Proposed Approximation Method Research Articles

Quasi-P-Wave Reverse Time Migration in TTI Media with a Generalized Fractional Convolution Stencil

In seismic modeling and reverse time migration (RTM), incorporating anisotropy is crucial for accurate wavefield modeling and high-quality images. Due to the trade-off between computational cost and simulation accuracy, the pure quasi-P-wave equation has good accuracy to describe wave propagation in tilted transverse isotropic (TTI) media. However, it involves a fractional pseudo-differential operator that depends on the anisotropy parameters, making it unsuitable for resolution using conventional solvers for fractional operators. To address this issue, we propose a novel pure quasi-P-wave equation with a generalized fractional convolution operator in TTI media. First, we decompose the conventional pure quasi-P-wave equation into an elliptical anisotropy equation and a fractional pseudo-differential correction term. Then, we use a generalized fractional convolution stencil to approximate the spatial-domain pseudo-differential term through the solution of an inverse problem. The proposed approximation method is accurate, and the wavefield modeling method based on it also accurately describes quasi-P-wave propagation in TTI media. Moreover, it only increases the computational cost for calculating mixed partial derivatives compared to those in vertical transverse isotropic (VTI) media. Finally, the proposed wavefield modeling method is utilized in RTM to correct the anisotropic effects in seismic imaging. Numerical RTM experiments demonstrate the flexibility and viability of the proposed method.

Saddlepoint approximations for the P-values and probability mass functions of some bivariate sign tests

Bivariate analysis is essential in several applied fields, such as medicine, engineering, biology, and econometrics. Many parametric and non-parametric procedures can be used for bivariate analysis. Non-parametric procedures require fewer conditions than their parametric counterparts. Therefore, this article proposes an accurate approximation of the p-values and the probability mass functions of some non-parametric bivariate tests, including bivariate sign tests and medians. The proposed approximation, saddlepoint approximation, is compared to the traditional approximation method, asymptotic normal approximation method, and Edgeworth expansion through three real data examples and a simulation study. The numerical results show that the proposed approximation method is more accurate than the asymptotic method and Edgeworth expansion. Furthermore, it is computationally less demanding than the simulation method, which is permutation-based and time-consuming.

Probabilistic Reachability Prediction of Unbounded Petri Nets: A Machine Learning Method

Unbounded Petri nets (UPNs) can describe and analyze discrete event systems with infinite states (DESIS). Due to the infinite state space and the combination explosion problem, the reachability analysis of UPNs is an NP-Hard problem. The existing reachability analysis methods cannot achieve an accurate result at reasonable costs (computational time and space) due to the finite reachability tree with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\omega$</tex-math> </inline-formula> -numbers. Based on the idea of approximating infinite space with finite states, given some limited reachable markings of a UPN, we propose a method that can quantitatively solve the UPN’s reachability problem with machine learning. Firstly, we define the probabilistic reachability of markings and transform the UPN’s reachability problem into the prediction problem of markings. The proposed method based on positive and unlabeled learning (PUL) and bagging trains a classifier to predict the probabilistic reachability of unknown markings. Finally, to predict the markings outside the positive sample set and unlabeled sample set, an iterative strategy is designed to update the classifier. Based on seven general UPNs, the results of the experiments show that the proposed method has a good performance in the accuracy and time consumption for the UPN’s reachability problem. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —In discrete event systems, the reachability problem mainly studies reachable states of the system and the relationship between states, which is the basis of the system’s states, behaviors, attributes and performance analysis. For discrete event systems with infinite states, it is hard to analyze the reachable relationship between states within a finite time due to the infinite state space and the combination explosion problem. The main motivation of the paper is to propose a method that can predict the reachable relationship between the states with a probability value within a finite time. By machine learning algorithms, the method learns the feature information of the known reachable states. The reachability of unknown states in the infinite state space can be predicted approximately. The proposed approximation method can be applied to analyze the reachability properties of general discrete event systems with infinite states, such as checking whether a fault occurs in operating systems, whether a message is delivered in communication and so on.

Parameter Determination Method in Analysis of Stratum Deformation Caused by Tunnel Excavation

Based on the analytical method of formation deformation caused by tunnel excavation proposed by Verruijt and Sagaseta, combined with the basic principles of soil mechanics and elastic mechanics, a method for determining the pending parameters required in the analysis process is proposed. This method does not require the actual measurement of deformation values in the field, and the required parameters can be determined from geological data, so as to obtain the deformation values of any points of the whole stratum. According to engineering examples, when the pending parameters are obtained by this method, the difference between the resolved value of maximum surface settlement and the measured value can be controlled within 5.0%, and the analytical results can splendidly reflect the formation deformation law caused by tunnel excavation. Further combined with the analysis of the numerical simulation results, the proposed approximation method has certain shortcomings and deficiencies, mainly in the lack of lateral constraints on the displacement. When the measured data in the field are insufficient, this method can be used to estimate the deformation, and the analysis results are slightly larger than the measured values, which is safer.

An integer programming model for the deployment of mobile health clinics

Facility location and resource allocation play an essential role in the operational management of mobile health clinics (MHCs) as optimization of both can help medical service providers improve their service coverage under limited resources. In this paper, we consider the issue of maximizing the coverage of immunization services provided by MHCs in vulnerable communities such as school zones and census tracts. Since demand in communities overlaps with neighboring communities, this leads to the so-called mobile facility location with overlapped demand (MFL-OD). To capture the overlapped demand in different communities, we introduce a novel integer programming model with linear and bilinear constraints (BLCIP) and show that the BLCIP model is NP-hard. A polynomial-time approximation algorithm is proposed for the BLCIP, and the approximation ratio of the obtained solution is estimated. We implement the proposed model and approximation algorithm for a case study in the Houston Independent School District and its associated communities. Preliminary experiments demonstrate the efficacy of the proposed model and the efficiency of the proposed approximation method.

ReAFM: A Reconfigurable Nonlinear Activation Function Module for Neural Networks

Deep neural networks (DNNs) with various nonlinear activation functions (NAFs) have achieved unprecedented successes, sparking interest in efficient DNN hardware implementation. However, most existing NAF implementations focus on one type of functions with dedicated architectures, which is not suited for supporting versatile DNN accelerators. In this brief, based on a proposed reciprocal approximation optimization (RAO) method, an efficient reconfigurable nonlinear activation function module (ReAFM) is devised to implement various NAFs. The computational logic and dataflow of certain NAFs are merged and reused to minimize hardware consumption by leveraging the correlations among different NAFs. In addition, a precision adjustable exponential unit (PAEU) is developed to obtain a good tradeoff between the approximation accuracy and hardware cost. Compared to the prior art, the experimental results demonstrate that the proposed ReAFM can support many more NAF types with comparable or even better performance. Furthermore, evaluation results on some prevalent neural networks show that the proposed approximation method causes negligible accuracy loss (< 0.1%).

Statistical Inference of the Class of Nonparametric Tests for the Panel Count and Current Status Data from the Perspective of the Saddlepoint Approximation

Many statisticians resort to using the asymptotic normal approximation method to carry out statistical inference for many statistical tests, especially nonparametric ones. In this article, the saddlepoint approximation method is proposed as an alternative to the asymptotic normal approximation method to carry out statistical inference for a number of nonparametric tests for an important type of data that appears frequently in many clinical studies such as cancer and tumorigenicity studies. In clinical trials, there are many strategies through which treatments are assigned to patients. Equal allocation of both treatments is a largely prevalent approach in clinical trials to eliminate experimental bias and increase power. Accordingly, the statistical analysis is carried out based on the truncated binomial design, which is one of the designs that achieve a perfect balance between the two treatments. To clarify the accuracy of the proposed approximation method, two sets of real data are analyzed, and for the same purpose, a comprehensive simulation study is carried out.

Consistency of Approximation of Bernstein Polynomial-Based Direct Methods for Optimal Control

Bernstein polynomial approximation of continuous function has a slower rate of convergence compared to other approximation methods. “The fact seems to have precluded any numerical application of Bernstein polynomials from having been made. Perhaps they will find application when the properties of the approximant in the large are of more importance than the closeness of the approximation.”—remarked P.J. Davis in his 1963 book, Interpolation and Approximation. This paper presents a direct approximation method for nonlinear optimal control problems with mixed input and state constraints based on Bernstein polynomial approximation. We provide a rigorous analysis showing that the proposed method yields consistent approximations of time-continuous optimal control problems and can be used for costate estimation of the optimal control problems. This result leads to the formulation of the Covector Mapping Theorem for Bernstein polynomial approximation. Finally, we explore the numerical and geometric properties of Bernstein polynomials, and illustrate the advantages of the proposed approximation method through several numerical examples.

Improved constant phase fractional order approximation method for induction motor FOPI speed controller

SummaryThis paper presents the digital realization of the fractional‐order PI (FOPI) controller fractional integrator element using the improved constant phase approximation method. A new pole‐zero rationalization approach is proposed within the bandwidth of the Bode phase plot. The voltage source inverter fed induction motor fractional‐order model has been identified. A novel mathematical FOPI controller design procedure for the identified FO plant model has been proposed. The designed FOPI controller fractional integrator element is implemented using the constant phase approximation method. A detailed comparative performance study of the proposed approximation method with the existing Charef, Oustaloup, and refined Oustaloup methods has been carried out using the Bode, Nyquist, and Nichols plots. The proposed FOPI controller is placed in the speed feedback loop of the rotor field‐oriented control algorithm of IM. To confirm the proposed FOPI controller approximation method's robustness, a hardware study is carried out on the laboratory hardware‐in‐loop (HIL) platform. In addition, variation in the number of pole‐zero pair's effect on the speed tracking performance, robustness against parameter and load variations are also analyzed.

Vibration control and calculating inverse dynamics of the rigid-flexible two-link manipulator T-R

The dynamics and control of flexible link manipulators (FLMs) are increasing in industrial robotics research. In which the problem of inverse dynamics of the flexible link manipulators is a problem that has been studied little and has no general solution. In this paper, an approximation method is presented to calculate the reverse dynamics of the serial manipulators with rigid-flexible links. The linearization of the motion equations for a rigid-flexible translation and rotation two-link manipulator (manipulator T-R) is addressed. The vibration control and calculating inverse dynamics of a periodic rigid-flexible two-link manipulator T-R are studied. The Taguchi method is used for the design of gain values of the controller PD for the manipulator. The results of numerical simulation show the efficiency and usefulness of the proposed approximation method.

Approximation of 3D trapezoidal fuzzy data using radial basis functions

We present a new methodology to approximate a trapezoidal fuzzy numbers set by using smoothing radial basis functions (RBFs). The methodology uses different error and similarity indices to determine and compare the accuracy of the approximation of the given trapezoidal fuzzy data. For the proposed approximation method a fuzzy radial basis functions type are defined, called fuzzy smoothing radial basis functions under tension. The computation of one of these approximation functions from a given trapezoidal fuzzy data set is described and some convergence results are proved. Finally, some examples in two-dimensions are given to compare the behavior of the presented method by using the proposed error and similarity indices for different configurations of the fuzzy smoothing radial basis functions under tension.

Derivation of the Empirical Bayesian method for the Negative Binomial-Lindley generalized linear model with application in traffic safety

The expected crash frequency is the long-term average crash count for a specific site. It is extensively used to systematically evaluate the crash risk associated with roadway elements. To estimate the expected crashes, the Empirical Bayesian (EB) approach is typically employed. The EB method is a computationally convenient approximation to the Full Bayesian (FB) method, which gained popularity due to its simple interpretation, computational efficiency, and the ability to account for the regression to the mean bias. However, the common EB method used in traffic safety analysis is only applicable when the traditional Negative Binomial (NB) model is used. The NB model, however, is not a suitable choice when data is highly dispersed, skewed, or has a large number of zero observations. The Negative Binomial-Lindley (NB-L) model is a mixture of the NB and Lindley distributions and has shown superior fit compared to the NB model, especially when the dataset is characterized by excess zero observations. Even though several studies have used the NB-L in developing crash prediction models, the application of the NB-L in other safety-related tasks (e.g., hot spot identification) is largely neglected. This study proposed a framework to develop the EB method for the NB-L model and subsequently estimate the expected crash values. A comparison between the EB and FB estimates was performed to validate the approximation framework in general. The results indicated that the proposed EB framework is able to estimate expected crashes with comparable precision to the FB estimate, but with much less computational cost. In addition, a site ranking analysis using the EB estimates was conducted to validate the proposed approximation method in safety studies. However, it should be noted that any other type of safety analysis that requires access to the expected crashes can benefit from the proposed EB method. This study concluded that the proposed EB framework can properly approximate the underlying FB approach and can reasonably be considered as an alternative to the traditional EB formula derived from the NB model. The results of this study can help to extend the application of the advanced predictive models beyond predicting crashes to other safety-related tasks, with no additional computational efforts.

Local Modal Frequency Improvement with Optimal Stiffener by Constraints Transformation Method

Local modal vibration could adversely affect the dynamical environment, which should be considered in the structural design. For the mode switching phenomena, the traditional structural optimization method for problems with specific order of modal frequency constraints could not be directly applied to solve problems with local frequency constraints. In the present work, a novel approximation technique without mode tracking is proposed. According to the structural character, three reasonable assumptions, unchanged mass matrix, accordant modal shape, and reversible stiffness matrix, have been used to transform the optimization problem with local frequency constraints into a problem with nodal displacement constraints in the local area. The static load case is created with the modal shape equilibrium forces, then the displacement constrained optimization is relatively easily solved to obtain the optimal design, which satisfies the local frequency constraints as well. A numerical example is used to verify the feasibility of the proposed approximation method. Then, the method is further applied in a satellite structure optimization problem.

Generating fractal rough surfaces with the spectral representation method

The application of the spectral representation method in generating Gaussian and non-Gaussian fractal rough surfaces is studied in this work. The characteristics of fractal rough surfaces simulated by the spectral representation method and the conventional Fast Fourier transform filtering method are compared. Furthermore, the fractal rough surfaces simulated by these two methods are compared in the simulation of contact and lubrication problems. Next, the influence of low and high cutoff frequencies on the normality of the simulated Gaussian fractal rough surfaces is investigated with roll-off power spectral density and single power-law power spectral density. Finally, a simple approximation method to generate non-Gaussian fractal rough surfaces is proposed by combining the spectral representation method and the Johnson translator system. Based on the simulation results, the current work gives recommendations on using the spectral representation method and the Fast Fourier transform filtering method to generate fractal surfaces and suggestions on selecting the low cutoff frequency of the power-law power spectral density. Furthermore, the results show that the proposed approximation method can be a choice to generate non-Gaussian fractal surfaces when the accuracy requirements are not high. The MATLAB codes for generating Gaussian and non-Gaussian fractal rough surfaces are provided.

Mean field approximations to a queueing system with threshold-based workload control scheme

In this paper, motivated by considerations of server utilization and energy consumption in cloud computing, we investigate a homogeneous queueing system with a threshold-based workload control scheme. In this system, a virtual machine will be turned off when there are no tasks in its buffer upon the completion of a service by the machine, and turned on when the number of tasks in its buffer reaches a pre-set threshold value. Due to complexity of this system, we propose approximations to system performance measures by mean field limits. An iterative algorithm is suggested for the solution to the mean field limit equations. In addition, numerical and simulation results are presented to justify the proposed approximation method and to provide a numerical analysis on the impact of the system performances by system parameters, which suggests that the proposed system saves about % of the energy consumption compared to the ordinary supermarket models.

Pricing VIX Options with Realized Volatility

It is well known that realized measures of volatility, which are computed from high-frequency intraday data, provide accurate measurements of the latent volatility process. We investigate the role of realized volatility in pricing VIX options using the GARV model (Christoffersen et al., 2014), and the Realized GARCH model (Hansen et al., 2012). A closed-form pricing formula for the (affine) GARV model is developed. For the (non-affine) log-linear Realized GARCH model, we introduce a novel approximation approach to derive its analytical pricing formula. We show that the newly proposed approximation method is fast, with a high degree of accuracy. An extensive empirical application on VIX options from 2006 to 2020 shows that models with realized volatility significantly outperform conventional GARCH-type models based on daily returns only. Among these, the Realized GARCH model provides the best pricing performance due to its fewer constraints and a more flexible modeling structure. Our results hold both in-sample and out-of-sample.

Convergence results for a zero of the sum of a finite family of maximal monotone mappings in Banach spaces

ABSTRACT The purpose of this paper is to study the method of approximation for zeros of the sum of a finite family of maximal monotone mappings in the setting of Banach spaces. Under some mild conditions, we establish strong convergence results of the proposed approximation method. The assumptions that one of the mappings is single valued and α-inverse strongly monotone are dispensed with. In addition, we give some applications to the minimization problems. Finally, we provide a numerical example which supports our main result. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

A strong convergence theorem for a zero of the sum of a finite family of maximally monotone mappings

AbstractThe purpose of this article is to study the method of approximation for zeros of the sum of a finite family of maximally monotone mappings and prove strong convergence of the proposed approximation method under suitable conditions. The method of proof is of independent interest. In addition, we give some applications to the minimization problems and provide a numerical example which supports our main result. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

Completion of right-hand side in the frame of inverse Cauchy problem of elliptic type equation through homogenization meshless collocation method

In this study, the well-known Cauchy problem of elliptic type equation possibly with variable coefficient is contemplated while a part of right-hand side source is unknown as well whereas overspecified boundary data is imposed on boundary. It is a supposition that the right-hand side source can be observed as a sum of two-parts which are independent of each other and at the same time, each of them being in terms of own one-variable. It is proved that such an inverse problem possesses unique solution. To approximate this unique solution, a kind of domain type meshless collocation method is proposed so that the boundary data are imposed directly. This is not troublesome because the original problem, by variable transforming through homogenization function, is converted to an inverse problem with homogeneous Cauchy boundary conditions. This surprisingly diminishes the ill-posedness of the right-hand side construction of Cauchy problem. As a result, it does not require any regularization algorithms and therefore reduces the computational time. The convergence and error analysis of the proposed approximation method is fully discussed. It is worth-mentioning that the considered domain is of arbitrary shape and discussed in the polar coordinate for simplicity and, it does not matter how scattered points are chosen, therefore the method is truly meshless one. The accuracy and robustness of this homogenization meshless collocation method (HMCM) is tested on several numerical examples.

Solving linear parabolic rough partial differential equations

We study linear rough partial differential equations in the setting of [Friz and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear parabolic partial differential equation driven by a deterministic rough path W of Hölder regularity α with 1/3<α≤1/2. Based on a stochastic representation of the solution of the rough partial differential equation, we propose a regression Monte Carlo algorithm for spatio-temporal approximation of the solution. We provide a full convergence analysis of the proposed approximation method which essentially relies on the new bounds for the higher order derivatives of the solution in space. Finally, we present a simulation study showing the applicability of the proposed algorithm.