Trapezoidal Rule Research Articles

Design of Discrete and Hybrid Nonlinear Control Systems

In this article the new method of discrete control systems design for nonlinear plants with differentiable nonlinearities is suggested. The increasing demands on the quality of control processes and the widespread use of computer technology provide ample opportunities for the design and implementation of digital control systems. However, discrete models of control plants are needed to solve this problem. In the case of linear plants, such models are created on the basis of z-transformation, Euler or Tustin formulas. In the case of nonlinear plants, these transformations are not applicable, so a large number of approximate discretization methods have been developed to date. Euler and Runge-Kutt transformations are used for these purposes most often, but they lead to satisfactory results only with very small period of discretization. In the case of automatic control systems, this requires the use of digital automation tools with very high speed, which is often economically impractical. Methods of discretization with a long period were most often developed on the basis of decomposition into series of the right-hand sides of the differential equations, transformed on Euler. Here, firstly, the problem of selecting the number of the series members, which to be retained arises, and secondly, already in the third or fourth order of the plant, the calculating ratios turn out to be extremely complex. The discretization method suggested below differs in that it is not the equations of nonlinear plants in the Cauchy form that are discretized, but the corresponding quasilinear model. In this case, a modified trapezoid method is used, and the discretization purpose is not the most accurate approximation of the original equations of the plant, but the stability of a closed nonlinear control system with rather big period. This system is designed using the algebraic polynomial-matrix method for designing of the nonlinear control systems. As a result, a hybrid nonlinear system with fairly simple algebraic calculation expressions is formed. The suggested approach makes it possible to create the control systems for nonlinear controlled plants using conventional computational automation tools.

Symmetric trapezoidal approximations of fuzzy numbers under a general condition

Symmetric trapezoidal approximations of fuzzy numbers under a general condition

Design of 10 kW, 400 kHz AC Power Supply for Ferrite Inductively Coupled Plasma

This article describes the design and implementation of a 10 kW, 400 kHz ac power supply for a ferrite inductively coupled plasma (ICP) reactor. The developed power supply is composed of an ac inverter and a front-end converter not only to offer stable dc input voltage for the inverter but also to respond to the dynamic load characteristic of the plasma reactor. A brief introduction of a series-parallel resonant converter (SPRC) with a trapezoidal approximation and determining a design point considering a wide load range are described. In the case of the inverter, a low-loaded quality factor design method based on the analysis of a series-parallel resonant inverter (SPRI) to obtain high efficiency is proposed. With respect to the plasma ignition, a control logic without an additional plasma ignition circuit is proposed. Finally, the performance of the developed power supply is demonstrated under various experimental conditions including an experiment with a ferrite ICP load.

Integral equation method for the 1D steady-state Poisson-Nernst-Planck equations

An integral equation method is presented for the 1D steady-state Poisson-Nernst-Planck equations modeling ion transport through membrane channels. The differential equations are recast as integral equations using Green’s 3rd identity yielding a fixed-point problem for the electric potential gradient and ion concentrations. The integrals are discretized by a combination of midpoint and trapezoid rules, and the resulting algebraic equations are solved by Gummel iteration. Numerical tests for electroneutral and non-electroneutral systems demonstrate the method’s 2nd order accuracy and ability to resolve sharp boundary layers. The method is applied to a 1D model of the K^+ ion channel with a fixed charge density that ensures cation selectivity. In these tests, the proposed integral equation method yields potential and concentration profiles in good agreement with published results.

A variable step‐size extrapolated Crank–Nicolson method for option pricing under stochastic volatility model with jump

This paper studies the numerical solution of the stochastic volatility model with jumps under European options. This model can be transformed into a partial integro‐differential equation (PIDE) with spatial mixed derivative terms. Due to the nonsmoothness of the initial function, the variable step‐size extrapolated Crank–Nicolson (CN) method, which explicitly discretizes the jump term and implicitly the rest, is proposed to solve this model. The finite difference method is used to discretize the spatial differential operator, the composite trapezoidal rule to calculate the jump integral, and then a linear system with a nine‐diagonal coefficient matrix is obtained, which is easy to solve. The stability of the variable step‐size extrapolated CN method is then proved. Based on realistic regularity assumptions on the data, the consistency error and global error bounds of the variable step‐size extrapolated CN method are derived in norm. Compared with the variable step‐size IMEX BDF2 method and the variable step‐size IMEX MP method, the numerical results show the efficiency of the proposed variable step‐size extrapolated CN method.

TrapezoidalNet: A new network architecture inspired from the numerical solution of ordinary differential equations

The architecture of deep neural networks is commonly determined via trial and error, resulting in inefficiency and a lack of architecture interpretability. Recent research shows that numerical solutions of ordinary differential equations have demonstrated great potential for designing network architectures. Implicit methods are generally more stable and accurate than explicit methods but are yet to be applied to neural network design. Unlike explicit methods that directly calculate the current state based on the previous state, implicit methods have to solve a nonlinear equation to get the approximate solution, which discourages using implicit methods in network design. In this paper, we propose replacing Euler's method with the trapezoidal rule, a scheme that is a second-order implicit method with much lower local truncation errors. The crux of our approach lies in first altering the trapezoidal rule from its implicit form to the explicit form and then designing the corresponding neural network. By introducing the multi-channel convolution and layer-sharing mechanism, we propose a new architecture called TrapezoidalNet. We expect TrapezoidalNet can outperform other deep neural networks based on explicit methods such as ResNet, FitResNet, FractalNet, and LM-ResNet on image recognition tasks. Experimental results on CIFAR10/100 datasets verify the effectiveness of TrapezoidalNet.

Development of a Composite Implicit Time Integration Scheme for Three-Dimensional Discontinuous Deformation Analysis

Discontinuous deformation analysis (DDA) is a discontinuum-based and implicit method for investigating the deformational behavior of block systems. The constant acceleration integration (CAI) scheme characterized by unconditional stability is employed in the traditional DDA. In this study, the problems of the CAI scheme regarding the time step in DDA are pointed out. A too large or too small time step size adopted in the CAI scheme will have adverse effects on the DDA computation. To overcome the weaknesses, an alternative composite implicit time integration (CITI) scheme, which is a combination of the trapezoidal rule and the three-point backward Euler method, is implemented in the three-dimensional (3D) DDA method. Verification examples and slope numerical simulations are presented to illustrate the accuracy and effectiveness of the proposed methodology. The results showed that the CITI scheme can overcome the numerical error caused by the large time step size, and the algorithm damping is closely related to the time step size and the selected splitting parameter.

Bounce-averaged drifts: Equivalent definitions, numerical implementations, and example cases

In this article, we provide various analytical and numerical methods for calculating the average drift of magnetically trapped particles across field lines in complex geometries, and we compare these methods against each other. To evaluate bounce integrals, we use a generalization of the trapezoidal rule which is able to circumvent integrable singularities. We contrast this method with more standard quadrature methods in a parabolic magnetic well and find that the computational cost is significantly lower for the trapezoidal method, though at the cost of accuracy. With numerical routines in place, we next investigate conditions on particles which cross the computational boundary, and we find that important differences arise for particles affected by this boundary, which can depend on the specific implementation of the calculation. Finally, we investigate the bounce-averaged drifts in the optimized stellarator NCSX. From investigating the drifts, one can readily deduce important properties, such as what subset of particles can drive trapped-particle modes and in what regions radial drifts are most deleterious to the stability of such modes.

Influence of CYP3A5 and ABCB1 polymorphisms on the pharmacokinetics of vincristine in adult patients receiving CHOP therapy.

This study aims to clarify the impact of CYP3A5 and ABCB1 polymorphisms on the pharmacokinetics of vincristine (VCR) in adult patients receiving CHOP therapy. Plasma samples were collected immediately after the end of VCR administration and at 1.5, 2.5, 3.5, 5.5, 9.5, 13.5, and 25.5h after the start of administration. Areas under the plasma concentration-time curves of VCR in the elimination phase (AUC1.5-25.5) were calculated using the linear trapezoidal rule. Half-lives of VCR during the early phase (1.5-5.5h) and terminal phase (5.5-25.5h; t1/2γ) were determined according to the log-linear regression of the concentration-time data for at least 3 sampling points. A total of 41 adult patients were enrolled in this study. The median t1/2γ and AUC1.5-25.5 were significantly longer and higher in CYP3A5 non-expressers (CYP3A5*3/*3) than in CYP3A5 expressers (CYP3A5*1/*1 or *1/*3) (21.3 vs 13.8h, P = 0.005 and 35.5 vs 30.0ng･h/mL, P = 0.006, respectively). Conversely, there were no significant differences in pharmacokinetic parameters among the ABCB1 c.1236C>T, c.2677G>A/T, c.3435C>T genotype groups. A stepwise selection multiple linear regression analysis showed that the dose of VCR administered and CYP3A5 non-expresser status were independent factors influencing the AUC1.5-25.5 (partial R2 = 0.212, P = 0.002 and partial R2 = 0.143, P = 0.010, respectively). The CYP3A5*3 polymorphism was found to be an indicator for predicting exposure to VCR in adult patients receiving CHOP therapy. This information may be useful for the individualization of VCR dosages.

Comparison of numerical-integration-based methods for blood flow estimation in diffuse correlation spectroscopy

Background and ObjectiveDiffuse correlation spectroscopy (DCS) is an optical blood flow monitoring technology that has been utilized in various biomedical applications. In signal processing of DCS, nonlinear fitting of the experimental data and the theoretical model can be a hindrance in real-time blood flow monitoring. As one of the approaches to resolve the issue, INISg1, the inverse of numerical integration of squared g1 (a normalized electric field autocorrelation function), that could surpass the state-of-the-art technique at the time in terms of signal processing speed, has been introduced. While it is possible to implement INISg1 using various numerical integration methods, no relevant studies have been performed. Meanwhile, INISg1 was only tested within limited experimental conditions, which cannot guarantee the robustness of INISg1 in various experimental conditions. Thus, this study aims to introduce variants of INISg1 and perform a thorough comparison of the original INISg1 and its variants. MethodsIn this study, based on the right Riemann sum (RR) and trapezoid rule (TR) of numerical integration, INISg1_RR and INISg1_TR are suggested. They are thoroughly compared with the original INISg1 using model-based simulations that offer us control of most of the experimental conditions, including integration time, β, and photon count rate. ResultsExcept for some extreme cases, INISg1 performed more robustly than INISg1_RR and INISg1_TR. However, in extreme conditions, variants of INISg1 performed better than INISg1. With the same condition, the signal processing speed of INISg1 was 1.63 and 1.98 times faster than INISg1_RR and INISg1_TR, respectively. ConclusionThis study shows that INISg1 is robust in most cases and the study can be a guide for researchers using INISg1 and its variants in different types of DCS applications.

Tacrolimus population pharmacokinetics in adult heart transplant patients.

Tacrolimus is an immunosuppressant largely used in heart transplantation. However, the calculation of its exposure based on the area under the curve (AUC) requires the use of a population pharmacokinetic (PK) model. The aims of this work were (i) to develop a population PK model for tacrolimus in heart transplant patients, (ii) to derive a maximum a posteriori Bayesian estimator (MAP-BE) based on a limited sampling strategy (LSS) and (iii) to estimate probabilities of target attainment (PTAs) for AUC and trough concentration (C0). Forty-seven PK profiles (546 concentrations) of 18 heart transplant patients of the Pharmacocinétique des Immunosuppresseurs chez les patients GREffés Cardiaques study receiving tacrolimus (Prograf®) were included. The database was split into a development (80%) and a validation (20%) set. PK parameters were estimated in MONOLIX® and based on this model a Bayesian estimator using an LSS was built. Simulations were performed to calculate the PTA for AUC and C0. The best model to describe the tacrolimus PK was a two-compartment model with a transit absorption and a linear elimination. Only the CYP3A5 covariate was kept in the final model. The derived MAP-BE based on the LSS (0-1-2h postdose) yielded an AUC bias ± SD =2.7 ±10.2% and an imprecision of 9.9% in comparison to the reference AUC calculated using the trapezoidal rule. PTAs based on AUC or C0 allowed new recommendations to be proposed for starting doses (0.11 mg·kg-1 ·12 h-1 for the CYP3A5 nonexpressor and 0.22 mg·kg1 ·12 h-1 for the CYP3A5 expressor). The MAP-BE developed should facilitate estimation of tacrolimus AUC in heart transplant patients.

Convergence of trapezoid rule to rough integrals

Convergence of trapezoid rule to rough integrals

GENERAL SOLUTION OF TWO-DIMENSIONAL PROJECTILE MOTION WITH AIR RESISTANCE

In this study, two-dimensional projectile motion is considered under the effect of a general power law model of air resistance. Classically, a projectile is treated as a point mass with mass m moving in a uniform gravitational field. The projectile is launched from the ground with an angle α to horizon. the drag force is assumed to be proportional to the speed raised to the power n. The analysis of the problem is performed using Cartesian coordinates. A general exact parametrical solution (with respect to the angle of motion) is derived for any power n, following simple steps: 1) find the speed in the direction of the axis x (horizontal – no gravity) 2) find the vertical component of the speed 3) find the time 4) find the horizontal position of the projectile and finally,5) find the vertical position of the projectile. Steps 1) and 2) give explicit closed form equations and the rest are given by exact integrals which can be solved numerically. In this study spreadsheet calculation are performed using trapezoidal rule of integration. The cases of motion in avacuum and linear drag law are used to check the accuracy of the numerical calculations.The importance of the proposed study is three-fold: a) The method of the derived solution is new, and couldnt be found elsewhere b) The derived equations make it possible to use spreadsheets for presenting the subject (no programming is required), and thus, serve as a tool to enhance teaching c) The derived equations are general for any power n thus, the same procedure could be used to find the position of the projectile at any time.

A Numerical Study for Solving the Systems of Fuzzy Fredholm Integral Equations of the Second Kind Using the Adomian Decomposition Method

In this paper, the Adomian decomposition method (ADM) is successfully applied to find the approximate solutions for the system of fuzzy Fredholm integral equations (SFFIEs) and we also study the convergence of the technique. A consistent way to reduce the size of the computation is given to reach the exact solution. One of the best methods adopted to determine the behavior of the approximate solutions. Finally, the problems that have been addressed confirm the validity of the method applied in this research using a comparison by combining numerical methods such as the Trapezoidal rule and Simpson rule with ADM.

Unified analysis of finite-size error for periodic Hartree-Fock and second order Møller-Plesset perturbation theory

Despite decades of practice, finite-size errors in many widely used electronic structure theories for periodic systems remain poorly understood. For periodic systems using a general Monkhorst-Pack grid, there has been no comprehensive and rigorous analysis of the finite-size error in the Hartree-Fock theory (HF) and the second order Møller-Plesset perturbation theory (MP2), which are the simplest wavefunction based method, and the simplest post-Hartree-Fock method, respectively. Such calculations can be viewed as a multi-dimensional integral discretized with certain trapezoidal rules. Due to the Coulomb singularity, the integrand has many points of discontinuity in general, and standard error analysis based on the Euler-Maclaurin formula gives overly pessimistic results. The lack of analytic understanding of finite-size errors also impedes the development of effective finite-size correction schemes. We propose a unified analysis to obtain sharp convergence rates of finite-size errors for the periodic HF and MP2 theories. Our main technical advancement is a generalization of the result of Lyness [Math. Comp. 30 (1976), pp. 1–23] for obtaining sharp convergence rates of the trapezoidal rule for a class of non-smooth integrands. Our result is applicable to three-dimensional bulk systems as well as low dimensional systems (such as nanowires and 2D materials). Our unified analysis also allows us to prove the effectiveness of the Madelung-constant correction to the Fock exchange energy, and the effectiveness of a recently proposed staggered mesh method for periodic MP2 calculations (see X. Xing, X. Li, and L. Lin [J. Chem. Theory Comput. 17 (2021), pp. 4733–4745]). Our analysis connects the effectiveness of the staggered mesh method with integrands with removable singularities, and suggests a new staggered mesh method for reducing finite-size errors of periodic HF calculations.

High-order corrected trapezoidal rules for a class of singular integrals

We present a family of high-order trapezoidal rule-based quadratures for a class of singular integrals, where the integrand has a point singularity. The singular part of the integrand is expanded in a Taylor series involving terms of increasing smoothness. The quadratures are based on the trapezoidal rule, with the quadrature weights for Cartesian nodes close to the singularity judiciously corrected based on the expansion. High-order accuracy can be achieved by utilizing a sufficient number of correction nodes around the singularity to approximate the terms in the series expansion. The derived quadratures are applied to the implicit boundary integral formulation of surface integrals involving the Laplace layer kernels.

Decomposition and conformal mapping techniques for the quadrature of nearly singular integrals

Gauss–Legendre quadrature, Clenshaw–Curtis quadrature and the trapezoid rule are powerful tools for numerical integration of analytic functions. For nearly singular problems, however, these standard methods become unacceptably slow. We discuss and generalize some existing methods for improving on these schemes when the location of the nearby singularity is known. We conclude with an application to some nearly singular surface integrals that arise in three-dimensional viscous fluid flow.

Enhanced trapezoidal rule for discontinuous functions

In many applications, data to be integrated is available only in the form of function values at predetermined equispaced points. For smooth periodic functions, the trapezoidal rule gives very accurate approximations to the integral, but is only second order accurate in more commonly arising non-periodic cases. The Gregory approach for end corrections can improve the accuracy order to 9 before the onset of negative weights, and of rapidly increasing ill-conditioning. In recent work, this has been extended to accuracy orders around 20.This present study focuses on the more general case when one or both end points of the integration interval do not coincide with any of the equispaced grid points. We show that accuracy orders up to around 10 can also then be achieved, with all quadrature weights still remaining non-negative. This method can be utilized for example when functions to be integrated feature discontinuities at locations that can be separately determined (at or in between the equispaced grid points).

Inaccuracies due to the frequency warping in simulation of electrical systems using combined state–space nodal analysis

The simulation of electromagnetic transients may suffer from inaccuracies due to a phenomenon known as frequency warping (FW). This paper presents an analysis of the effects of FW on the accuracy of digital simulations, demonstrating that the use of the trapezoidal integration rule (TR), commonly employed in many electromagnetic transients simulators, is the root cause of such inaccuracies. Although FW is considered a major problem in digital signal processing, it is often overlooked when simulating electrical transients. The analysis is carried out in a fourth-order RLC circuit, from which the analytic solution is derived. The circuit is solved using the combined state–space nodal method, considering the TR or recursive convolutions as solution methods for the state–space representation. It was observed that the FW caused a change in the natural oscillation frequency of the system, causing a pulsating behavior of absolute error. The accumulation of errors over time can result in deteriorated solutions when either the time steps are not sufficiently small or the simulation runs for a long enough duration. This paper emphasizes the significance of accounting for the FW phenomenon in digital simulations that rely on integration methods, such as the TR.

Multi-Objective Optimization for Controlling the Dynamics of the Diabetic Population

To limit the adverse effects of diabetes, a personalized and long-term management strategy that includes appropriate medication, exercise and diet has become of paramount importance and necessity. Compartment-based mathematical control models for diabetes usually result in objective functions whose terms are conflicting, preventing the use of single-objective-based models for obtaining appropriate personalized strategies. Taking into account the conflicting aspects when controlling the diabetic population dynamics, this paper introduces a multi-objective approach consisting of four steps: (a) modeling the problem of controlling the diabetic population dynamics using a multi-objective mathematical model, (b) discretizing the model using the trapezoidal rule and the Euler–Cauchy method, (c) using swarm-intelligence-based optimizers to solve the model and (d) structuring the set of controls using soft clustering methods, known for their flexibility. In contrast to single-objective approaches, experimental results show that the multi-objective approach obtains appropriate personalized controls, where the control associated with the compartment of diabetics without complications is totally different from that associated with the compartment of diabetics with complications. Moreover, these controls enable a significant reduction in the number of diabetics with and without complications, and the multi-objective strategy saves up to 4% of the resources needed for the control of diabetes without complications and up to 18% of resources for the control of diabetes with complications.