Integral Approximation Research Articles

Features of the Expansion of Multiple Stochastic Stratonovich Integrals Using Walsh and Haar Functions

The problem of the root-mean-square convergence for approximations of multiple stochastic Stratonovich integrals based on the generalized multiple Fourier series method using Walsh and Haar functions is considered. It is shown that when they are chosen to expand multiple stochastic integrals, the proof of the root-mean-square convergence of a subsequence of series partial sums, which is formed in a way that is quite natural for these functions, does not require the explicit fulfillment of any additional conditions, except for the condition of the existence of the multiple stochastic Stratonovich integral.

Mean-square Approximation of Iterated Ito and Stratonovich Stochastic Integrals: Method of Generalized Multiple Fourier Series. Application to Numerical Integration of Ito Sdes and Semilinear Spdes (third Edition)

This is the third edition of the monograph (first edition 2020, second edition 2021) devoted to the problem of mean-square approximation of iterated Ito and Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. The mentioned problem is considered in the book as applied to the numerical integration of non-commutative Ito stochastic differential equations and semilinear stochastic partial differential equations with nonlinear non-commutative trace class noise. The book opens up a new direction in researching of iterated stochastic integrals. For the first time we use the generalized multiple Fourier series converging in the sense of norm in Hilbert space for the expansion of iterated Ito stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process (Chapter 1). Sections 1.11-1.13 (Chapter 1) are new and generalize the results of Chapter 1 obtained earlier by the author and are also closely related to the multiple Wiener stochastic integral introduced by Ito in 1951. The convergence with probability 1 as well as the convergence in the sense of n-th (n=2, 3,...) moment for the expansion of iterated Ito stochastic integrals have been proved (Chapter 1). Moreover, the rate of both types of convergence has been established. The main difference between the third and second editions of the book is that the third edition includes original material (Chapter 2, Sections 2.10-2.19) on a new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process. The above approach allowed us to generalize some of the author's earlier results and also to make significant progress in solving the problem of series expansion of iterated Stratonovich stochastic integrals. In particular, for iterated Stratonovich stochastic integrals of the fifth and sixth multiplicity, series expansions based on multiple Fourier-Legendre series and multiple trigonometric Fourier series are obtained. In addition, expansions of iterated Stratonovich stochastic integrals of multiplicities 2 to 4 were generalized. These results (Chapter 2) adapt the results of Chapter 1 for iterated Stratonovich stochastic integrals. Two theorems on expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k based on generalized iterated Fourier series with pointwise convergence are formulated and proved (Chapter 2). The results of Chapters 1 and 2 can be considered from the point of view of the Wong-Zakai approximation for the case of a multidimensional Wiener process and the Wiener process approximation based on its series expansion using Legendre polynomials and trigonometric functions. The integration order replacement technique for iterated Ito stochastic integrals has been introduced (Chapter 3). Exact expressions are obtained for the mean-square approximation error of iterated Ito stochastic integrals of arbitrary multiplicity k (Chapter 1) and iterated Stratonovich stochastic integrals of multiplicities 1 to 4 (Chapter 5). Furthermore, we provided a significant practical material (Chapter 5) devoted to the expansions and mean-square approximations of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the unified Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4). These approximations were obtained using Legendre polynomials and trigonometric functions. The methods constructed in the book have been compared with some existing methods (Chapter 6). The results of Chapter 1 were applied (Chapter 7) to the approximation of iterated stochastic integrals with respect to the finite-dimensional approximation of the Q-Wiener process (for integrals of multiplicity k) and with respect to the infinite-dimensional Q-Wiener process (for integrals of multiplicities 1 to 3).

Asymptotic distributions of a new type of design‐based incomplete U‐statistics

The U‐statistic has been an important part of the arsenal of statistical tools. Meanwhile, the computation of it could easily become expensive. As a remedy, the idea of incomplete U‐statistics has been adopted in practice, where only a small fraction of combinations of units are evaluated. Recently, researchers proposed a new type of incomplete U‐statistics called ICUDO, which needs substantially less time of computing than all existing methods. This paper aims to study the asymptotic distributions of ICUDO to facilitate the corresponding statistical inference. This is a non‐trivial task due to the restricted randomization in the sampling scheme of ICUDO. The bootstrap approach for the finite sample distribution of ICUDO is also discussed. Lastly, we observe some intrinsic connections between U‐statistics and computer experiments in the context of integration approximation. This allows us to generalize some existing theoretical results in the latter topic.

Numerical integration approximations to estimate the Weitzman overlapping measure: Weibull distributions

This paper deals with the problem of estimating the overlapping (OVL) Weitzman measure (?) when two independent random variables ?? and ?? are given by two-parameter Weibull distribution. The measure ? has been studied in the literature in the case of two Weibull distributions under the assumption that the two shape parameters are equal. In this work, a general expression for the Weitzmans measure is provided under the Weibull distribution without using any assumptions about the distribution parameters. Some new estimators for ? are developed depending on three numerical integration rules known as trapezoidal, Simpson 1/3 and Simpson 3/8 rules. The performance of the proposed estimators were investigated and compared with some existing estimators via simulation technique and real data. The results demonstrated the superiority of the proposed estimators over the existing one in almost all considered cases.

Bayesian estimation of nonlinear models parameters in the description of growth coffee fruits

Coffee is one of the main products of Brazilian agriculture and the country is currently the largest producer and exporter in the world. The coffee fruit has a double sigmoidal growth pattern, however, as well as in other fruits that also show such a growth pattern, the authors generally do not estimate parameters of regression models to describe such curve. In the study of fruit growth curves, the sample size is generally small, so the estimation of the parameters should preferably be done by the Bayesian methodology, since a priori information is incorporated, reducing the effects of having few observations. The Markov Chain Monte Carlo algorithms are the most used computational tool in Bayesian statistics. However, these generate dependent samples, can be complicated to implement and, mainly, to teach. There are also other alternatives to the MCMC algorithms to obtain approximations of integrals of interest in Bayesian inference, the main ones are based on the importance resampling techniques. The objective of this work is to use Bayesian inference with the weighted importance resampling technique in the estimation of parameters of double sigmoidal nonlinear regression models to the description of coffee fruit growth. The double nonlinear logistic model was used in the description of the accumulation of fresh weight in coffee fruits. All prioris used have Beta distribution and were obtained by the called prior of specialist technique. Bayesian methodology was efficient, since it provided parameters with practical interpretation to coffee fruit growth, consistent with the reality. Thus, Bayesian inference by weighted importance resampling was a good alternative for the parameters estimation of nonlinear double sigmoid regression models. The logistic model showed that the growth of coffee fruits is more intense in the first sigmoid (until 162 DAF)of the growth curve and stabilizes in its final weight after 262 daf.

Numerical solution of Atangana–Baleanu–Caputo time-space fractional diffusion equation

In this article, the time-space fractional diffusion equation is solved by using the fractional operator in Atangana–Baleanu–Caputo (ABC) sense based on the Mittag-Leffler function involving non-singular and non-local kernels. This study mainly develop the numerical method for time-space linear and non-linear ABC fractional diffusion equation by utilising centred difference approximation with second-order accuracy of Riesz fractional derivative in space and trapezoidal formula for fractional integral approximation for a system of ABC integral equations. Also, we examine the stability and show that the proposed scheme converges at the rate of having space mesh size h and time step size t. Numerical simulations are presented based on the presented method. We finished the article with a conclusion.

Path integral description and direct interaction approximation for elastic plate turbulence

In this work, we apply the Martin–Siggia–Rose path integral formalism to the equations of a thin elastic plate. Using a diagrammatic technique, we obtain the direct interaction approximation (DIA) equations to describe the evolutions of the correlation function and the response function of the fields. Consistent with previous results, we show that DIA equations for elastic plates can be derived from a non-markovian stochastic process and that in the weakly nonlinear limit, the DIA equations lead to the kinetic equation of wave turbulence theory. We expect that this approach will allow a better understanding of the statistical properties of wave turbulence and that DIA equations can open new avenues for understanding the breakdown of weakly nonlinear turbulence for elastic plates.

Numerical Integral Approximation to Estimate Matusita Overlapping Coefficient for Normal Distributions

The overlapping coefficients are defined as the measures of similarity and agreement between two or more distributions. Estimation of the well-known overlapping coefficient; Matusita ρ under normal distributions is the main aim of this paper. Given that we have two independent random samples each following a normal distribution, a new method is proposed to estimate ρ without using any assumptions about the equality of the location or scale parameters. Three numerical integration methods are suggested and used to achieve our main objective. The maximum likelihood estimation method is used to estimate the interesting parameters. The properties of the resulting estimators are investigated and compared with some corresponding estimators that have been developed in the literature by using the simulation method. The simulation results show the effectiveness of the proposed technique over the existing ones for almost all considered cases in this study.

Kernel-Based Methods for Solving Time-Dependent Advection-Diffusion Equations on Manifolds

In this paper, we extend the class of kernel methods, the so-called diffusion maps (DM) and ghost point diffusion maps (GPDM), to solve the time-dependent advection-diffusion PDE on unknown smooth manifolds without and with boundaries. The core idea is to directly approximate the spatial components of the differential operator on the manifold with a local integral operator and combine it with the standard implicit time difference scheme. When the manifold has a boundary, a simplified version of the GPDM approach is used to overcome the bias of the integral approximation near the boundary. The Monte-Carlo discretization of the integral operator over the point cloud data gives rise to a mesh-free formulation that is natural for randomly distributed points, even when the manifold is embedded in high-dimensional ambient space. Here, we establish the convergence of the proposed solver on appropriate topologies, depending on the distribution of point cloud data and boundary type. We provide numerical results to validate the convergence results on various examples that involve simple geometry and an unknown manifold.

A time-domain POD approach based on numerical implicit and explicit Green’s functions for 3D elastodynamic analysis

This paper is concerned with the development of time integration methods based on numerical Green’s functions applied in conjunction with the proper orthogonal decomposition (POD) technique to solve three-dimensional elastodynamic problems discretized by the FEM. The convolution integral in the velocity expression of these methods is firstly carried out through integration by parts, and then approximated numerically, resulting in a simpler and more accurate convolution than that reported in the literature. To compute the solution samples, required by the POD technique, the Implicit Green’s functions Approach (ImGA)-Newmark method rewritten in terms of the ultimate spectral radius is employed. The proposed ImGA scheme is truly self-starting and easy to implement with just one free parameter. To solve the reduced-order model, a new highly accurate time integration scheme based on the Explicit Green’s functions Approach (ExGA)-Runge–Kutta method is proposed, considering the Gauss–Lobatto-Legendre (GLL) points to perform both the time integration of the Green’s functions and the approximation of the convolution integrals. To verify the accuracy and efficiency of the proposed formulation, the numerical results are compared with those from the full-order model. Moreover, undamped solution samples are utilized to build a reduced-order model that can take into account a Rayleigh damping matrix.

An Explicit Wavelet Method for Solution of Nonlinear Fractional Wave Equations

An explicit method for solving time fractional wave equations with various nonlinearity is proposed using techniques of Laplace transform and wavelet approximation of functions and their integrals. To construct this method, a generalized Coiflet with N vanishing moments is adopted as the basis function, where N can be any positive even number. As has been shown, convergence order of these approximations can be N. The original fractional wave equation is transformed into a time Volterra-type integro-differential equation associated with a smooth time kernel and spatial derivatives of unknown function by using the technique of Laplace transform. Then, an explicit solution procedure based on the collocation method and the proposed algorithm on integral approximation is established to solve the transformed nonlinear integro-differential equation. Eventually the nonlinear fractional wave equation can be readily and accurately solved. As examples, this method is applied to solve several fractional wave equations with various nonlinearities. Results show that the proposed method can successfully avoid difficulties in the treatment of singularity associated with fractional derivatives. Compared with other existing methods, this method not only has the advantage of high-order accuracy, but it also does not even need to solve the nonlinear spatial system after time discretization to obtain the numerical solution, which significantly reduces the storage and computation cost.

Classification logit two-sample testing by neural networks for differentiating near manifold densities.

The recent success of generative adversarial networks and variational learning suggests that training a classification network may work well in addressing the classical two-sample problem, which asks to differentiate two densities given finite samples from each one. Network-based methods have the computational advantage that the algorithm scales to large datasets. This paper considers using the classification logit function, which is provided by a trained classification neural network and evaluated on the testing set split of the two datasets, to compute a two-sample statistic. To analyze the approximation and estimation error of the logit function to differentiate near-manifold densities, we introduce a new result of near-manifold integral approximation by neural networks. We then show that the logit function provably differentiates two sub-exponential densities given that the network is sufficiently parametrized, and for on or near manifold densities, the needed network complexity is reduced to only scale with the intrinsic dimensionality. In experiments, the network logit test demonstrates better performance than previous network-based tests using classification accuracy, and also compares favorably to certain kernel maximum mean discrepancy tests on synthetic datasets and hand-written digit datasets.

Comparative analysis of different FOPI approximations and number of terms used on simulations of a battery-powered, field-oriented induction motor based electric vehicle traction system

Because of their enhanced performance, the fractional order proportional-integral (FOPI) controllers are becoming an appealing choice for controlling induction motor speed. To implement FOPI controllers, several fractional order integral approximations are available in the literature. The approximation used, and the order of approximation affects the speed tracking, transient response, and induction motor power consumption. This further affects the energy consumption analysis if simulations are conducted based on such approximations. In this paper an electric vehicle (EV) traction system is simulated to investigate the effect of such approximations on the simulations of a battery powered, induction motor driven EV system. The system consists of an indirect field-oriented induction motor, a lithium-ion battery bank, and a three-phase inverter. This work presents a quantitative analysis of the performance of FOPI controllers using different approximations, and order of approximations is presented. The controllers are evaluated based on speed tracking, transient response, computational time, and power consumption. Both step functions and standard drive cycles are used as the speed reference signal to evaluate the effects of using different approximations and different orders of approximation, when different references are used. This work establishes a reference set of simulations that can be used to infer the amount of error in battery state of charge, and state of health analysis conducted on such an EV system, when dealing with FOPI controllers under different approximations and related settings.

Correction of High-Order $$L_k$$ Approximation for Subdiffusion

The subdiffusion equations with a Caputo fractional derivative of order \(\alpha \in (0,1)\) arise in a wide variety of practical problems, which describe the transport processes, in the force-free limit, slower than Brownian diffusion. In this work, we derive the correction schemes of the Lagrange interpolation with degree k (\(k\le 6\)) convolution quadrature, called \(L_k\) approximation, for the subdiffusion. The key step of designing correction algorithm is to calculate the explicit form of the coefficients of \(L_k\) approximation by the polylogarithm function or Bose-Einstein integral. To construct a \(\tau _8\) approximation of Bose-Einstein integral, the desired \((k+1-\alpha )\)th-order convergence rate can be proved for the correction \(L_k\) scheme with nonsmooth data, which is higher than kth-order BDFk method in [Jin, Li, and Zhou, SIAM J. Sci. Comput., 39 (2017), A3129–A3152; Shi and Chen, J. Sci. Comput., (2020) 85:28]. The numerical experiments with spectral method are given to illustrate theoretical results.

A numerical method based on quadrature rules for [formula omitted]-fractional differential equations

This paper presents a numerical method for the solution of a class of ψ-fractional differential equations involving Caputo derivative with respect to a function. Initial value problem for the certain ψ-fractional differential equation is converted into equivalent second kind of Volterra integral equation. A combination of Simpson’s and Trapezoidal rule is used to transform the Volterra equation into a system of algebraic equations. The numerical solutions to the original problem are recovered from a solution of an algebraic system. We also give an error estimate for the function approximation and fractional integral approximation. Error bound for the numerical approximation of solutions is also derived. The numerical method is tested for various specific problems.

Sinc Numeric Methods for Fox-H, Aleph (ℵ), and Saxena-I Functions

The purpose of this study is to offer a systematic, unified approach to the Mellin-Barnes integrals and associated special functions as Fox H, Aleph ℵ, and Saxena I function, encompassing the fundamental features and important conclusions under natural minimal assumptions on the functions in question. The approach’s pillars are the concept of a Mellin-Barnes integral and the Mellin representation of the given function. A Sinc quadrature is used in conjunction with a Sinc approximation of the function to achieve the numerical approximation of the Mellin-Barnes integral. The method converges exponentially and can handle endpoint singularities. We give numerical representations of the Aleph ℵ and Saxena I functions for the first time.

General relativity on the multiverse and nature’s hierarchies

We define ``third-derivative'' general relativity by promoting the integration measure in Einstein-Hilbert action to be an arbitrary 4-form field strength. We project out its local fluctuations by coupling it to another 4-form field strength. This ensures that the gravitational sector contains only the usual massless helicity-2 propagating modes. Adding the charges to these 4-forms allows for discrete variations of the coupling parameters of conventional general relativity: ${G}_{N},\mathrm{\ensuremath{\Lambda}},{H}_{0}$, and even $⟨\mathsf{Higgs}⟩$ are all variables which can change by jumps. Hence, de Sitter spacetime is unstable to membrane nucleation. Using this instability, we explain how the cosmological constant problem can be solved. The scenario utilizes the idea behind the irrational axion, but instead of an axion it requires one more 4-form field strength and corresponding charged membranes. When the membrane charges satisfy the constraint $\frac{2{\ensuremath{\kappa}}_{\mathsf{eff}}^{2}{\ensuremath{\kappa}}^{2}|{\mathcal{Q}}_{i}|}{3{\mathcal{T}}_{i}^{2}}<1$, the theory which ensues exponentially favors a huge hierarchy $\mathrm{\ensuremath{\Lambda}}/{M}_{\mathrm{Pl}}^{4}\ensuremath{\ll}1$ instead of $\mathrm{\ensuremath{\Lambda}}/{M}_{\mathrm{Pl}}^{4}\ensuremath{\simeq}1$. The discharges produce the distribution of the values of $\mathrm{\ensuremath{\Lambda}}$ described by the saddle point approximation of the Euclidean path integral.

Invariance of convex sets: An alternative proof and application to Black-Scholes operator

An alternative proof of invariance of convex sets by the solution of non autonomous Cauchy problem is given. The proof is based on the recent integral approximation of time dependent operators A(t) acting on Hilbert space when they are associated with smooth sesquilinear forms a(t,.,.) defined on common dense domain and the known Chernoff Product Formula. An application to positivity of Black-Scholes operator is given.

Efficient Static-Driven Integration for Step-Function Transient Simulation

Step-function excitations are convenient abstractions for circuit turn on/off and for switching activities in general. However, they are ill-handled by SPICE-derived circuit simulators that employ implicit integration approximations and substitute steep ramps for steps. An ACES-derived explicit integration method efficiently handles step-functions, manages potential instability, and converges to the steady state. The resulting circuit simulator can be employed to find an initial state size and excursion-limited accuracy control. It also is able to accommodate resistance, capacitance, inductance, and nonlinear elements efficiently and accurately.

A quadratic Wiener path integral approximation for stochastic response determination of multi-degree-of-freedom nonlinear systems

A Wiener path integral (WPI) technique is developed for determining the stochastic response of multi-degree-of-freedom (MDOF) nonlinear systems. Specifically, the nonlinear system response joint transition probability density function (PDF) is expressed as a WPI over the space of paths satisfying the initial and final conditions in time. Next, a functional series expansion is considered for the WPI and a quadratic approximation is employed. Further, relying on a variational principle yields a functional optimization problem to be solved for the most probable path, which is used for determining approximately the joint response transition PDF. It is shown that compared to the standard (semiclassical) WPI solution approach, which accounts only for the most probable path, the quadratic approximation developed herein exhibits enhanced accuracy. This is due to the fact that fluctuations around the most probable path are also accounted for by considering a localized state-dependent factor in the calculation of the WPI. Furthermore, the PDF normalization step of the most probable path approach is bypassed, and thus, probabilities of rare events (e.g., failures) can be determined in a direct manner without the need for obtaining the complete joint response PDF first. The herein developed technique can be construed as an extension of earlier efforts in the literature to account for MDOF systems. Several numerical examples are considered for demonstrating the accuracy of the technique. These pertain to various dynamical systems exhibiting diverse nonlinear behaviors. Comparisons with pertinent Monte Carlo simulation data are included as well.