Iterative Least-squares Method Research Articles

Iterative total least-squares image reconstruction algorithm for optical tomography by the conjugate gradient method

We present an iterative total least-squares algorithm for computing images of the interior structure of highly scattering media by using the conjugate gradient method. For imaging the dense scattering media in optical tomography, a perturbation approach has been described previously [Y. Wang et al., Proc. SPIE 1641, 58 (1992); R. L. Barbour et al., in Medical Optical Tomography: Functional Imaging and Monitoring (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 87-120], which solves a perturbation equation of the form W delta x = delta I. In order to solve this equation, least-squares or regularized least-squares solvers have been used in the past to determine best fits to the measurement data delta I while assuming that the operator matrix W is accurate. In practice, errors also occur in the operator matrix. Here we propose an iterative total least-squares (ITLS) method that minimizes the errors in both weights and detector readings. Theoretically, the total least-squares (TLS) solution is given by the singular vector of the matrix [W/ delta I] associated with the smallest singular value. The proposed ITLS method obtains this solution by using a conjugate gradient method that is particularly suitable for very large matrices. Simulation results have shown that the TLS method can yield a significantly more accurate result than the least-squares method.

A study on the adaptive method of control model for tandem cold rolling mill

The control model in the tandem cold rolling mill consists of many mathematical theories and is used to calculate the reference values such as the roll gap and the rolling speed for good operation of rolling mill. But, the control model used presently has a problem causing inaccurate prediction of the rolling force. By the parameter identification, it was found that the main factor causing inaccurate prediction of the rolling force was incorrect modeling of the friction coefficient and the flow stress. To get rid of the erroneous factor new adaptive schemes are suggested in this work. Those are a long-time adaptation by the iterative least-square method and a short-time adaptation by the recursive weighted least-square method respectively. The new equations for the friction coefficient and the flow stress are derived by applying the suggested adaptive algorithms. Through the on-line test in an actual mill, it is proved that the rolling force predicted by the new equations is more accurate than the one by the existing equations ever used.

Error Analysis of Total Field Models Derived from POGS Data

Scalar magnetic data from the POGS satellite are analysed for the 1991-1993 period. Models of the main geomagnetic field are computed using an iterative least-squares method. Vector components of the field derived from these models display large errors known as the Backus effect. A method for estimating the noise level in global data in the form of an equivalent gaussian noise is introduced and POGS data are shown to have a standard deviation of about σ = 45 nT. Despite this high noise level, it is shown that POGS data can be used to improve models derived from observatory vector data. It is also shown that the Backus effect can be significantly reduced by using some accurate vector ground-based measurements of the main field at observatories located near the equator. Monthly models are computed using both POGS and observatory data for which monthly mean values are available at POGS period. Local corrections applied to observatory measurements are computed by comparing observatory monthly mean values with Magsat models. The results are shown to be improved when these corrections are applied to the observatory data, confirming that they correspond essentially to stationary short-wavelength crustal anomalies.

Damping–undamping strategies for the Levenberg–Marquardt nonlinear least-squares method

The speed of the Levenberg–Marquardt (LM) nonlinear iterative least-squares method depends upon the choice of damping strategy when the fitted parameters are highly correlated. Additive damping with small damping increments and large damping decrements permits LM to efficiently solve difficult problems, including those that otherwise cause stagnation. © 1997 American Institute of Physics.

Evaluation of spherical form errors—Computation of sphericity by means of minimum zone method and some examinations with using simulated data

Because of the practical difficulties of measuring whole spherical surface form errors, no concrete three-dimensional (3-D) verification has yet been developed. This article deals with the calculation of the value of spherical form errors; that is, sphericity. The iterative least-squares method in which the problem is linearized and the minimum zone method in which the downhill simplex method, one of the nonlinear optimization techniques, is applied are considered. The data to be analyzed are not obtained by actual measurement of a spherical surface, because there is no such a measuring system in my laboratory, but simulated by applying surface harmonics (Laplace's spherical function) with a computer. Then, the application conditions for downhill simplex method are investigated. Furthermore, the roundness values of the spherical surface are compared with the sphericity by means of the minimum-zone method.

Determining reliable edge isopachic data from interior thermoelastic measurements

The maximum stresses in a plane-stressed component typically occur on the boundary. However, it is generally difficult to obtain reliable experimental data at an edge and thermoelastic stress analysis is no exception. The inability to measure reliable edge isopachic stresses has caused many previous thermoelastic stress analyses to be more qualitative than quantitative. This paper develops and implements an effective iterative least-squares method for calculating reliable edge isopachic stresses from measured interior values. The method is based upon the plane-stress isotropic compatibility equation. A regularization scheme is employed to minimize the sensitivity to measurement error and to improve the stability of the algorithm by controlling the rate of convergence. An illustrative example with actual measured thermoelastic data is included. The processes thermoelastically determined results compare well with those obtained using strain gages.

A new linear method to calculate monomer reactivity ratios by using high-conversion copolymerization data: penultimate model with r2=0

The new iterative linear least-squares method for calculation of monomer reactivity ratios for the terminal model, proposed by Mao and Huglin, is applied to the penultimate model system with r 2=0. The principle and the calculation procedure are similar for the two different models. All advantages of the new method when applied to the terminal model, i.e. fast convergence, exact calculation and the fact that any initial estimation of reactivity ratios always leads to the same correct results, remain valid when applied to the penultimate model. The superiority of the new method over the Kelen-Tüdős and extended Kelen-Tüdős procedures is even more evident for the penultimate model than for the terminal model.

Cottonwood Site-Index Growth Curves

Cottonwood Site-Index Growth Curves

Substituent Effect Studies of Aryl-Assisted Solvolyses. II. The Acetolysis of 2-Phenyl-2-(substituted phenyl)ethyl p-Toluenesulfonates

Abstract The substituent effect on the acetolysis of 2-phenyl-2-(substituted phenyl)ethyl p-toluenesulfonates had a nonlinear LArSR correlation and was explicable in terms of a competitive aryl-assisted mechanism involving the X-substituted phenyl-assisted () pathway and the unsubstituted phenyl-assisted () pathway. By the application of the iterative nonlinear least-squares method based on the LArSR Eq., the substituent effect on the overall kt was dissected into the best-fit kM correlation of ρM = −3.53 with rM = 0.60, and the kN correlation of ρN = −0.88 with σ0. The ρM and rM values for the effects of assisting aryl substituents are quite close to those of the 2-methyl-2-phenylpropyl system and the small ρN value with unexalted σ0 constant for the unassisting aryls is compatible to the remote β-aryl effect. The relative rates of competing pathways dissected based on the substituent effect analysis agreed completely with the ratio of respective aryl migration products determined by the 13C-tracer method. Exact rate-product correlation demonstrates that this system involves two discrete aryl-assisted pathways, and , which do not cross over.

Conductance-Depth Imaging of Airborne TEM Data

Recently a number of conductivity-depth imaging techniques have been developed for processing transient EM data. These techniques require the magnetic field data due to a step current source. However, current commercial airborne TEM systems employ a half-sine pulse source. To apply the conductivity-depth imaging techniques to airborne data, deconvolution is needed to transform the data into the required step function response. This deconvolution may be unstable with noise in the data. Furthermore, in areas of conductive overburden, the system response is mainly a function of the conductance of the overburden. The conductivity and thickness of the overburden can not be resolved uniquely. The system response may also be so high that “halfspace apparent conductivity” is undefined or the conductivity image has to be unphysically extended above the earth surface to fit the data. To overcome the above problems, a conductance-depth imaging method is introduced. The method is based on the horizontal thin-sheet model which has a simple analytical solution. By fitting the data measured at two adjacent time channels, the conductance and the depth of a thin sheet are found using the iterative least-squares method. Transient electromagnetic profile data can then be presented as a conductance-depth section for preliminary interpretation. The technique has been tested and found to be robust on model and field data. An example of processing field data is used to demonstrate the capabilities of the conductance-depth imaging technique in minerals exploration.

Least-squares palinspastic restoration of regions of normal faulting—application to the Campos basin (Brazil)

We have developed an iterative least-squares method for palinspastic restoration of regions of normal faulting. The method closes the gaps (cut-off lenses) across normal faults on a structure-contour map of a given stratigraphie horizon. Such data are often available for sedimentary basins covered by seismic surveys. Before restoration, we first produce a fault-block map, where each block is completely surrounded by faults, whether real or artificial. We then apply an algorithm to minimize D, the sum of the squares of the distances across cut-off lenses, with respect to unknown rigid block translations and rotations. Minimization results in non-linear equations, which we solve by an iterative method. We deem the solution to have converged when D falls below a given critical level. In general, restoration not only closes the original gaps, but also results in strike-slip displacements between blocks. Thus the deformation need not be a plane strain and displacement vectors need not be parallel. We have applied the method to an area of the Campos basin, on the Atlantic margin of Brazil, where the stratigraphy and structure are well-known from seismic surveys and from wells. Cretaceous and Tertiary sediments have glided downslope on a lubricating layer of Aptian evaporites, producing an uppermost domain of normal faulting. The normal faults occupy a strip, lying between undisturbed continental crust to the west and relatively unstructured sediments to the east. Assuming the western boundary to be stationary, we have restored an Upper Oligocene stratigraphie level, the Marco Azul. Convergence was achieved rapidly and the degree of fit is good. According to our restoration, the eastern boundary of the area has bowed out seawards since the Oligocene. Block displacement vectors form a fan-like pattern, with an element of mirror symmetry about a line trending NW-SE. Maximum displacements occur in an apical domain, bounded by a horseshoe pattern of faults. There is a general pattern of block rotations, counterclockwise in the northeast, clockwise in the southwest. Some notable exceptions to this general pattern are due to local domino motions. We attribute the overall displacement field to divergent gravitational gliding off a local salient in the sea-bottom topography. Because of the three-dimensional nature of the displacement field, restoration of vertical cross-sections alone would yield gross errors. This demonstrates the value of palinspastic methods such as ours.

Substituent Effects. XVIII. The Resonance Demand in the Acetolysis of Neophyl Brosylates

Abstract The substituent effect in the acetolysis of neophyl brosylates was analyzed statistically based on the Yukawa–Tsuno LArSR equation. Though the Brown ρ+σ+ equation did not provide a linear correlation, the LArSR equation gave an excellent linear correlation with an r value of 0.57, substantiating our basic LArSR concept of varying resonance demands with different systems. For a statistical examination of the reliability of this nonunity r value, the change in the standard deviation of the LArSR correlation was followed as a function of varying r by an iterative least-squares method. The r value of the neophyl solvolysis must have sufficient statistical significance to distinguish between the LArSR and simple Brown ρ+σ+ correlations of this system. The significance of the best-fit correlation was found to strongly depend upon the number and combination as well as the range of the effective substituents. The study was extended to various aryl-assisted processes of complex mechanisms, solvolyses of 2-arylethyl, 2-aryl-1-methylethyl, 2-aryl-1-methylpropyl, and (1-arylcyclobutyl)methyl arenesulfonates. A precise dissection analysis of non-crossover kΔ and ks processes has been realized by the Gauss–Newton type nonlinear least-squares method based on the assumption of LArSR correlations for both processes. The generality of the unique rΔ value of neophyl solvolysis was verified regarding these β-aryl-assisted processes. The substituent effect on the kΔ process can be characterized far more appropriately in terms of the \barσΔ of r=0.6, rather than σ+ of r=1.00.

Origin of soluble chemical species in bulk precipitation collected in Tokyo, Japan: Statistical evaluation of source materials

An iterative least-squares method with a receptor model was applied to the analytical data of the precipitation samples collected at 23 points in the suburban area of Tokyo, and the number and composition of the source materials were determined. Thirty-nine monthly bulk precipitation samples were collected in the spring and summer of 1987 from the hilly and mountainous area of Tokyo and analyzed for Na +, K +, NH 4 +, Mg 2+, Ca 2+, F −, Cl −, Br −, NO 3 − and SO 4 2− by atomic absorption spectrometry and ion chromatography. The pH of the samples was also measured. A multivariate ion balance approach (Tsurumi, 1982, Anal. Chim. Acta 138, 177–182) showed that the solutes in the precipitation were derived from just three major sources; sea salt, acid substance (a mixture of 53% HNO 3, 39% H 2SO 4 and 8% HCl in equivalent) and CaSO 4. The contributions of each source to the precipitation were calculated for every sampling site. Variations of the contributions with the distance from the coast were also discussed.

Iterative network magnitude estimation and uncertainty assessment with noisy and clipped data

We briefly discuss the similarities and differences between two iterative estimators that are suitable for the network m b estimation problem, namely a modification of the Iterative Least-Squares method (ILS) developed by Schmee and Hahn (1979) and the Maximum-Likelihood Estimator (MLE). Both methods reduce to the usual Least-Squares Multiple Factors (LSMF) method when the censored data are deleted from the network observational data. For censored case, the standard deviation ( σ ) of the obscuring noise has to be solved through iteration along with the event magnitudes and the station corrections. An extra constraint on σ is necessary to determine which optimal estimation scheme is of interest. The final value of σ for each iterative scheme can be used as a good approximation to the unbiased estimate of the standard deviation of the perturbing noise. By scaling this σ value by the square root of the number of observations associated with each unknown parameter, the uncertainty in each estimated parameter can be approximated efficiently. These error estimates seem to differ from the unbiased standard errors only by a common multiplying constant across all stations and all event m b s. The bootstrap method is reviewed and adapted to the case of multivariate estimation with doubly censored data. The Monte Carlo resampling is carried out among the collection of residuals instead of the observational data. The pool of residuals is enlarged to include all censored residuals for random drawing. The bootstrap result confirms the aforementioned scaling relationship between the individual error estimates and the global σ of the perturbing noise. As a result, the bootstrap/jackknife technique might not be really worth the considerable computational effort they require on this specific application.

A Least-Squares Method for Computing Rate Constants of Reversible, First-Order, Triangular Network Reactions and Its Application to the Intermolecular Rearrangements of Acetylacetonatobis(4,4,4-trifluoro-1-phenyl-1,3-butanedionato)ruthenium(III)

Abstract A useful method for computing the rate constants of reversible, first-order, triangular network reactions (B1\ightleftarrowsB2, B2\ightleftarrowsB3, B3\ightleftarrowsB1) is developed. It gives the best-fit set of the rate constants from the experimental data of the equilibrium constants and the mole fraction-time curves through an iterative least-squares method. This type of reaction can be classified into three categories for different shapes of the mole fraction-time curves. The method was successfully applied to determining the rate constants of the rearrangement reactions among the three isomers of [Ru(acac)(tfpb)2] (acac−=acetylacetonate ion, tfpb−=4,4,4-trifluoro-1-phenyl-1,3-butanedionate ion) in N,N-dimethylformamide at 90 °C. The direct conversion between the de,fc-bis(tfpb) isomer and the cf,ed-bis(tfpb) isomer was forbidden. The twist mechanisms are incompatible with this fact.

A new iterative least-squares method for the extraction of NMR parameters from nonideal powder patterns

A new approach to fitting powder patterns via nonlinear least-squares analysis is described. This fitting method is designed to be particularly sensitive to the frequencies of important features and insensitive to discrepancies in intensity between calculated and observed spectra. A description of an optimized powder pattern calculation routine is given with a discussion of the calculation of chemical-shift, dipole-coupled chemical-shift, and nonaxially symmetric quadrupolar powder spectra. Examples of fits of chemical-shift and dipole-coupled chemical-shift powder patterns are shown. The potential applications and limitations of the technique are discussed.

Evaluating Hackberry Sites

Site-index curves are presented for evaluating the growth potential of hackberry in native timber stands in Kansas. A single guide curve was computed from data gathered in the 1964-1965 Kansas forest inventory and harmonized curves were calculated. Hackberry (Celtis occidentalis L.) is one of the most widely distributed tree species in the woodlands of Kansas. Hackberry ranks third in total standing biomass and harvested sawtimber from commercial forest land (Raile and Spencer, 1984), and is used principally for furniture and containers. Although found on a variety of sites, hackberry grows best on rich soil along streams. In the eastern third of the state it grows in mixture with other hardwood species but occurs in almost pure stands in the central part of the state. Hackberry is drought resistant and has survived extremely dry periods in the Great Plains (Fowells, 1965). Evaluation of site productivity (wood producing potential of an area) is an important aspect of management for attaining maximum benefits from forest lands. The site-index method is the most widely accepted means for determining site quality. It uses tree height/age relationships and correlates well with timber yields. By definition, site index is the average height at age 50 years of dominant (overstory) trees in a fully stocked, even-aged stand. Curves or tables have been published for many commercial tree species in the United States. In Kansas, only curves for black walnut (Juglans nigra L.) have been published (Geyer and Lynch, 1987). The purpose of this paper is to provide another in a series of site-index equations, curves, and height tables for important tree species growing in natural stands in Kansas. PROCEDURE Forest survey data from Kansas (Chase and Strickler, 1968) provided the basic input for the construction of the site-index curves. As part of a stand inventory and analysis program, 776 temporary, variable-radius plots randomly located in the forest areas of the state were established at 10 points uniformly placed over each sample. This content downloaded from 157.55.39.43 on Fri, 30 Sep 2016 05:29:42 UTC All use subject to http://about.jstor.org/terms VOLUME 91, NUMBERS 3-4 159 Table 1. Distribution of sample trees by age and height classes. Height classes Age classes 25 35 45 55 65 75+ Totals 20 2 9 5 1 1 18 30 2 13 19 14 1 1 50 40 5 13 6 1 25 50 3 6 5 4 18 60 1 1 7 2 11 70+ 4 4 8 Totals 4 22 33 35 24 12 130 Tree measurements included height and age of the first tree in each plot that was one of the major species represented and whose crown was in the upper canopy. Total tree height to the nearest foot was measured with a relascope, and age to the nearest year was determined by increment borings at breast height (4.5 ft.). Additional years were added for seedling development to estimate total age. Hackberry was found at 130 sample areas. The distribution of these trees by height and age classes is shown in Table 1 and by geographic location and physiographic site classes in Table 2. A non-linear growth model (Richards, 1959) was used to express heightgrowth patterns. The model used was H = bo + bl(1l e-b2Age), where: H = dominant and codominant height (feet), Age = total age in years, b = coefficients of the model estimated by the iterative method of least squares, e = base of natural logarithms. A guide curve was calculated using SAS (Barr et al., 1976) and a family of site-index curves was developed utilizing the harmonized curve approach (Spurr, 1952). This approach assumes that additional curves are proportional to the guide curve, differing only by a fixed percentage. Aspen in the Central Table 2. Distribution of sample plots by geographic location and physiographic sites. Physiographic sites Geographic location Upland Lower slope Intermittent Bottomland Totals Northeast (22 counties) 4 15 28 16 63 Southeast (21 counties) 5 26 18 49 Western (62 counties) 3 12 3 18 Totals 4 23 66 37 130 This content downloaded from 157.55.39.43 on Fri, 30 Sep 2016 05:29:42 UTC All use subject to http://about.jstor.org/terms 160 TRANSACTIONS OF THE KANSAS ACADEMY OF SCIENCE Table 3. Estimated total height (ft.) of upper canopy hackberry trees in Kansas with base age 50 years.

Stability assessment of highly stressed power systems using the sparse formulation of the direct method

In the stability assessment of large-scale power systems using the direct method, slow convergence and ill conditioning of computation often occur when highly stressed systems are analyzed. The authors describe a robust computerized technique to assess the transient stability of such systems using the sparse formulation of the transient energy function method. This technique combines a direct method of Gaussian elimination and a least-squares iterative method to calculate the Newton step needed in the calculation process of the unstable equilibrium point. The technique is applied to a 108 generator, 252 bus highly stressed utility system, characterized by large power flows over weak transmission links. The superiority of the sparse formulation to handle large-scale ill-conditioned systems is demonstrated.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Asymptotic behaviour of transfer function synthesis methods

This paper presents an analysis of the asymptotic behaviour of the solutions to various iterative linear least-squares methods that synthesize transfer functions from frequency-response data. The methods of Sanathanan and Koerner and Lawrence and Rogers are shown to possess asymptotic solutions that do not coincide with the solution to a fundamental non-linear least-squares criterion. Several methods with more appropriate asymptotic behaviour are then suggested. The analytical tools presented should allow analysis of conditions for the convergence of iterative linear least-squares methods.

Moment analysis of intravenous, intraduodenal, buccal, rectal and percutaneous nifedipine in rats.

The pharmacokinetics and bioavailability of intravenous, intraduodenal, buccal, rectal and percutaneous nifedipine were studied in rats to evaluate the influence of route of administration. Plasma concentrations of nifedipine were determined by electron capture gas-liquid chromatography with preliminary oxidation of the drug to its pyridine analog. Pharmacokinetic evaluations were carried out by noncompartmental analysis of plasma concentration-time curves based on the statistical moment theory. The moments were computed by fitting a polyexponential function to the discrete time-course data of plasma concentration using the iterative least-squares method. A good computer fit to biexponential kinetics and a short mean residence time were observed after intravenous administration (i.v.). The linear distribution and disposition of nifedipine was found within the dosing range tested (0.025-0.100 mg/kg, i.v.). The systemic availability of nifedipine after intraduodenal administration was on the average between 52% and 57%, indicating that nifedipine is extensively metabolized during first passage through the liver. Avoidance of first-pass metabolism was investigated following buccal, rectal and percutaneous administration to rats. These nonportal routes of administration gave approximately 10-30% increases in systemic availability compared to that from intraduodenal nifedipine delivery. Moment analysis revealed variations of mixing condition of nifedipine at the dosing site. It was shown that the absorption occurs in a mammillary manner after intraduodenal and rectal administration. On the other hand, there is steady-state absorption of the drug from the buccal route of administration. Following percutaneous administration, the drug seems to be absorbed catenary to some degree.(ABSTRACT TRUNCATED AT 250 WORDS)