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This chapter proposes M-estimators of a fractional response model with an endogenous count variable under the presence of time-constant unobserved heterogeneity. To address the endogeneity of the right-hand-side count variable, I use instrumental variables and a two-step procedure estimation approach. Two methods of estimation are employed: quasi-maximum likelihood (QML) and nonlinear least squares (NLS). Using these methods, I estimate the average partial effects, which are shown to be comparable across linear and nonlinear models. Monte Carlo simulations verify that the QML and NLS estimators perform better than other standard estimators. For illustration, these estimators are used in a model of female labor supply with an endogenous number of children. The results show that the marginal reduction in women's working hours per week is less as women have one additional kid. In addition, the effect of the number of children on the fraction of hours that a woman spends working per week is statistically significant and more significant than the estimates in all other linear and nonlinear models considered in the chapter.

This paper aims to investigate the application of adaptive integration in element-free Galerkin methods for solving problems in structural and solid mechanics to obtain accurate reference solutions.

An adaptive quadrature algorithm which allows user control over integration accuracy, previously developed for integrating boundary value problems, is adapted to elasticity problems. The algorithm allows the development of a convergence study procedure that takes into account both integration and discretisation errors. The convergence procedure is demonstrated using an elasticity problem which has an analytical solution and is then applied to accurately solve a soft-tissue extension problem involving large deformations.

The developed convergence procedure, based on the presented adaptive integration scheme, allows the computation of accurate reference solutions for challenging problems which do not have an analytical or finite element solution.

This paper investigates the application of adaptive quadrature to solid mechanics problems in engineering analysis using the element-free Galerkin method to obtain accurate reference solutions. The proposed convergence procedure allows the user to independently examine and control the contribution of integration and discretisation errors to the overall solution error. This allows the computation of reference solutions for very challenging problems which do not have an analytical or even a finite element solution (such as very large deformation problems).

Integral equation methods are very often used for high‐voltage field computation. This paper describes an adaptive Gaussian quadrature method for solving potential and field related integrals. Based on computer‐aided design, the high‐voltage configuration is modelled using bi‐cubic spline functions for the description of the boundary surfaces. This problem definition requires a numerical solution of the governing integral equations. Standard Gauss‐Legendre quadrature is improved by an adaptive approach. The describing surfaces are sub‐divided under distance‐dependant aspects. Hence, different rules of Gaussian quadrature are examined and compared to analytically obtained results. The solution is used to compute the potential and field strength on the electrode or insulator contour. The accuracy and the speed‐up of the new method is demonstrated by several calculations.

In empirical research, panel (and multinomial) probit models are leading examples for the use of maximum simulated likelihood estimators. The Geweke–Hajivassiliou–Keane (GHK) simulator is the most widely used technique for this type of problem. This chapter suggests an algorithm that is based on GHK but uses an adaptive version of sparse-grids integration (SGI) instead of simulation. It is adaptive in the sense that it uses an automated change-of-variables to make the integration problem numerically better behaved along the lines of efficient importance sampling (EIS) and adaptive univariate quadrature. The resulting integral is approximated using SGI that generalizes Gaussian quadrature in a way such that the computational costs do not grow exponentially with the number of dimensions. Monte Carlo experiments show an impressive performance compared to the original GHK algorithm, especially in difficult cases such as models with high intertemporal correlations.

This article presents a locally conservative projection method which aims to preserve the integral of a function and one operator among grad, div, or curl.

After a theoretical description of the projection methods, the locally conservative projection is analytically tested and compared with the orthogonal method. In the second part, the implementation of the methods is described, and improvements are proposed. An industrial application of the present work, consisting in a magneto-thermal coupled problem, is then presented.

The implementation of the conservative method is simpler than the implementation of the orthogonal method while presenting similar behaviour in terms of accuracy and conservation.

The locally conservative method is extended to curl-conform and div-conform elements. Furthermore, three-dimensional studies are proposed.

A fast algorithm is proposed to calculate the lightning electromagnetic field over a perfectly conducting earth surface.

The channel base current is approximated by a number of sub‐domain quadratic functions using the proposed adaptive sampling technique, and the derivative and integral of the channel base current with respect to time can be analytically expressed. With the help of these approximations, the ideal electromagnetic field of the lightning channel can be evaluated along the lightning channel with respect to the height.

The computational time can be greatly reduced using the proposed approach to evaluate the electromagnetic field of a lightning channel in the time domain.

The adaptive sampling technique is a general‐purposed approach, which can be potentially used in other applications to fit a function with the minimal number of intervals.

The paper aims to develop expressions for calculating the mutual impedance between isolated conductors buried in homogeneous earth. The conductors have finite length and arbitrary position.

The conductors are represented by the use of elementary electric dipoles. Well‐known existing expressions are employed for the electric field of these dipoles. The induced voltages are evaluated and the final expressions for the mutual impedance are derived. The resulting expressions involve infinite double integrals, evaluated by using adaptive quadratures that are, however, time consuming. Therefore, an alternative approach is followed involving Sommerfeld integrals (SI) for representing the electric field of a dipole and a recently devised method for computing the SI, in the spatial domain, by using calculation of discrete complex images.

Final expressions for parallel and perpendicular conductors were derived and numerical results for several values of frequency, conductors' length and horizontal distance between them, were produced. Comparison to results produced with the well‐known Pollaczek formula showed excellent agreement.

For future research, it is possible to use the developed expressions for earthing systems study, where the grounding grid is discretized and the moment method is invoked.

Currently, the formulas used for calculating mutual impedance are valid for parallel conductors of infinite length. With the present work, accurate expressions are given for finite length and arbitrary horizontal positioned conductors. In addition, the use of SI and the discrete complex image method results in a rapid and efficient tool for massive impedance calculations.

Bayesian cubature (BC) has emerged to be one of most competitive approach for estimating the multi-dimensional integral especially when the integrand is expensive to evaluate, and alternative acquisition functions, such as the Posterior Variance Contribution (PVC) function, have been developed for adaptive experiment design of the integration points. However, those sequential design strategies also prevent BC from being implemented in a parallel scheme. Therefore, this paper aims at developing a parallelized adaptive BC method to further improve the computational efficiency.

By theoretically examining the multimodal behavior of the PVC function, it is concluded that the multiple local maxima all have important contribution to the integration accuracy as can be selected as design points, providing a practical way for parallelization of the adaptive BC. Inspired by the above finding, four multimodal optimization algorithms, including one newly developed in this work, are then introduced for finding multiple local maxima of the PVC function in one run, and further for parallel implementation of the adaptive BC.

The superiority of the parallel schemes and the performance of the four multimodal optimization algorithms are then demonstrated and compared with the k-means clustering method by using two numerical benchmarks and two engineering examples.

Multimodal behavior of acquisition function for BC is comprehensively investigated. All the local maxima of the acquisition function contribute to adaptive BC accuracy. Parallelization of adaptive BC is realized with four multimodal optimization methods.

In this article we propose a multivariate dynamic probit model. Our model can be viewed as a nonlinear VAR model for the latent variables associated with correlated binary time-series data. To estimate it, we implement an exact maximum likelihood approach, hence providing a solution to the problem generally encountered in the formulation of multivariate probit models. Our framework allows us to study the predictive relationships among the binary processes under analysis. Finally, an empirical study of three financial crises is conducted.

The purpose of this paper is to propose a Chebyshev collocation method (CCM) for Hallén’s equation of thin wire antennas.

Since the current induced on the thin wire antennas behaves like the square root of the distance from the end, a smoothed current is used to annihilate this end effect. Then the CCM adopts Chebyshev polynomials to approximate the smoothed current from which the actual current can be quickly recovered. To handle the difficulty of the kernel singularity and to realize fast computation, a decomposition is adopted by separating the singularity from the exact kernel. The integrals including the singularity in the linear system can be given in an explicit formula while the others can be evaluated efficiently by the fast cosine transform or the fast Fourier transform.

The CCM convergence rate is fast and this method is more efficient than the other existing methods. Specially, it can attain less than 1 percent relative errors by using 32 basis functions when a/h is bigger than 2×10−5 where h is the half length of wire antenna and a is the radius of antenna. Besides, a new efficient scheme to evaluate the exact kernel has been proposed by comparing with most of the literature methods.

Since the kernel evaluation is vital to the solution of Hallén’s and Pocklington’s equations, the proposed scheme to evaluate the exact kernel may be helpful in improving the efficiency of existing methods in the study of wire antennas. Due to the good convergence and efficiency, the CCM may be a competitive method in the analysis of radiation properties of thin wire antennas. Several numerical experiments are presented to validate the proposed method.