Modified Levenberg-Marquardt Algorithm Research Articles

SPIN: An Inversion Code for the Photospheric Spectral Line

Inversion codes are the most useful tools to infer the physical properties of the solar atmosphere from the interpretation of Stokes profiles. In this paper, we present the details of a new Stokes Profile INversion code (SPIN) developed specifically to invert the spectro-polarimetric data of the Multi-Application Solar Telescope (MAST) at Udaipur Solar Observatory. The SPIN code has adopted Milne–Eddington approximations to solve the polarized radiative transfer equation (RTE) and for the purpose of fitting a modified Levenberg–Marquardt algorithm has been employed. We describe the details and utilization of the SPIN code to invert the spectro-polarimetric data. We also present the details of tests performed to validate the inversion code by comparing the results from the other widely used inversion codes (VFISV and SIR). The inverted results of the SPIN code after its application to Hinode/SP data have been compared with the inverted results from other inversion codes.

Dynamic model identification method of manipulators for inside DEMO engineering

In the inside engineering of DEMO, the robotic machines and manipulators are envisaged to be widely employed, which often have to deal with the demanding working conditions. The construction of the dynamic model of the robot and manipulator can be incorporated into the control system design to gain the adaptive control performance. A method of constructing the dynamic model with the unknown parts is proposed. The method can identify the unknown parts of the dynamic system by incorporating a BP neural network that will eventually substitute the unknown parts in the system dynamics after the well training. A modified Levenberg-Marquardt algorithm is developed for the training of BP neural network, which can back propagate the errors between entire actual system and the constructed dynamic model into the training process of the neural network. An example of constructing the dynamic model for a general Stewart structure mechanism is presented, in which the unknown parts, as well the entire dynamics are successfully identified. The proposed method can be extrapolated to the dynamics modeling of the blanket transporter in DEMO design, as well as the other general manipulators.

Modified Levenberg–Marquardt Algorithm for Backpropagation Neural Network Training in Dynamic Model Identification of Mechanical Systems

For modeling a dynamic system in practice, it often faces the difficulty in improving the accuracy of the constructed analytical model, since some components of the dynamic model are often ignored deliberately due to the difficulty of identification. It is also unwise to apply the neural network to approximate the entire dynamic system as a black box, when the comprehensive knowledge of most components of the dynamics of a large system are available. This paper proposes a method that utilizes the backpropagation (BP) neural network to identify the unknown components of the dynamic system based on the experimental front-end inputs–outputs data of the entire system. It can avoid the difficulty in getting the direct training data for the unknown components, and brings great benefits in the practical application, since to get the front-end inputs–outputs data of the entire dynamic system is easier and cost-effective. In order to train such neural network for the unknown components of dynamics, a modified Levenberg–Marquardt algorithm, which can utilize the front-end inputs–outputs data of the entire dynamic system, has been developed in the paper. Three examples from different application points of view are presented in the paper, and the results show that the proposed modified Levenberg–Marquardt algorithm is efficient to train the neural network for the unknown components of the system based on the data of entire system. The constructed dynamics model, in which the unknown components are substituted by the neural network, can satisfy the requisite accuracy successfully in the computation.

Local atomic structure of Fe/Cr multilayers: Depth-resolved method

A depth-resolved method for the investigation of the local atomic structure by combining data of X-ray reflectivity and angle-resolved EXAFS is proposed. The solution of the problem can be divided into three stages: 1) determination of the element concentration profile with the depth z from X-ray reflectivity data, 2) determination of the X-ray fluorescence emission spectrum of the element i absorption coefficient μia(z,E) as a function of depth and photon energy E using the angle-resolved EXAFS data Iif(E,ϑl), 3) determination of partial correlation functions gij(z,r)as a function of depth from μi(z,E).All stages of the proposed method are demonstrated on a model example of a multilayer nanoheterostructure Cr/Fe/Cr/Al2O3. Three partial pair correlation functions are obtained. A modified Levenberg-Marquardt algorithm and a regularization method are applied.

Assessment of objects equatorial coordinates on the digital frames for long-focus and short-focus telescopes

Computational methods for determining equatorial coordinates of celestial objects based on an assessment of their positions on the digital frame are described. These methods are based on the possibility of representing errors in celestial objects positions mappings in the form of reduction equation which depends only on the object’s location in the frame. To assess the coefficients of the reduction model the modified Levenberg-Marquardt algorithm was used that allowed to eliminate the problem with accumulation of computational errors for ill-conditioned matrix of reduction model partial derivatives. The developed methods take into account the peculiarities of astronomical reduction in long-focus and short-focus optical systems. These systems have different types of optical aberrations that negatively affect accuracy of objects positions determination that leads to the problem of choosing optimal model for each one. With using of digital images obtained from both types of optical systems the comparative analysis of the accuracy of the position of objects in case of application of considered methods have been carried out. As the investigated objects in the analysis the set of reference stars as well as stars identified with UCAC4 have been used. The results showed reasonableness of applying direct coordinate reduction methods for frames obtained by using long-focus optical systems. According to the analysis results for short-focus optical systems it is advisable to use the reverse coordinate reduction methods

Correlation of capacity fading processes and electrochemical impedance spectra in lithium/sulfur cells

The capacity fading of lithium/sulfur (Li/S) cells is one major challenge that has to be overcome for a successful commercialization of this electrochemical storage system. Therefore it is essential to detect the major fading mechanisms for further improvements of this system. In this work, the processes leading to fading are analyzed in terms of a linear four state model and correlated to the distribution of relaxation times calculated with a modified Levenberg-Marquardt algorithm. Additionally, the Warburg impedance and the solution resistance are also obtained by the same algorithm. The detailed analysis of intermediate states during the first cycle gives the distinction between relaxation processes at the sulfur cathode and at the lithium anode. The influence of the polysulfides on the impedance parameters was evaluated using symmetric cells; this yields a good correlation with the results obtained from the first discharge/charge experiment. A fast and a slow capacity fading process are observed for the charge and the discharge during 50 cycles. The fast fading process can be assigned to Faradaic reactions at the lithium anode.

A modified Levenberg–Marquardt algorithm for simultaneous estimation of multi-parameters of boundary heat flux by solving transient nonlinear inverse heat conduction problems

Despite numerous studies of Levenberg–Marquardt (LM) algorithms for solving inverse heat conduction problems, sensitivity coefficients are mainly evaluated by numerical differentiation methods. However, sensitivity coefficients are difficult to be precisely calculated by numerical differentiation methods, if multi-dimensions, transient states and nonlinearities are involved. To the best knowledge of the authors, there has not been a general method for accurately calculating sensitivity coefficients for a LM algorithm, except numerical differentiation methods. In this study, a modified LM algorithm is presented by introducing the complex-variable-differentiation method for sensitivity analysis, and multi-parameters of boundary heat flux are simultaneously recovered by solving transient nonlinear inverse heat conduction problems. The results show that the modified LM algorithm has the advantages of the conventional LM algorithm, that are effective, accurate and robust for simultaneous estimation of multi-parameters of boundary heat flux. Meanwhile, the efficiency and the convergence stability of the modified LM algorithm are improved, which are attributed to accurate evaluation of sensitivity coefficients, compared with the conventional LM algorithm.

Hydraulic Tomography: Continuity and Discontinuity of High-K and Low-K Zones.

Hydraulic tomography is an emerging field and modeling method that provides a continuous hydraulic conductivity (K) distribution for an investigated region. Characterization approaches that rely on interpolation between one-dimensional (1D) profiles have limited ability to accurately identify high-K channels, juxtapositions of lenses with high K contrast, and breaches in layers or channels between such profiles. However, locating these features is especially important for groundwater flow and transport modeling, and for design and operation of in situ remediation in complex hydrogeologic environments. We use transient hydraulic tomography to estimate 3D K in a volume of 15-m diameter by 20-m saturated thickness in a highly heterogeneous unconfined alluvial (clay to sand-and-gravel) aquifer with a K range of approximately seven orders of magnitude at an active industrial site in Assemini, Sardinia, Italy. A modified Levenberg-Marquardt algorithm was used for geostatistical inversion to deal with the nonlinear nature of the highly heterogeneous system. The imaging results are validated with pumping tests not used in the tomographic inversion. These tests were conducted from three of five clusters of continuous multichannel tubing (CMTs) installed for observation in the tomographic testing. Locations of high-K continuity and discontinuity, juxtaposition of very high-K and very low-K lenses, and low-K "plugs" are evident in regions of the investigated volume where they likely would not have been identified with interpolation from 1D profiles at the positions of the pumping well and five CMT clusters. Quality assessment methods identified a suspect high-K feature between the tested volume and a lateral boundary of the model.

A modified two steps Levenberg–Marquardt method for nonlinear equations

The modified Levenberg–Marquardt (MLM) method to solve nonlinear equations was introduced by Fan (2012). This method uses an addition of the Levenberg–Marquardt step and an approximate LM step as the trial step at every iteration. Using a trust region technique, the global and cubic convergence of the MLM method under the local error bound condition is proved (Fan, 2012). Recently, Fan proposed an accelerated MLM algorithm by using a line search strategy to generate a modified LM step and showed that the convergence rate of the algorithm is min{1+2δ,3} which results the cubic convergence for δ≥1 (Fan, 2014). In this paper, by introducing an adaptive LM parameter for AMLM algorithm, we propose an efficient AMLM algorithm. The cubic convergence of the new algorithm is presented while numerical experiments show the new algorithm is promising.

Uniform Mesoporous Silicoaluminophosphate Derived by Vapor Phase Treatment: Its Catalytic and Kinetic Studies in Hydroisomerization of 1-Octene

Ordered mesoporous silicoaluminophosphate (MESO-SAPO-34) assembled from microporous SAPO-34 precursor was stabilized by post synthesis vapor phase treatment. The resultant MESO-SAPO-34 possessed uniform mesoporosity, high surface area, pore volume, and strong acidity. The resultant materials were studied as catalysts for the hydroisomerization of 1-octene and showed excellent conversion, and a maximum branched isomer selectivity of about 52% was achieved at low weight hourly space velocity (WHSV). Kinetic models based on Langmuir–Hinshelwood–Hougen–Watson (LHHW) mechanism were developed to describe the hydroisomerization of 1-octene, and the kinetics and adsorption parameters were estimated. A nonlinear regression based on modified Levenberg–Marquardt algorithm was used to estimate the parameters, and the model values were found to compare well with the experimental values. Parameters obtained from the LHHW model show that the conversion of 1-octene to other linear octenes requires low activation energy, whereas the conversion of linear octene to branched octene demands the highest activation energy. The enthalpy and entropy of adsorption obtained from Arrhenius plots were found to be consistent with thermodynamics.

Numerical Shape Optimization in Industrial Glass Blowing

Industrial glass blowing is an essential stage of manufacturing hollow glass containers, e.g., bottles, jars. A glass preform is brought into a mold and inflated with compressed air until it reaches the mold shape. A simulation model for blowing glass containers based on finite element methods, which adopts a level set method to track the glass–air interfaces, has previously been developed [Giannopapa and Groot, 2007, “A Computer Simulation Model for the Blow–Blow Forming Process of Glass Containers,” Paper No. PVP2007-26408, pp. 79–86; Giannopapa, C. G., 2008, “Development of a Computer Simulation Model for Blowing Glass Containers,” ASME J. Manuf. Sci. Eng., 130(4), p. 041003; Giannopapa and Groot, 2011, “Modeling the Blow–Blow Forming Process in Glass Container Manufacturing: A Comparison Between Computations and Experiments,” ASME J. Fluids Eng., 133(2), p. 021103]. A considerable challenge in glass blowing is the inverse problem: to determine an optimal preform from the desired container shape. In previous work of the authors [Groot et al., 2009, “Numerical Optimisation of Blowing Glass Parison Shapes,” ASME Paper No. PVP2009-77946; Groot et al., 2011, “Development of a Numerical Optimization Method for Blowing Glass Parison Shapes,” ASME J. Manuf. Sci. Eng., 133(1), p. 011010] a numerical method was introduced for optimizing the shape of the preform. The optimization method described the shape of the preform by parametric curves, e.g., Bezier-curves or splines, and employed a modified Levenberg–Marquardt algorithm to find the optimal positions of the control points of the curves. A combined finite difference and Broyden method was used to compute the Jacobian of the residual with respect to changes in the positions of the control points. The objective of this paper is to perform an error analysis of the optimization method previously introduced and to improve its accuracy and performance. The improved optimization method is applied to modeled containers of industrial relevance, which shows its usefulness for practical applications.

Improved modeling of in vivo kinetics of slowly diffusing radiotracers for tumor imaging.

Large-molecule tracers, such as labeled antibodies, have shown success in immuno-PET for imaging of specific cell surface biomarkers. However, previous work has shown that localization of such tracers shows high levels of heterogeneity in target tissues, due to both the slow diffusion and the high affinity of these compounds. In this work, we investigate the effects of subvoxel spatial heterogeneity on measured time-activity curves in PET imaging and the effects of ignoring diffusion limitation on parameter estimates from kinetic modeling. Partial differential equations (PDE) were built to model a radially symmetric reaction-diffusion equation describing the activity of immuno-PET tracers. The effects of slower diffusion on measured time-activity curves and parameter estimates were measured in silico, and a modified Levenberg-Marquardt algorithm with Bayesian priors was developed to accurately estimate parameters from diffusion-limited data. This algorithm was applied to immuno-PET data of mice implanted with prostate stem cell antigen-overexpressing tumors and injected with (124)I-labeled A11 anti-prostate stem cell antigen minibody. Slow diffusion of tracers in linear binding models resulted in heterogeneous localization in silico but no measurable differences in time-activity curves. For more realistic saturable binding models, measured time-activity curves were strongly dependent on diffusion rates of the tracers. Fitting diffusion-limited data with regular compartmental models led to parameter estimate bias in an excess of 1,000% of true values, while the new model and fitting protocol could accurately measure kinetics in silico. In vivo imaging data were also fit well by the new PDE model, with estimates of the dissociation constant (Kd) and receptor density close to in vitro measurements and with order of magnitude differences from a regular compartmental model ignoring tracer diffusion limitation. Heterogeneous localization of large, high-affinity compounds can lead to large differences in measured time-activity curves in immuno-PET imaging, and ignoring diffusion limitations can lead to large errors in kinetic parameter estimates. Modeling of these systems with PDE models with Bayesian priors is necessary for quantitative in vivo measurements of kinetics of slow-diffusion tracers.

Parameter estimation of the exponentially damped sinusoids signal using a specific neural network

The problem of estimating the parameters of exponentially damped sinusoids (EDSs) signal has received very much attention in many fields. In this paper, a specific neural network termed EDSNN for parameter estimation of the EDSs has been proposed. Aiming at effectively evaluating the parameters of the EDSs signal, we construct a specific topology of EDSNN strictly following the mathematic formulation of EDSs signal. Then, what should be further done is how to train EDSNN using the data-set sampled from the EDSs signal. For this purpose, a modified Levenberg–Marquardt algorithm is derived for iteratively solving the weights of EDSNN by optimizing the pre-defined objective function. Profiting from good performance in fault tolerance of neural network, the proposed algorithm possesses a good performance in resistance to noise. Several computer simulations have been conducted to apply this method to some EDSs signal models. The results substantiate that the proposed EDSNN can synchronously obtain a higher precision for the damped factors, frequencies, also amplitudes and initial phases of all the EDSs than the state-of-the-art algorithm for noise free or noise case.

The use of parameter optimization in the design of VLSI library cells and custom interfaces of nanoelectronic devices

The modified Levenberg-Marquardt algorithm applied to the parametric optimization of VLSI library cells and custom interfaces of nanoelectronic devices is presented.

A new mechanistic model for liquid–liquid phase transfer catalysis: Reaction of benzyl chloride with aqueous ammonium sulfide

A new mechanistic model for reactions involving two liquid phases and a homogeneous phase transfer catalyst (PTC) has been developed based on extraction mechanism considering the equilibrium of catalyst and active catalysts at the interface. The proposed kinetic model considers thermodynamic framework based aqueous phase ionic equilibrium and the separate contributions of non-PTC and PTC-enhanced reactions towards the overall reactions. The developed model was then applied to an industrially important reaction of benzyl chloride with aqueous ammonium sulfide for synthesis of dibenzyl sulfide and benzyl mercaptan. The kinetic parameters of the developed model were estimated at different temperatures using an indigenously developed non-linear regression technique based on modified Levenberg–Marquardt algorithm. Sensitivity analysis was then performed under various experimental conditions using the estimated parameters and the results were compared with experimental observations. A good agreement was observed between experimental and calculated values with proper trends of the results.

Alternative phase-diverse phase retrieval algorithm based on Levenberg-Marquardt nonlinear optimization

A modified Levenberg-Marquardt (MLM) algorithm is proposed to substitute for modified G-S (MGS) algorithm in some situations of phase-diverse phase retrieval wavefront sensing (WFS), such as the obstructed pupil, in which the second derivative information is specifically employed to eliminate the local minimum stagnation. Experiments have been performed to validate MLM algorithm in WFS accuracy (less than lambda/30 RMS) referring to ZYGO interferometer results and in WFS repeatability (less than lambda/200 RMS), even the dynamic range is more than 7 lambda PV. Moreover, experiments have shown the MLM algorithm is superior to the MGS algorithm both in WFS accuracy and repeatability.

Constitutive parameter estimation of a viscoelastic model with internal variables

The present work is aimed at estimating the constitutive parameters of viscoelastic materials. The constitutive equation addressed takes internal dissipation into account by means of internal variables. The identification method is built on an optimization problem where the minimum of an error function is sought. The error function is based on the difference between the experimental response and the model one and on the parameter constraints as well. The constitutive parameters are estimated by means of a modified Levenberg–Marquardt algorithm. The viscoelastic material characterization was assessed through an experimental set-up consisted of a sandwich beam with a viscoelastic core.

A Modified Levenberg‐Marquardt Method for Nonsmooth Equations with Finitely Many Maximum Functions

For solving nonsmooth systems of equations, the Levenberg‐Marquardt method and its variants are of particular importance because of their locally fast convergent rates. Finitely many maximum functions systems are very useful in the study of nonlinear complementarity problems, variational inequality problems, Karush‐Kuhn‐Tucker systems of nonlinear programming problems, and many problems in mechanics and engineering. In this paper, we present a modified Levenberg‐Marquardt method for nonsmooth equations with finitely many maximum functions. Under mild assumptions, the present method is shown to be convergent Q‐linearly. Some numerical results comparing the proposed method with classical reformulations indicate that the modified Levenberg‐Marquardt algorithm works quite well in practice.

Efficient Gate Capacitance and Current Modeling of High-k Gate Stacks

Computationally efficient and accurate physically-based self-consistent compact models of CMOS gate-capacitance (Cg-Vg) and gate-current (Ig-Vg) for both ultra-thin SiO2 and high-k gate-stacks down to ~ 0.5 nm EOT are reviewed. The compact model produces results comparable to those obtained via computationally intense simulators. A modified Levenberg-Marquardt algorithm has been used in combination with these models to provide an efficient material and device parameter extraction capability. In a few seconds, parameters such as effective oxide thickness, surface substrate doping concentrations, flat-band voltages and poly-Si doping concentrations can be obtained from measured Cg-Vg, and physical thicknesses, band offsets, dielectric constants and tunneling masses for the gate-dielectrics can be extracted from measured Ig-Vg.

Predicting hydrate stability zones of petroleum fluids using sound velocity data of salt aqueous solutions

Predicting hydrate stability zones of petroleum fluids from the aqueous phase properties can have a practical application as measuring these properties is normally easier than hydrate phase equilibrium measurement and can reduce experimental costs and efforts. In this work, the possibility of estimating hydrate stability zone from sound velocity data of salt aqueous solutions is investigated using a feed-forward artificial neural network method with a modified Levenberg–Marquardt algorithm. The method considers the changes of sound velocity in salt (NaCl, KCl, NaBr, KBr, BaCl 2, MgCl 2, Na 2SO 4, HCOONa) aqueous solution with respect to sound velocity in pure water and therefore there is no need to have a quantitative analysis of the aqueous solution. Independent data (not used in training and developing of the method) are used to examine the reliability of this tool. The predictions of this method are in acceptable agreement with independent experimental data, demonstrating the reliability of this tool for estimating the hydrate stability zone in the presence of salt aqueous solutions.