoted as Q(CCSDT), has an average error of 0.23 mhartree and has only an N 7 basis set dependence. This method offers a convenient estimate of the principal effects due to T 4.

Comparison of statistical methods for the analysis of limiting dilution assays.

This study reports the results of a critical comparison of five statistical methods for estimating the density of viable cells in a limiting dilution assay (LDA). Artificial data were generated using Monte Carlo simulation. The performance of each statistical method was examined with respect to the accuracy of its estimator and, most importantly, the accuracy of its associated estimated standard error (SE). The regression method was found to perform at a level that is unacceptable for scientific research, due primarily to gross underestimation of the SE. The maximum likelihood method exhibited the best overall performance. A corrected version of Taswell's weighted-mean method, which provides the best performance among all noniterative methods examined, is also presented.

Rapid approximate vapour—liquid equilibrium calculations

-Methods are presented for effectively noniterative approximate methods for calculating vapour—liquid equilibria in the three main variations: specified vapour fraction, specified enthalpy and specified temperature together in each case with a given system pressure. Accuracy is typically around 1% at a first estimate for near ideal systems. Adjustable parameters in the approximate model used may be refitted to increase accuracy to any degree required.

Accurate autoregressive spectrum estimation at low signal-to-noise ratio using a phase matching technique

This paper describes a new method of accurately estimating the parameters of an autoregressive (AR) process contaminated by high-level white noise. Based on the phase matching technique, it minimizes the difference between the phase of the all-zero model and the phase of the maximum phase signal reconstructed from the power spectrum of the observed signal. The parameters of the AR model are obtained from the finite length sequence of the estimated all-zero model. The proposed method works only when the order of the AR model is known a priori at present. However, since the phase matching technique satisfies the conditions needed to apply the least mean-square method, the AR parameters are estimated accurately even at a low signal-to-noise ratio. With the iterative or noniterative methods as discussed in the recent literature, it is not possible to reconstruct the all-zero model from the power spectrum when there are dips and peaks having no correlation with the poles of original AR signal in the power spectrum. The method proposed in this paper allows one to accurately reconstruct the phase from the power spectrum in such cases. Finally, it is confirmed with computer simulations and experiments that the proposed method is useful for accurate estimation of the AR parameters.

Direct least-squares fitting of algebraic surfaces

In the course of developing a system for fitting smooth curves to camera input we have developed several direct (i.e. noniterative) methods for fitting a shape (line, circle, conic, cubic, plane, sphere, quadric, etc.) to a set of points, namely exact fit, simple fit, spherical fit, and blend fit. These methods are all dimension-independent, being just as suitable for 3D surfaces as for the 2D curves they were originally developed for.Exact fit generalizes to arbitrary shapes (in the sense of the term defined in this paper) the well-known determinant method for planar exact fit. Simple fit is a naive reduction of the general overconstrained case to the exact case. Spherical fit takes advantage of a special property of circles and spheres that permits robust fitting; no prior direct circle fitters have been as robust, and there have been no previous sphere fitters. Blend fit finds the best fit to a set of points of a useful generalization of Middleditch-Sears blending curves and surfaces, via a nonpolynomial generalization of planar fit.These methods all require ( am + bn ) n 2 operations for fitting a surface of order n to m points, with a = 2 and b = 1/3 typically, except for spherical fit where b is larger due to the need to extract eigenvectors. All these methods save simple fit achieve a robustness previously attained by direct algorithms only for fitting planes. All admit incremental batched addition and deletion of points at cost an 2 per point and bn 3 per batch.

Parametric Random Excitation. II: White‐Noise Parameters

The results of a companion paper (1) are extended here for the case of white‐noise parameters. The problem is that of a harmonic oscillator with random force. These random processes are assumed to be stationary and Gaussian. The cases considered are: (1) In the limit as the correlation functions approach a delta function and thus the processes have white spectra; (2) where an iterative method is applied to the white‐noise case; and (3) where a noniterative, direct method is applied to the white‐noise problem. It is shown that white‐noise results can be obtained directly from the previous analysis using a limiting procedure. In addition, an exact solution is obtained using iterative and noniterative methods. This permits a simple error analysis by comparing the results of a truncated series with those of the exact solution. Qualitatively, it is possible to say that the better the white‐noise spectra approximates the physical processes, the more accurate will be the truncated solution of Ref. 1.

Optimality of certain iterative and non-iterative data extrapolation procedures

A fundamental aspect of many modern spectral analysis and image restoration procedures is the extrapolation of the measured data to achieve improved resolution. Iterative extrapolation poses some mathematical questions about convergence and about the optimality of the resulting limit. We consider these questions and discuss briefly the optimality of certain non-iterative extrapolation methods.

Internally iterative improvement of optical tomographic reconstructions

An iterative improvement of minimum-norm least-squares estimates for values in limited-angle reconstruction is presented. Implementation provides considerable improvement over noniterative methods for reconstructions with total angles less than 90 degrees with redundant data and requires considerably less computation time than do globally iterative schemes.

Adaptive noise cancellation for a class of nonlinear, dynamic reference channels

The computational complexity of nonlinear adaptive noise cancellation can be reduced by restricting the nonlinearity expected in the reference path to the noise canceler. The class of zero memory nonlinearities preceded by linear processors in the reference path is considered. Noniterative and iterative methods of system identification are applied to the determination of processor parameters in the noise canceler. The computational requirements of each of the algorithms are compared, and the iterative method is modified for improved convergence. Experimental results are presented for the modified iterative algorithm.

Implicit methods of second-order accuracy for the Euler equations

This paper presents a study of noniterative implicit finite difference methods for hyperbolic systems. New space-centered methods involving two time levels are considered. An analysis is made of various properties such as solvability, stability, dissipation, dispersion, and efficient solution of the algebraic systems. The computation of shock waves with a CFL number larger than the unity is discussed. Unconditional stability results are proved for a case having several space variables. Accurate solutions of the Euler equations are obtained that offer major reductions in computing costs over explicit methods.

Regularized iterative and noniterative procedures for object restoration in the presence of noise: an error analysis

An analysis is carried out, using the prolate spheroidal wave functions, of certain regularized iterative and noniterative methods previously proposed for the achievement of object restoration (or, equivalently, spectral extrapolation) from noisy image data. The ill-posedness inherent in the problem is treated by means of a regularization parameter, and the analysis shows explicitly how the deleterious effects of the noise are then contained. The error in the object estimate is also assessed, and it is shown that the optimal choice for the regularization parameter depends on the signal-to-noise ratio. Numerical examples are used to demonstrate the performance of both unregularized and regularized procedures and also to show how, in the unregularized case, artefacts can be generated from pure noise. Finally, the relative error in the estimate is calculated as a function of the degree of superresolution demanded for reconstruction problems characterized by low space–bandwidth products.