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Many dynamic problems in economics are characterized by large state spaces which make both computing and estimating the model infeasible. We introduce a method for approximating the value function of high-dimensional dynamic models based on sieves and establish results for the (a) consistency, (b) rates of convergence, and (c) bounds on the error of approximation. We embed this method for approximating the solution to the dynamic problem within an estimation routine and prove that it provides consistent estimates of the modelik’s parameters. We provide Monte Carlo evidence that our method can successfully be used to approximate models that would otherwise be infeasible to compute, suggesting that these techniques may substantially broaden the class of models that can be solved and estimated.

To develop a new computational tool for predicting failure probability of structural/mechanical systems subject to random loads, material properties, and geometry.

High dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for capturing the high‐dimensional relationships between sets of input and output model variables. It is a very efficient formulation of the system response, if higher order variable correlations are weak and if the response function is dominantly of additive nature, allowing the physical model to be captured by the first few lower order terms. But, if multiplicative nature of the response function is dominant then all right hand side components of HDMR must be used to be able to obtain the best result. However, if HDMR requires all components, which means 2^N number of components, to get a desired accuracy, making the method very expensive in practice, then factorized HDMR (FHDMR) can be used. The component functions of FHDMR are determined by using the component functions of HDMR. This paper presents the formulation of FHDMR approximation of a multivariate limit state/performance function, which is dominantly of multiplicative nature. Given that conventional methods for reliability analysis are very computationally demanding, when applied in conjunction with complex finite element models. This study aims to assess how accurately and efficiently HDMR/FHDMR based approximation techniques can capture complex model output uncertainty. As a part of this effort, the efficacy of HDMR, which is recently applied to reliability analysis, is also demonstrated. Response surface is constructed using moving least squares interpolation formula by including constant, first‐order and second‐order terms of HDMR and FHDMR. Once the response surface form is defined, the failure probability can be obtained by statistical simulation.

Results of five numerical examples involving structural/solid‐mechanics/geo‐technical engineering problems indicate that the failure probability obtained using FHDMR approximation for the limit state/performance function of dominantly multiplicative in nature, provides significant accuracy when compared with the conventional Monte Carlo method, while requiring fewer original model simulations.

This is the first time where application of FHDMR concepts is explored in the field of reliability and system safety. Present computational approach is valuable to the practical modeling and design community, where user often suffers from the curse of dimensionality.

In acceptance sampling, the hypergeometric operating characteristic (OC) function (so called type-A OC) is used to be approximated by the binomial or Poisson OC function, which actually reduce computational effort, but do not provide suffcient approximation results. The purpose of this paper is to examine binomial- and Poisson-type approximations to the hypergeometric distribution, in order to find a simple but accurate approximation that can be successfully applied in acceptance sampling.

The authors present a new binomial-type approximation for the type-A OC function, and derive its properties. Further, the authors compare this approximation via an extensive numerical study with other common approximations in terms of variation distance and relative efficiency under various conditions on the parameters including limiting cases.

The introduced approximation generates best numerical results over a wide range of parameter values, and ensures arithmetic simplicity of the binomial distribution and high accuracy to meet requirements regarding acceptance sampling problems. Additionally, it can considerably reduce the computational effort in relation to the type-A OC function and therefore is strongly recommended for calculating sampling plans.

The newly presented approximation provides a remarkably close fit to the type-A OC function, is discrete and needs no correction for continuity, and is skewed in the same direction by roughly the same amount as the exact OC. Due to less factorials, this OC in general involves lower powers than the type-A OC function. Moreover, the binomial-type approximation is easy to fit to the conventional statistical computing packages.

High‐dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for capturing the high‐dimensional relationships between sets of input and output model variables. It is an efficient formulation of the system response, if higher‐order cooperative effects are weak, allowing the physical model to be captured by the lower‐order terms. The paper's aim is to develop a new computational tool for estimating probabilistic sensitivity of structural/mechanical systems subject to random loads, material properties and geometry.

When first‐order HDMR approximation of the original high‐dimensional limit state is not adequate to provide the desired accuracy to the sensitivity analysis, this paper presents an enhanced HDMR (eHDMR) method to represent the higher‐order terms of HDMR expansion by expressions similar to the lower‐order ones with monomial multipliers. The accuracy of the HDMR expansion can be significantly improved using preconditioning with a minimal number of additional input‐output samples without directly invoking the determination of second‐ and higher‐order terms. As a part of this effort, the efficacy of HDMR, which is recently applied to uncertainty analysis, is also demonstrated. The method is based on computing eHDMR approximation of system responses and score functions associated with probability distribution of a random input. Surrogate model is constructed using moving least squares interpolation formula. Once the surrogate model form is defined, both the probabilistic response and its sensitivities can be estimated from a single probabilistic analysis, without requiring the gradients of performance functions.

The results of two numerical examples involving mathematical function and structural/solid‐mechanics problems indicate that the sensitivities obtained using eHDMR approximation provide significant accuracy when compared with the conventional Monte Carlo method, while requiring fewer original model simulations.

This is the first time where application of eHDMR concepts is explored in the stochastic sensitivity analysis. The present computational approach is valuable to the practical modelling and design community.

The mathematical complexity of the B_J(x) Brillouin function makes it unsuitable for most calculations and its application difficult for computer programming in magnetism. Here, its approximation with the tanh function is proposed to ease the mathematical operations for most cases. The approximation works with good accuracy, acceptable in practical calculations. This approximation has already formed the foundation of the “hyperbolic model” in magnetism for the study of hysteretic phenomena. The reversal of the Brillouin function is an important but difficult mathematical problem for practical purposes. Here, a proposal has been put forward for an easy approximation using an analytical expression. This provides a good workable solution for the B_J(x)⁻¹ function dependent on J, the angular momentum quantum number of the material used. The proposed approximation is applicable within the working range of practical applications. The paper aims to discuss these issues.

The multi-variant Brillouin function is closely approximated by the tanh function to ease calculations. Its mathematically unsolved reversed function is approximated by a simple analytical expression with a good working accuracy.

The Brillouin function and its reversal can be approximated for practical users mostly for professionals working in Magnetism.

Most if not all practical problems in Magnetism can be solved within the limitations of the two approximations.

Both proposed functions can ease the mathematical problems faced by researchers and other users in Magnetism.

Ease the frustration of most users working in the field of Magnetism.

The application of the tanh function for replacing the Brillouin function led to the creation of the hyperbolic model of hysteresis. To the author's knowledge, the reverse function was mathematically only solved in 2015 with a vastly complicated mathematics, and is hardly suitable for practical calculations in Magnetism. The proposed simple expression can be very useful for theorists and experimental scientists.

The investigation of the efficiency of optimisation technique based on approximation of objective function by multiquadric (MQ) function, used for induction heating devices was the aim of the paper.

The optimisation package based on Matlab language and using Flux2D commercial program for calculation of electromagnetic and thermal fields was built. It allows the use of different optimisation techniques for induction heating devices, e.g. based on MQ function approximation. In the paper two algorithms of approximated points generating have been tested.

The efficiency of MQ optimisation method strongly depends on the applied algorithm of approximated point generating. To ensure high efficiency of MQ optimisation method, the stochastic element of the algorithm of approximated point generating should have a significant role.

The efficiency of elaborated algorithms of MQ function approximated point generating should be proved in other applications.

The efficient optimisation technique of induction heating devices has been proposed.

The two new algorithms for generation of MQ function approximated points have been proposed. The paper could be useful for designers of induction heating devices.

Global optimization in electrical engineering using stochastic methods requires usually a large amount of CPU time to locate the optimum, if the objective function is calculated either with the finite element method (FEM) or the boundary element method (BEM). One approach to reduce the number of FEM or BEM calls using neural networks and another one using multiquadric functions have been introduced recently. This paper compares the efficiency of both methods, which are applied to a couple of test problems and the results are discussed.

This paper compares three different radial basis function neural networks, as well as the diffuse element method, according to their ability of approximation. This is very useful for the optimization of electromagnetic devices. Tests are done on several analytical functions and on the TEAM workshop problem 25.

The purpose of this paper is to present a constructive algorithm to design multilayer perceptron neural networks used as approximation models of electromagnetic devices.

The proposed procedure allows automatic determination of both the number of neurons and the synaptic weights of networks with a single hidden layer. The approximation model is used in design optimization problems. The inputs of the neural network correspond to the design parameters whereas the output corresponds to the objective function of the optimization problem. The neural model is then inverted in order to determine which input is associated to a prefixed output.

The performance of the algorithm has been tested on analytical function and on the TEAM workshop problem 25.

As the reliability of the optimum solution is strongly affected by the accuracy of the neural approximation model, the approximation error is kept as low as possible, especially in the maximum/minimum points.