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When the conjugate gradient (CG) method for solving linear algebraic systems was formulated about 70 years ago by Lanczos, Hestenes, and Stiefel, it was considered an iterative process possessing a mathematical finite termination property. With the deep insight of the original authors, CG was placed into a very rich mathematical context, including links with Gauss quadrature and continued fractions. The optimality property of CG was described via a normalized weighted polynomial least squares approximation to zero. This highly nonlinear problem explains the adaptation of CG iterates to the given data. Karush and Hayes immediately considered CG in infinite dimensional Hilbert spaces and investigated its superlinear convergence. Since then, the view of CG, as well as other Krylov subspace methods developed in the meantime, has changed. Today these methods are considered primarily as computational tools, and their behavior is typically characterized using linear upper bounds, or heuristics based on clustering of eigenvalues. Such simplifications limit the mathematical understanding of Krylov subspace methods, and also negatively affect their practical application. This paper offers a different perspective. Focusing on CG and the generalized minimal residual (GMRES) method, it presents mathematically important as well as practically relevant phenomena that uncover their behavior through a discussion of computed examples. These examples provide an easily accessible approach that enables understanding of the methods, while pointers to more detailed analyses in the literature are given. This approach allows readers to choose the level of depth and thoroughness appropriate for their intentions. Some of the points made in this paper illustrate well known facts. Others challenge mainstream views and explain existing misunderstandings. Several points refer to recent results leading to open problems. We consider CG and GMRES crucially important for the mathematical understanding, further development, and practical applications also of other Krylov subspace methods. The paper additionally addresses the motivation of preconditioning. [ABSTRACT FROM AUTHOR]

The stochastic actor oriented model (SAOM) is a method for modelling social interactions and social behaviour over time. It can be used to model drivers of dynamic interactions using both exogenous covariates and endogenous network configurations, but also the co-evolution of behaviour and social interactions. In its standard implementations, it assumes that all individual have the same interaction evaluation function. This lack of heterogeneity is one of its limitations. The aim of this paper is to extend the inference framework for the SAOM to include random effects, so that the heterogeneity of individuals can be modelled more accurately. We decompose the linear evaluation function that models the probability of forming or removing a tie from the network, in a homogeneous fixed part and a random, individual-specific part. We extend the algorithm so that the variance of the random parameters can be estimated with method of moments. Our method is applicable for the general random effect formulations. We illustrate the method with a random out-degree model and show the parameter estimation of the random components, significance tests and model evaluation. We apply the method to the Kapferer's Tailor shop study. It is shown that a random out-degree constitutes a serious alternative to including transitivity and higher-order dependency effects. • The paper allows effects to be individual-specific to model heterogeneity accurately. • The method of moments is generalized to estimate random effects in the SAOM. • The model evaluation procedure based on the score-test is also been developed. • We identify interesting random effects in the Kapferer's Tailor-shop dataset. [ABSTRACT FROM AUTHOR]

[Display omitted] • Gradient fibrous media was derived by modifying fiber packing density distribution of a uniform one. • Transport and deposition process of polydisperse particles were investigated using method of moments. • Particle deposition mass distributions across filter depth were analyzed and optimized. • There exists an optimal fiber packing density distribution ratio for gradient fibrous media. The goal of this work is to optimize the deposition mass distribution of polydisperse particles inside the fibrous media by modifying the distribution of the fiber packing density across filter depth. Two types of gradient fibrous media with linear and exponential fiber packing density distribution along the flow direction were derived. The method of moments was employed to study the transport and deposition characteristics of polydisperse aerosol particles inside the fibrous media. The reliability of the calculated results by the method of moments was confirmed by the numerical simulations based on the discrete phase model (DPM). Results indicate that the gradient fibrous media allows more particles to transport and deposit into the interior of the fibrous media, which improves the uniformity of particle deposition distribution across filter depth. There exists an optimal fiber packing density distribution ratio for the gradient fibrous media where the non-uniformity of deposition mass distribution of polydisperse particles across filter depth is minimized. In this way, uniform deposition distribution along the flow direction can be generally achieved by optimizing the fiber packing density distribution, which will contribute to the optimal design of microstructures for the fibrous filter. [ABSTRACT FROM AUTHOR]

We consider the binomial random set model [ n ] p where each element in { 1 , ... , n } is chosen independently with probability p : = p (n). We show that for essentially all regimes of p and very general conditions for a matrix A and a column vector b , the count of specific integer solutions to the system of linear equations A x = b with the entries of x in [ n ] p follows a (conveniently rescaled) normal limiting distribution. This applies among others to the number of solutions with every variable having a different value, as well as to a broader class of so-called non-trivial solutions in homogeneous strictly balanced systems. Our proof relies on the delicate linear algebraic study both of the subjacent matrices and the corresponding ranks of certain submatrices, together with the application of the method of moments in probability theory. [ABSTRACT FROM AUTHOR]

Orbit recovery problems are a class of problems that often arise in practice and various forms. In these problems, we aim to estimate an unknown function after being distorted by a group action and observed via a known operator. Typically, the observations are contaminated with a non-trivial level of noise. Two particular orbit recovery problems of interest in this paper are multireference alignment and single-particle cryo-EM modeling. In order to suppress the noise, we suggest using the method of moments approach for both problems while introducing deep neural network priors. In particular, our neural networks should output the signals and the distribution of group elements, with moments being the input. In the multireference alignment case, we demonstrate the advantage of using the NN to accelerate the convergence for the reconstruction of signals from the moments. Finally, we use our method to reconstruct simulated and biological volumes in the cryo-EM setting. [ABSTRACT FROM AUTHOR]

Continuous measurement of the material's relative moisture content (m.c) is an important duty in industrial process control. However, current techniques only provide the absolute amount of water in the sample container, and (m.c of corn) is obtained from an extra mass measurement. The investigation of the dielectric behavior of moist matter at microwave frequencies using an open-ended coaxial sensor, as well as the relationship of reflection coefficient and moisture content for frequencies ranging from 1 GHz to 5 GHz via MOM and measurement, and thus the comparison of results, confirms the superiority of Method of Moment (MOM), due to benefits such as low cost and non-harmful to the sample materials. The basic concept is explored, and the results of experiments using maize as the medium are shown. [Display omitted] • Application of electromagnetic propagation were explained. • Method of moment for coaxial sensor in reflection coefficient magnitude and moisture content. • Normalized calculations using the Method of Moment (MOM). [ABSTRACT FROM AUTHOR]

• Novel closure schemes for moment-transformed population balance are introduced. • The full distribution can be reconstructed in a stepwise manner. • The introduced method provides an easy access to pointwise values of PDF. • The method enables modelling of dissolution and Ostwald ripening using MOM. • First stages of batch silica precipitation are simulated. Ostwald ripening is a main phenomenon responsible for growth of particles manufactured using colloidal methods. This paper considers mathematical description of the process using population balance approach solved by method of moments. However, in the case of particles' dissolution which is a part of the process a closure problem is generated. A novel closure method for the moment – transformed population balance equations based on Gram – Charlier expansions is introduced. Due to its simplicity it allows to reconstruct the full particle size distribution in a stepwise manner giving an access to the pointwise values of the density function required to close the model. The introduced numerical scheme is further used to simulate several carefully chosen test cases including dissolution of solid particles and ageing of particles' population. The mathematical model presented in the paper is then applied to interpret experimental results for batch precipitation of silica. [ABSTRACT FROM AUTHOR]

• An analytical solution is derived for combined Brownian and shear coagulation. • This solution is the same as previous analytical solutions for limiting cases. • The present solution gives reasonable predictions compared with the LNMOM. • The coagulation rate ratio is a key factor for describing the evolution process. Brownian coagulation of aerosol particles can take place in both laminar and turbulent flows. Thus, the simultaneous Brownian and shear coagulation will always occur in practical applications. This study presents an analytical solution to describe the size evolution of polydisperse particles undergoing simultaneous Brownian and shear coagulation. The analytical solution is derived using the log-normal method of moments (LNMOM) with some approximations. Then, the analytical solution is validated by comparing with previous analytical solutions derived for limiting cases. The results show that the present analytical solution is consistent with previous analytical solutions for these limiting cases. Further, the time trajectories of the total particle number concentration, the geometric standard deviation and the geometric number mean particle volume predicted by the present analytical solution are compared with those of a numerical LNMOM model. The results show that the present analytical solution gives good predictions of the total particle number concentration but less accurate predictions of the geometric standard deviation and the geometric number mean particle volume. A dimensionless analysis shows that the coagulation rate ratio and the initial polydispersity are two important factors for characterizing the size evolution of simultaneous Brownian and shear coagulation. [ABSTRACT FROM AUTHOR]

The Method of Moments with Interpolative Closure (MOMIC) is a widely used approach for numerical modeling of aerosol dynamics, where soot formation is an important area of application. Two basic variants of the algorithm were proposed in the past: In the original method, transport equations are solved for several non-negative order integer moments of the soot Number Density Function and one for the moment of order minus infinity. Based on these moments, two interpolation functions are constructed and used for the evaluation of positive and negative fractional order moments required for closure of the moment equation source terms. In the other variant, which is a simplification of the first, equations are solved only for non-negative order integer moments. Negative order moments then need to be evaluated by extrapolation. In addition to the choice between these two algorithms, it is not clear a-priori what is the optimal choice regarding the number of transported moments and hence the order of the interpolation polynomial. In this work, the effects of different choices in the setup of the MOMIC algorithm on the evaluation of soot source terms and the prediction of the soot number density and volume fraction are assessed systematically. To this end, MOMIC is combined with a state-of-the-art physico-chemical soot model and applied to the simulation of soot formation in a laminar premixed flame and along Lagrangian trajectories extracted from a Direct Numerical Simulation of a turbulent sooting flame. Monte Carlo simulations serve as reference solutions for a combined a-priori and a-posteriori study. Including the moment of order minus infinity is shown to be beneficial for an accurate calculation of the coagulation source term and hence the soot number density, because moments of negative order can be evaluated more accurately. Furthermore, a high interpolation order yields a more accurate prediction of the surface growth source term. This source term is systematically overpredicted, but errors are substantially reduced to a few percent with increasing interpolation order. [ABSTRACT FROM AUTHOR]

The method of moments is widely used for the reduction of kinetic equations into fluid models. It consists in extracting the moments of the kinetic equation with respect to a velocity variable, but the resulting system is a priori underdetermined and requires a closure relation. In this paper, we adapt the φ -divergence based closure, recently developed for rarefied gases i.e. with a velocity variable describing R d , to the radiative transfer equation where velocity describes the unit sphere S 2. This closure is analyzed and a numerical method to compute it is provided. Eventually, it provides the main desirable properties to the resulting system of moments: Similarly to the entropy minimizing closure (M N), it dissipates an entropy and captures exactly the equilibrium distribution. However, contrarily to M N , it remains computationally tractable even at high order and it relies on an exact quadrature formula which preserves exactly symmetry properties, i.e. it does not trigger ray effects. The purely anisotropic regimes (beams) are not captured exactly but they can be approached as close as desired and the closures remains again tractable in this limit. [ABSTRACT FROM AUTHOR]

Under certain mild conditions, some limit theorems for functionals of two independent Gaussian processes are obtained. The results apply to general Gaussian processes including fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. A new and interesting phenomenon is that, in comparison with the results for fractional Brownian motion, extra randomness appears in the limiting distributions for Gaussian processes with nonstationary increments, say sub-fractional Brownian motion and bi-fractional Brownian. The results are obtained based on the method of moments, in which Fourier analysis, the chaining argument introduced in [11] and a pairing technique are employed. [ABSTRACT FROM AUTHOR]

We derive the asymptotic behavior for an additive functional of two independent self-similar Gaussian processes when their intersection local time exists, using the method of moments. [ABSTRACT FROM AUTHOR]

Abstract In this contribution, we present a Calderón preconditioner for a novel single-source equation to efficiently model electromagnetic scattering problems involving high magnetic contrasts. Through analysis of the spectral properties of the system matrix after discretization, it is shown that this formulation does not break down when high permeabilities are present, which was an unresolved problem of the Calderón preconditioned Poggio–Miller–Chan–Harrington–Wu–Tsai method. The adopted discretization scheme, which involves Rao–Wilton–Glisson and Buffa–Christiansen basis functions, allows for an easy integration in existing commercial Method of Moments software. The efficiency and accuracy of the presented method is corroborated by numerical examples. [ABSTRACT FROM AUTHOR]

Abstract An efficient Volume Integral Equation Method is proposed for the analysis of radially inhomogeneous tapered dielectric rod antennas (DRAs). This technique results in radiation pattern formulas that need only common integrations. Radiation patterns for DRAs of different radii and permittivities are calculated. The method is validated by comparison with the data obtained from CST Microwave Studio integral equation solver. Furthermore, radiation pattern of different permittivity profiles are evaluated and compared to achieve the optimal design parameters. The design formulas for choosing the maximum and minimum radii for different permittivity profiles are presented and compared to achieve the desired radiation parameters. An accurate formula for the calculation of antenna bandwidth is proposed. It is shown that decreasing the dielectric constant of tapered rod in the radial direction results in higher antenna gain and wider operating bandwidth. [ABSTRACT FROM AUTHOR]

Abstract Several method of moments (MOM) models were developed recently to describe the evolution of soot particle populations. However, especially soot oxidation is still very challenging in MOM, as pointwise information of the underlying soot particle number density function (NDF) is usually unknown and thus, the prediction of smallest particles that oxidize completely is not easy. The recently proposed Extended Quadrature Method of Moments (EQMOM) (Yuan, Laurent, & Fox, 2012) has the potential to resolve this issue providing a continuous NDF reconstruction based on the set of transported moments. While its general suitability for soot was already demonstrated in the literature, the EQMOM moment inversion procedure reveals several numerical difficulties. In this work, we propose an EQMOM modification called split-based EQMOM that builds upon the general idea of EQMOM to represent the NDF by a weighted sum of superimposed, non-negative, continuous kernel density functions (KDFs), avoiding, however, the numerical problems of its original version. The model is based on a split of the entire NDF into a sum of overlaying density functions, whose evolution is governed by individual population balance equations (PBEs). These PBEs are derived and implemented in a novel Monte Carlo (MC) framework which is applied to simulate two fuel-rich burner-stabilized laminar premixed flames with different sooting behaviours. Results are compared to a classical MC model to demonstrate the suitability. Next, the MC data is used to assess the suitability of lognormal, gamma and inverse Gaussian distributions to approximate the NDF shape by employing the Wasserstein metric as criterion. This information is finally used to formulate an improved EQMOM soot model, which is then evaluated for both fuel-rich and oxidizing conditions. For the latter, the two-stage burner experiments of Echavarria, Jaramillo, Sarofim, & Lighty (2011) is considered, where particle oxidation is the dominant process. It is demonstrated that the proposed MOM model allows an accurate and numerically robust description of soot formation, growth and also oxidation. Highlights • A novel method of moments with number density function reconstruction is proposed. • The model is based on the EQMOM approach avoiding its numerical difficulties. • The suitability of the novel model is shown for soot formation, growth and oxidation. [ABSTRACT FROM AUTHOR]

Abstract A new Multi-Moment Sectional Method (MMSM) is proposed for tracking the evolution of the soot Number Density Function (NDF). Unlike conventional sectional methods, in MMSM, multiple statistical moments are solved within each section, and the size distribution within each section is reconstructed from a polynomial profile or, for the last section, an exponential profile. MMSM can be reduced to a method of moments with a single section to minimize the computational cost and reduced to a sectional method when the number of moments per section is one. The MMSM approach is combined with a detailed univariate soot model and applied to the computation of the soot NDF in an ethylene laminar premixed flame. Compared to a pure sectional method, MMSM is shown to achieve a higher rate of convergence for the predicted total number density and particle size distribution. [ABSTRACT FROM AUTHOR]

Abstract The simulation of turbulent sooting flames requires a host of models, of which the two critical components are the chemical kinetics that describe soot precursor evolution and the description of the soot population. The purpose of this study is to understand the sensitivity of soot predictions in a realistic aircraft combustor to model choices for these components. Two different chemistry mechanisms, three different statistical approaches, and two different soot inception models are considered. The simulations show that acetylene-based soot inception produces very high soot volume fraction, with the soot particles present predominantly in the inner recirculation zone of the swirl-stabilized combustor. The PAH-based nucleation models lead to soot generation in the shear layers emanating from fuel injection. The two advanced statistical approaches (Hybrid and Conditional Quadrature Method of Moments) also show significant differences. While the Hybrid method produces lower soot number density, it also generates larger soot particles due to a faster predicted rate of coagulation. The Conditional Quadrature approach produces much higher soot number density, but its particle sizes are smaller compared to the Hybrid method for all kinetic mechanisms considered. This experimental combustor is strongly dominated by surface growth based soot mass addition. As a result, even if nucleation/condensation rates are different, the final soot mass yield is comparable for PAH-based soot models. These results indicate the importance of not only the chemical mechanism, which may be less important in this surface growth dominated combustor, but also the soot statistical model, to which coagulation and the soot surface area are relatively sensitive. [ABSTRACT FROM AUTHOR]

• The failure probability is updated via the posterior moments of the limit-state function. • The limit-state function evaluations and inverse analysis results can be reused. • Only simple arithmetic algorithms by BDRM and PEM are needed for subsequent updates. • The efficiency and accuracy of the proposed method are validated. Reliability updating of structures is computationally demanding, especially when the likelihood function involves inverse analysis and needs to be updated multiple times. In this study, a novel efficient approach for updating the failure probability based on the method of moments and information reuse is proposed. Instead of updating the failure probability using newly obtained posterior distributions, the proposed method transforms the estimate of posterior failure probability into a computation of the reliability index based on the posterior moments of the limit-state function at each update. When calculating these posterior moments, the terms containing the likelihood function are separated from the limit-state function analysis and then the bivariate dimension-reduction method (BDRM) in combination with the point estimate method (PEM) [1,2] is used to conduct the calculation. In this way, the results evaluated in the first updating time can be reused to update the probability of failure in the subsequent updates, avoiding the need to reconduct limit-state function analysis and inverse analysis using the posterior samples, thus improving computational efficiency. Four numerical examples are presented to validate the efficiency and accuracy of the proposed method, and the results indicate that the proposed method has obvious computational advantages for multiple reliability updating problems without loss of accuracy. [ABSTRACT FROM AUTHOR]

A modified robust design optimization approach is presented, which uses the first-order second-moment method to compute the mean value and the standard deviation for arbitrary objective functions. Existing approaches compute the gradient of the variance using the adjoint method, direct differentiation or finite differences, respectively. These approaches either access to the FE-code and/or have high computational cost. In this paper, a new approach for the computation of the gradient of the variance is provided. It can be easily implemented as a non-intrusive method, which behaves similar to finite differences with the cost of only one additional objective evaluation, independent of the number of variables. Here, a step-size has to be chosen carefully and therefore, a procedure to determine a problem-independent step-size is provided. As an alternative, the approach can be implemented as an analytic method with the same cost like the adjoint method, but providing wider applicability (e.g. eigenvalue problems). The provided approach is derived, analyzed and applied to several benchmark examples. [ABSTRACT FROM AUTHOR]

New estimators for the Fréchet distribution based on the method of logarithmic moments are proposed. These are the first estimators for the Fréchet distribution taking closed forms and applicable for all parameter values. Large sample properties of the proposed estimators are derived. The proposed estimators are compared to the maximum likelihood estimators by simulation. [ABSTRACT FROM AUTHOR]

Abstract Accurate estimation of extreme wind speeds for different return periods is necessary to avoid extensive costs or large damages. To achieve this aim, the probability distribution of the wind speed data should be well defined and its parameters should be more precisely estimated. In this study, the commonly used probability distributions, including Gamma, Generalized Extreme Value, Logistic, Lognormal, Normal and Weibull, are fitted to annual maximum wind speed data in Turkey. Parameters of the fitted distributions are estimated using method of moments (MOM), method of maximum likelihood (MLM) and method of probability weighted moments (PWMs). Based on various model selection criterions (Akaike Information Criterion, Bayesian Information criterion, Anderson-Darling, Cramér-von-Mises, and Kolmogorov–Smirnov tests), the Generalized Extreme Value and Logistic, which provided the best fit for 40% and 30% of the series, respectively, were mostly found to be the most suitable distributions. Additionally, the Lognormal, Normal and Gamma distributions showed the best fit for 15%, 10% and 5% of the series, respectively. Moreover, the MLM and PWMs provided better parameter estimations for 57% and 30% the best fitted distributions, respectively. Furthermore, wind speed quantiles with the standard errors in various return periods were estimated using the best fitted distributions. Highlights • Significant trends were observed for the wind speed of the eight (8) stations. • The GEV and LOGIS were mostly found to be the best fitting distributions. • The MLM and PWM provided more accurate parameter estimates than the MOM. • Wind speeds in various return periods were estimated by the best fitted distributions. [ABSTRACT FROM AUTHOR]

Abstract Probabilistic predictions from hydrological models, including rainfall-runoff models, provide valuable information for water and environmental resource risk management. However, traditional "deterministic" usage of rainfall-runoff models remains prevalent in practical applications, in many cases because probabilistic predictions are perceived to be difficult to generate. This paper introduces a simplified approach for hydrological model inference and prediction that bridges the practical gap between "deterministic" and "probabilistic" techniques. This approach combines the Least Squares (LS) technique for calibrating hydrological model parameters with a simple method-of-moments (MoM) estimator of error model parameters (here, the variance and lag-1 autocorrelation of residual errors). A case study using two conceptual hydrological models shows that the LS-MoM approach achieves probabilistic predictions with similar predictive performance to classical maximum-likelihood and Bayesian approaches—but is simpler to implement using common hydrological software and has a lower computational cost. A public web-app to help users implement the simplified approach is available. Highlights • New simplified approach for producing probabilistic hydrological predictions. • Similar performance to maximum-likelihood approach, at lower computational cost. • Web-app available to facilitate uptake of probabilistic predictions. [ABSTRACT FROM AUTHOR]

Abstract A detailed model is proposed for predicting soot formation from complex solid fuels. The proposed model resolves two particle size distributions, one for soot precursors and another for soot particles. The precursor size distribution is represented with a sectional approach while the soot particle-size distribution is represented with the method of moments and an interpolative closure method is used to resolve fractional methods. Based on established mechanisms, this model includes submodels for precursor coagulation, growth, and consumption, as well as soot nucleation, surface growth, agglomeration, and consumption. The model is validated with comparisons to experimental data for two systems: coal combustion over a laminar flat-flame burner and biomass gasification. Results are presented for soot yield for three coals at three temperatures each, and for soot yield from three types of biomass at two temperatures each. These results represent a wide range of fuels and varying combustion environments, demonstrating the broad applicability of the model. [ABSTRACT FROM AUTHOR]

Recently improved knowledge about key processes for the formation and growth of soot particles lead to very extensive soot models describing several kinetic pathways for particle nucleation and growth in detail. One of these models has been proposed by D’Anna and coworkers (D’Anna et al., 2010, Sirignano et al., 2010). In these original studies, the multivariate formulation was solved with a sectional approach, while a more recent study (Salenbauch et al., 2017) also showed the suitability of moment methods such as the Conditional Quadrature Method of Moments (CQMOM) (Yuan et al., 2011). However, being a moment method, CQMOM does not allow to directly access the soot particle size distribution (PSD). This prevents the consistent comparability of CQMOM results to soot measurements based on a scanning mobility particle sizer (SMPS). Furthermore, the comparison to laser-induced fluorescence (LIF) experiments is also limited, as the LIF signal only represents small particles and their concentration is only extractable from the simulation results if the PSD shape is known. The aim of this study is to extend the previously developed CQMOM soot model by a PSD reconstruction step applying the concept of entropy maximization. Maintaining the efficiency of the CQMOM moment inversion algorithm to close the moment equations, the model extension enables to evaluate the diameter-based PSD in a post-processing step without prescribing a specific shape as input. The updated algorithm is applied to simulate soot formation in two different burner-stabilized premixed C 2 H 4 /O 2 /N 2 flames with a very lightly sooting (C/O = 0.67) and a heavily sooting (C/O = 0.77) character. Numerical results are compared to recently published LIF, laser-induced incandescence (LII) and SMPS measurement data. The analysis investigates the model’s capability to predict phenomena such as uni- or bimodal distribution shapes and the transition of small nanostructures to agglomerates in the two target flames both of which exhibit very different sooting behaviour. [ABSTRACT FROM AUTHOR]

The classic method of moment is extended to account the slips of velocity and temperature between the liquid phase and the gas phase in condensing flows. The moments of droplet size distribution are defined on the liquid phase, instead of on the mixture of liquid and gas, so that the gas phase and the liquid phase can be treated separately. The interphase interaction is realized through source terms which include the mass, momentum and energy exchanges. As a demonstration, this new method of moments is integrated into a numerical method and is applied to a vortex shedding problem with condensation. Due to the very strong velocity slip, a dry zone is found in the vortex core region. This phenomenon was experimentally observed but not discovered by the classic moment method. [ABSTRACT FROM AUTHOR]

This paper presents a hybrid method of moments with interpolation closure–Taylor-series expansion method of moments (MoMIC–TEMoM) scheme for solving the Smoluchowski coagulation equation. In the proposed scheme, the exponential function, which arises in the conversion from a particle size distribution space to a space of moments, is expressed in an additive form using the third-order Taylor-series expansion; the implicit moments are approximated using two Lagrange interpolation functions, namely the newly defined normalized moment function and the normalized moment function defined by Frenklach and Harris (1987). The new hybrid scheme allows implementation of the method of moments with an arbitrary type of moment sequence, and it overcomes the shortcomings of the Taylor-series expansion moment method proposed by Frenklach and Harris. The proposed scheme is verified with three aerosol dynamics, namely Brownian coagulation in the free molecular regime, Brownian coagulation in the continuum-slip regime, and turbulence coagulation. The results reveal that the hybrid MoMIC–TEMoM scheme has similar accuracy to currently recognized methods including the quadrature method of moments, MoMIC, and TEMoM, and its accuracy can be further enhanced as the fractional moment sequence type is used for Brownian coagulation in the free molecular regime. Thus, the proposed scheme is a reliable for solving the Smoluchowski coagulation equation. [ABSTRACT FROM AUTHOR]

Graph tries are a new and interesting data structure proposed by Jacquet in 2014. They generalize the classical trie data structure which has found many applications in computer science and is one of the most popular data structure on words. For his generalization, Jacquet considered the size (or space requirement) and derived an asymptotic expansion for the mean and the variance when graph tries are built from n independently chosen random labelings of a rooted M -ary tree. Moreover, he conjectured a central limit theorem for the (suitably normalized) size as the number of labelings tends to infinity. In this paper, we verify this conjecture with the method of moments. [ABSTRACT FROM AUTHOR]

[Display omitted] • An efficient and accurate kinetic model for linear polymer degradation. • A deterministic sensitivity analysis of the parameters for linear polymer degradation. • Both average and distributed properties simulated by the model. • Kinetic insights into alcoholysis of polylactic acid gained through the model. In the field of polymer reaction engineering, accurately simulating polymer degradation process is beneficial to understand the reaction kinetics, which guides the design, operation, and control of the degradation reactor. Herein, a sectional strategy based on the method of moments with a newly defined critical chain length was proposed to simulate the degradation process of linear polymers with low computational cost. By accounting for the concentration of species with low molar mass (e.g., dimer), the evolution of average properties and concentration polymer species during linear polymer degradation were simulated with high accuracy through this model. In addition, the evolution of molar mass distribution was approximated by using the as-developed model. Sensitivity analysis was performed for simultaneous random scission and irreversible/reversible chain-end scission processes. Results show that monomer yields, average and distributed molecular properties can be tuned by the rate coefficients of random scission (k r), chain-end scission (k e), and its reverse reaction (k -e). Finally, kinetic insights into the polylactic acid alcoholysis system were gained by the as-developed MoM-based model, which demonstrates the accuracy and applicability of the model. This work provides a new method for simulating linear polymer degradation, and thus revealing the kinetic evolution during degradation. [ABSTRACT FROM AUTHOR]

A predictive modelling for the simulation of bioreactors must account for both the biological and hydrodynamics complexities. Population balance models (PBM) are the best approach to conjointly describe these complexities, by accounting for the adaptation of inner metabolism for microorganisms that travel in a large-scale heterogeneous bioreactor. While being accurate for solving the PBM, the Class and Monte-Carlo methods are expensive in terms of calculation and memory use. Here, we apply Methods of Moments to solve a population balance equation describing the dynamic adaptation of a biological population to its environment. The use of quadrature methods (Maximum Entropy, QMOM or EQMOM) is required for a good integration of the metabolic behavior over the population. We then compare the accuracy provided by these methods against the class method which serves as a reference. We found that the use of 5 moments to describe a distribution of growth-rate over the population gives satisfactory accuracy against a simulation with a hundred classes. Thus, all methods of moments allow a significant decrease of memory usage in simulations. In terms of stability, QMOM and EQMOM performed far better than the Maximum Entropy method. The much lower memory impact of the methods of moments offers promising perspectives for the coupling of biological models with a fine hydrodynamics depiction. [ABSTRACT FROM AUTHOR]

A method is presented for reducing the required sample sizes for reporting energy savings with predetermined statistical accuracy in lighting retrofit measurement and verification projects, where the population of retrofitted luminaires is to be tracked over time. The method uses a Dynamic Generalised Linear Model with Bayesian forecasting to account for past survey sample sizes and survey results and forecast future population decay, while quantifying estimation uncertainty. A Genetic Algorithm is used to optimise multi-year sampling plans, and distributions are convolved using a new method of moments technique using the Mellin transform instead of a Monte Carlo simulation. Two cases studies are investigated: single population designs, and stratified population designs, where different kinds of lights are replaced in the same retrofit study. Results show significant cost reductions and increased statistical efficiency when using the proposed Bayesian framework. [ABSTRACT FROM AUTHOR]

We propose an algorithm for optimal control of univariate and multivariate population balance systems governed by a class of first-order linear partial differential equation of hyperbolic type involving size dependent particle growth and filtering. In particular, we investigate the impact of filtering on the optimal control outcomes. To this end, we apply a recently developed polynomial method of moments and a standard steepest descent gradient-based optimization scheme to batch crystallization benchmark problems involving external fines removal. Numerical evaluations for various case-studies aiming at minimizing the mass of grown nuclei in the context of crystal shape manipulation demonstrate the effectiveness of fines dissolution. We also validate the accuracy of the proposed method of moments by utilizing a known numerical moving grid discretization method as a reference. Our results are of interest for the control of a wide range of population balance systems in various applications such as pharmaceutics, chemical engineering, particle shape engineering. [ABSTRACT FROM AUTHOR]

Microemulsion systems may enable innovative and highly efficient synthesis paths for the chemical industry. When expensive catalysts are involved in the process, an efficient separation of products together with an extremely low catalyst leaching are of utmost importance. In order to optimize the separation process and improve the settler design, a thorough understanding of droplet-droplet interactions is required. In this work, computational fluid dynamics (CFD) has been used to study the multiphase flow in a horizontal settler using the Eulerian two-fluid framework. Droplet coalescence, which is the most important physical phenomenon in gravity-driven separation, has been modeled using a spatially inhomogeneous buoyancy-based kernel, and the resulting population balance equation has been solved using the quadrature-based method of moments (QMOM). After calibrating the model parameters using selected experiments, a very good description of the observed separation is obtained, opening the door for process understanding and optimization. [ABSTRACT FROM AUTHOR]

The stagnation flame synthesis of titanium dioxide nanoparticles from titanium tetraisopropoxide (TTIP) is modelled based on a simple one-step decomposition mechanism and one-dimensional stagnation flow. The particle model, which accounts for nucleation, surface growth, and coagulation, is fully-coupled to the flow and the gas phase chemistry and solved using the method of moments with interpolative closure (MoMIC). The model assumes no formation of aggregates considering the high temperature of the flame. In order to account for the free-jet region in the flow, the computational distance, H = 1.27 cm, is chosen based on the observed flame location in the experiment (for nozzle-stagnation distance, L = 3.4 cm). The model shows a good agreement with experimentally measured mobility particle size for stationary stagnation surface with varying TTIP loading, although the particle geometric standard deviation, GSD, is underpredicted for high TTIP loading. The particle size is predicted to be sensitive to the sampling location near the stagnation surface in the modelled flame. The sensitivity to the sampling location is found to increase with increasing precursor loading and stagnation temperature. Lastly, the effect of surface growth is evaluated by comparing the result with an alternative reaction model. It is found that surface growth plays an important role in the initial stage of particle growth which, if neglected, results in severe underprediction of particle size and overprediction of particle GSD. [ABSTRACT FROM AUTHOR]

An approximate scheme that permits a much simplified application of method of moments to the study of aerosol coagulation and gelation processes has been proposed and verified in this work. The central idea underlying the scheme is to express a fractional moment in terms of its adjacent integer moments. In this manner the moment equations are then decoupled, and can be solved separately and analytically. The crucial rules for approximating fractional moments are proposed. A method of moments using the new scheme is applied to the study of aerosol coagulation characterized by various collision kernels, including Brownian kernels in the free-molecular and continuum regime and parametric gelling kernel. The computational accuracy of the proposed scheme is validated with a sectional model. The simulational results are extensively compared with previously published results. [ABSTRACT FROM AUTHOR]

In this work, a detailed kinetic mechanism for soot formation and evolution (D'Anna et al. 2010) is implemented in a method of moments framework using the concept of ‘Conditional Quadrature Method of Moments’ (CQMOM) (Yuan and Fox, 2011). Particle nucleation and growth pathways are described in detail tracking moments of the number density function (NDF) for large PAHs, soot clusters and agglomerates in both stable and radical forms. Individual entities of these groups are characterized by both their size and their H/C ratio, the latter providing information on the chemical structure and the reactivity of the particles. This leads to a multivariate population balance statement. The CQMOM implementation of the soot model is then applied to predict particle formation in a lightly sooting burner-stabilized premixed ethylene flame with an equivalence ratio of ϕ = 2.1. Several experimental data sets for this target flame are available in the literature and they are used in this study for comparison with the simulation results. The suitability of the soot model to predict particle matter under such conditions is shown, as both the onset of soot formation by nucleation and the subsequent growth are captured. Modeling results offer further insights into the relevance of agglomeration and dehydrogenation in this lightly sooting flame. [ABSTRACT FROM AUTHOR]

Modeling the oxidation of soot particles in flames is a challenging topic both from a chemical point of view and regarding the statistical treatment of the evolution of the soot number density function (NDF). The method of moments is widely-used for the statistical modeling of aerosol dynamics in various applications, and a number of different moment methods have been established and successfully applied to the modeling of soot formation and growth. However, a shortcoming of existing moment methods is the lack of an accurate, numerically robust, and computationally efficient way to treat soot oxidation, especially regarding the prediction of the particle number density. In this work, the recently developed Extended Quadrature Method of Moments (EQMOM) is integrated with a physico-chemical soot model and combined with a treatment for particle removal by oxidation. This leads to a modeling framework for the simulation of coupled inception, growth, coagulation, and oxidation of soot in flames. In EQMOM, the moment equations are closed by reconstructing the soot NDF with a superposition of continuous kernel functions. Various standard distribution functions can be used as kernel functions, and the algorithm has been implemented here using gamma and lognormal distributions. It is shown that and discussed why gamma distributions are more suitable as kernel functions than lognormal distributions in order to accurately predict soot oxidation. The integrated model is validated by comparisons with analytical solutions for the NDF, results from Monte Carlo simulations of soot formation and oxidation in flames, and experimental data. [ABSTRACT FROM AUTHOR]

Conventional methods of SiPM characterization are mostly based on a resolved spectrum of photoelectron or dark electron detection events which represents a random number of fired SiPM cells. However, such methods are not applicable for unresolved spectra caused, for example, by a low single electron response to noise ratio or a high dark count rate, for example of irradiated SiPMs. This challenge is anticipated in modern developments of radiation-intense facilities like CERN CMS and applications of large-area SiPMs and SiPM arrays with noisy readout like Cherenkov Telescope Array. The method of statistical moments is a well-known technique for the estimation of parameters of a random variable distribution using sample moments of raw data. The first two moments have already been used for estimation of the gain and the number of photoelectrons by the mean and the variance of detected charges relying on a priori knowledge of the Excess Noise Factor of the SiPM or PMT. The third moment or skewness of the measured charge can also be involved to estimate three main SiPM parameters including crosstalk. The skewness-based characterization presumes Generalized Poisson distribution of fired cells which was found to be in good agreement with many experiments. Initial theoretical and experimental evaluations of the skewness method have shown promising results. This study focuses on the feasibility of using such method to characterize the parameters of SiPM with unresolved charge-amplitude spectrum. [Display omitted] • The method estimates a mean number of primary fired cells, gain, and crosstalk. • The method features simplest processing of raw data, acceptable accuracy, and immunity to noises. • The method is expected to be useful for mass testing and large-scale applications. [ABSTRACT FROM AUTHOR]

Background The cost of pharmaceuticals dosed by weight or body surface area (BSA) can be estimated in several ways for economic evaluations. A review of 20 recent National Institute for Health and Care Excellence appraisals showed that 17 of them took the mean weight or BSA of patients, 2 costed the individual patient data from trials, and 2 fitted a distribution to patient-level data. Objectives To investigate the estimated drug costs using different methodologies to account for patient characteristics for pharmaceuticals with a weight- or BSA-based posology. The secondary objective was to explore the suitability of general population data as a proxy for patient-level data. Methods Patient-level data were pooled from three clinical trials and used to calculate a hypothetical cost per administration of eight licensed pharmaceuticals, applying the three methods used in recent National Institute for Health and Care Excellence appraisals. The same analysis was performed using data from the Health Survey for England (in place of patient-level data) to investigate the validity of using general population data as a substitute for patient-level data. Results Compared with using patient-level data from clinical trials, the mean patient characteristics (weight or BSA) led to an underestimation of drug cost by 6.1% (range +1.5% to −25.5%). Fitting a distribution to patient-level data led to a mean difference of +0.04%. All estimates were consistent using general population data. Conclusions Estimation of drug costs in health economic evaluation should account for the distribution in weight or BSA to produce accurate results. When patient data are not available, general population data may be used as an alternative. [ABSTRACT FROM AUTHOR]

The two-phase flow dynamics involving mass transfer and coalescence is investigated. The model is specifically developed to describe the dynamics of bubble population dispersed in glass forming liquids. The amounts of gas dissolved in the liquid are determined using the chemical equilibrium involving oxidation–reduction reactions. The gravitational bubble rising is used to write the coalescence kernel for which a collision efficiency is also introduced. Based on a Direct Quadrature Method of Moments (DQMOM), a numerical method is developed. This numerical tool is applied to melting of borosilicate glass beads for which temperature and residence time of the sample in a crucible are investigated. The bubble density decreases sharply at short times. This early stage decrease is well explained and quantified when the coalescence is taken into account in numerical computations. The bubble size density is very well described with a log-normal distribution. Using the first three moments, the bubble size distribution obtained numerically is in good agreement with experimental data. Numerical computations are also applied to soda-lime-silica glass in which the bubble release is driven by the mass transfer between the two phases. The faster decrease of bubble density than would be expected by temperature is reproduced by the numerical computation. The enhancement of the bubble release rate is mainly due to the increase of dissolved gas species with temperature. [Display omitted] • A population balance equation on the bubble dynamics in a molten glass is developed. • A Direct Quadrature Method of Moments is applied to a glass melting in a crucible. • The sharp decrease of bubble density is explained by the coalescence process. • The experimental bubble size densities follow a log-normal distribution. • The numerical results allow to rebuild the bubble size distribution. [ABSTRACT FROM AUTHOR]