First Approximation Research Articles

Unveiling the Power of Implicit Six-Point Block Scheme: Advancing numerical approximation of two-dimensional PDEs in physical systems.

In the era of computational advancements, harnessing computer algorithms for approximating solutions to differential equations has become indispensable for its unparalleled productivity. The numerical approximation of partial differential equation (PDE) models holds crucial significance in modelling physical systems, driving the necessity for robust methodologies. In this article, we introduce the Implicit Six-Point Block Scheme (ISBS), employing a collocation approach for second-order numerical approximations of ordinary differential equations (ODEs) derived from one or two-dimensional physical systems. The methodology involves transforming the governing PDEs into a fully-fledged system of algebraic ordinary differential equations by employing ISBS to replace spatial derivatives while utilizing a central difference scheme for temporal or y-derivatives. In this report, the convergence properties of ISBS, aligning with the principles of multi-step methods, are rigorously analyzed. The numerical results obtained through ISBS demonstrate excellent agreement with theoretical solutions. Additionally, we compute absolute errors across various problem instances, showcasing the robustness and efficacy of ISBS in practical applications. Furthermore, we present a comprehensive comparative analysis with existing methodologies from recent literature, highlighting the superior performance of ISBS. Our findings are substantiated through illustrative tables and figures, underscoring the transformative potential of ISBS in advancing the numerical approximation of two-dimensional PDEs in physical systems.

Modified correlated measurement errors model for estimation of population mean utilizing auxiliary information

The existence of measurement errors cannot be avoided in practice. It is a prominent fact that the existence of measurement errors diminishes conventional properties of the estimators. A modified correlated measurement errors model has been proposed. Shalabh and Tsai (Commun Stat Simul Comput 46(7):5566–5593. 10.1080/03610918.2016.1165845, 2017) correlated measurement errors model is a particular member of the suggested modified model. In this article, we have tackled the estimation of population mean utilizing auxiliary information under modified correlated measurement errors model. We have developed ratio and product estimators and studied their properties in case of simple random sampling without replacement (SRSWOR) up to first order of approximation. It has been illustrated that suggested ratio and product estimators are more efficient than the conventional unbiased estimator as well as Shalabh and Tsai (Commun Stat Simul Comput 46(7):5566–5593. 10.1080/03610918.2016.1165845, 2017) ratio and product estimators under very realistic situations. An empirical study has also been performed to demonstrate the merits of the recommended estimators over other estimators.

Brownian motion in the Hilbert space of quantum states and the stochastically emergent Lorentz symmetry: A fractal geometric approach from Wiener process to formulating Feynman’s path-integral measure for relativistic quantum fields

This paper aims to provide a consistent, finite-valued, and mathematically well-defined reformulation of Feynman’s path-integral measure for quantum fields obtained by studying the Wiener stochastic process in the infinite-dimensional Hilbert space of quantum states. This reformulation will undoubtedly have a crucial role in formulating quantum gravity within a mathematically well-defined framework. In fact, this study is fundamentally different from any previous research on the relationship between Feynman’s path-integral and the Wiener stochastic process. In this research, we focus on the fact that the classic Wiener measure is no longer applicable in infinite-dimensional Hilbert spaces due to fundamental differences between displacements in low and extremely high dimensions. Thus, an analytic norm motivated by the role of the fractal functions in the Wilsonian renormalization approach is worked out to properly characterize Brownian motion in the Hilbert space of quantum states on a compact flat manifold. This norm, the so-called fractal norm, pushes the rougher functions (physically the quantum states with higher energies) to the farther points of the Hilbert space until the fractal functions as the roughest ones are moved to infinity. Implementing the Wiener stochastic process with the fractal norm, results in a modified form of the Wiener measure called the Wiener fractal measure, which is a well-defined measure for Feynman’s path-integral formulation of quantum fields. Wiener fractal measure has a complicated formula of non-local terms but produces the Klein–Gordon action at the first order of approximation. Using complex integrals to compensate for the removal of non-local terms appearing in higher orders of approximation, the Wiener fractal measure turns into a complex measure and generates Feynman’s path-integral formulation of scalar quantum fields. This brings us to the main objective of this study. Finally, some various significant aspects of quantum field theory (such as renormalizability, RG flow, Wick rotation, regularization, etc.) are revisited by means of the analytical aspects of the Wiener fractal measure.

Almost Unbiased Estimators for Population Coefficient of Variation Using Auxiliary Information

The objective of the paper is to propose an almost unbiased ratio estimator for the finite coefficient of variation (CV). In this paper, we have proposed an exponential ratio type and log ratio type estimators for estimating population coefficient of variation. Two real data sets and one simulation study is carried out in support of the theoretical results. Mean squared error and Percent relative efficiency criteria is used to assess the performance of the estimators. It has been shown that the proposed class of estimators are almost unbiased up to the first order of approximation. Also proposed estimators are better in efficiency to other estimators consider in this study.

Evolving Improved Sampling Protocols for Dose–Response Modelling Using Genetic Algorithms with a Profile-Likelihood Metric

Practical limitations of quality and quantity of data can limit the precision of parameter identification in mathematical models. Model-based experimental design approaches have been developed to minimise parameter uncertainty, but the majority of these approaches have relied on first-order approximations of model sensitivity at a local point in parameter space. Practical identifiability approaches such as profile-likelihood have shown potential for quantifying parameter uncertainty beyond linear approximations. This research presents a genetic algorithm approach to optimise sample timing across various parameterisations of a demonstrative PK-PD model with the goal of aiding experimental design. The optimisation relies on a chosen metric of parameter uncertainty that is based on the profile-likelihood method. Additionally, the approach considers cases where multiple parameter scenarios may require simultaneous optimisation. The genetic algorithm approach was able to locate near-optimal sampling protocols for a wide range of sample number (n = 3–20), and it reduced the parameter variance metric by 33–37% on average. The profile-likelihood metric also correlated well with an existing Monte Carlo-based metric (with a worst-case r > 0.89), while reducing computational cost by an order of magnitude. The combination of the new profile-likelihood metric and the genetic algorithm demonstrate the feasibility of considering the nonlinear nature of models in optimal experimental design at a reasonable computational cost. The outputs of such a process could allow for experimenters to either improve parameter certainty given a fixed number of samples, or reduce sample quantity while retaining the same level of parameter certainty.

A well-balanced second-order finite volume approximation for a coupled system of granular flow

A well-balanced second-order finite volume scheme is proposed and analyzed for a 2×2 system of non-linear partial differential equations which describes the dynamics of growing sandpiles created by a vertical source on a flat, bounded rectangular table in multiple dimensions. To derive a second-order scheme, we combine a MUSCL type spatial reconstruction with strong stability preserving Runge-Kutta time stepping method. The resulting scheme is ensured to be well-balanced through a modified limiting approach that allows the scheme to revert to a well-balanced first-order scheme if it fails to capture or preserve the discrete steady state solution during its time evolution. Further, the scheme maintains the second-order accuracy away from the steady state. The well-balance property of the scheme is proven analytically in one dimension and demonstrated numerically in two dimensions. Additionally, numerical experiments reveal that the second-order scheme reduces finite time oscillations and provides better resolutions of the physical properties of the state variables, as compared to the existing first-order schemes of the literature.

Tunable spatial and angular spin splitting of reflected vortex-beam off hyperbolic metasurface

This paper investigated the spin splitting of the reflected beam with a vortex beam irradiating on a hyperbolic metasurface. This metasurface structure is the periodically rectangular-groove array on the surface of a hyperbolic crystal with the optical axis normal to its surface, where the array is a sub-wavelength biaxially hyperbolic grating with its periodical direction situated in the surface plane. Based on the hexagonal nitride boron crystal, we theoretically and numerically analyzed the spatial and angular spin splittings of the reflected beam, containing those produced by the transverse and longitudinal shifts. We found that the spatial shifts or spin splitting can be obtained in the first-order approximation of beam deflection, but the angular shifts can be described in the second-order approximation so that it enormously varies with the beam waist. Due to the hyperbolic anisotropy of the metasurface and the intrinsic orbital angular momentum in the incident beam, the expressions of spatial and angular shifts are effectively modified, and a unique double reflection effect occurs on the surface. Near certain reflective intensity distribution singularities, extremely asymmetrical spin splitting can be observed. These shifts and spin splitting can be tuned by varying the orientation of the incident plane and the angle of incidence.

Estimating neutrosophic finite median employing robust measures of the auxiliary variable

Our study explores neutrosophic statistics, an extension of classical and fuzzy statistics, to address the challenges of data uncertainty. By leveraging accurate measurements of an auxiliary variable, we can derive precise estimates for the unknown population median. The estimators introduced in this research are particularly useful for analysing unclear, vague data or within the neutrosophic realm. Unlike traditional methods that yield single-valued outcomes, our estimators produce ranges, suggesting where the population parameter is likely to be. We present the suggested generalised estimator's bias and mean square error within a first-order approximation framework. The practicality and efficiency of these proposed neutrosophic estimators are demonstrated through real-world data applications and the simulated data set.

Estimation of population mean using ranked set sampling in the presence of measurement errors

Ranked set sampling is widely acknowledged for its superior efficiency compared with simple random sampling. Only a small amount of work has been conducted using ranked set sampling when measurement errors are present. This study introduces innovative estimators utilizing ranked set sampling to assess the population mean when faced with both correlated and uncorrelated measurement errors. The expressions for the bias and mean squared error of the proposed estimators are derived up to the first-order approximation, revealing their superior performance compared to the other examined estimators. The efficacy of the suggested estimators in handling measurement errors was demonstrated through numerical illustration and simulation study investigations. The recommended estimators are further compared to the existing ones using the percentage relative efficiency and mean squared error, and the impact of measurement errors on the results is highlighted through the percentage computation of measurement errors. The innovative estimators suggested were formulated by judiciously incorporating ratio, exponential, and log estimators. Numerical examples involving expenditure and income, as well as simulated data generated from a normal population using R software, affirm the superior performance of the proposed estimators over existing ones such as the usual mean estimator and those proposed by Vishwakarma and Singh (2022), as evidenced by the higher percent relative efficiency and lower mean squared error. The evaluation of the percentage contribution of measurement error values confirms the impact of measurement errors on the properties of the estimators.

Multiple scales method for analyzing a forced damped rotational pendulum oscillator with gallows

This study reports the analytical solution for a generalized rotational pendulum system with gallows and periodic excited forces. The multiple scales method (MSM) is applied to solve the proposed problem. Several types of rotational pendulum oscillators are studied and talked about in detail. These include the forced damped rotating pendulum oscillator with gallows, the damped standard simple pendulum oscillator, and the damped rotating pendulum oscillator without gallows. The MSM first-order approximations for all the cases mentioned are derived in detail. The obtained results are illustrated with concrete numerical examples. The first-order MSM approximations are compared to the fourth-order Runge–Kutta (RK4) numerical approximations. Additionally, the maximum error is estimated for the first-order approximations obtained through the MSM, compared to the numerical approximations obtained by the RK4 method. Furthermore, we conducted a comparative analysis of the outcomes obtained by the used method (MSM) and He-MSM to ascertain their respective levels of precision. The proposed method can be applied to analyze many strong nonlinear oscillatory equations.

Hybrid class of robust type estimators for variance estimation using mean and variance of auxiliary variable

This study is completed for the estimation of unknown population variance for the variable of mean and variance of interest. To accomplish this task, a new generalized class of robust kind of variance estimators proposed utilizing known descriptives of auxiliary variable, for example, Mid-range, Hodges-Lehmann Mean, Tri-mean, deciles mean, coefficient of skewness, interquartile range, first quartile, coefficient of kurtosis, semi-interquartile average, inter decile range and Mean, etc. These conventional measures of auxiliary variable improve the accuracy of the suggested class under simple random sampling without replacement (SRSWOR) scheme. The properties such as the bias, mean square errors (MSE), and least MSE of the suggested class are derived up to first order of approximation. The superiority conditions of the developed class of estimators over existing estimators are also made out theoretically. Finally, numerical representation is also completed for the motivations behind the article. The usual variance estimator is considered as a benchmark for comparing all considered estimators in numerical illustration. The results have been indicated that the suggested class is performing better than the usual variance estimator and all other thoughts about existing estimators.

Development of improved estimators of finite population mean in simple random sampling with dual auxiliaries and its application to real world problems

In general, the incorporation of supplementary information reduces the Mean Square Error (MSE) and, consequently, enhances the precision of estimating a population parameter. This improvement relies on the appropriate application of a suitable function, with careful consideration. This study introduces two innovative families of estimators for the finite population mean, both of which exhibit superior performance in scenarios involving dual auxiliary information in simple random sampling. Expressions up to the first-order approximation, for bias, and Mean Square Error were derived, and the conditions under which these proposed families surpassed the existing estimators. Our evaluation involved the use of both real and simulated data to compute the Mean Square Error and Percent Relative Efficiency (PRE) of the estimators. A comparative analysis revealed that under the specified conditions, both proposed families yielded more precise results.

POWER MEDIAN-BASED ESTIMATORS OF FINITE POPULATION MEAN

In this paper, median based mean estimators for estimating finite population mean are proposed. The proposed estimators were obtained by transforming estimators in literature utilizing mean of auxiliary variable into median based estimators with the aim of obtaining estimators with higher efficiency. The mean square error of the proposed estimators was obtained up to the first order of approximation using Taylor series approach and the optimum values of the unknown of the estimators were obtained by means of partial derivative of the mean square error and equating to zero. A Numerical study was carried out to support the fact that the proposed estimators are more efficient as compared to the existing ones, as the proposed estimators have the least mean squared error at optimum values of the unknown constants and have higher percentage relative efficiency (PRE). This implies that the proposed estimators are more efficient than the traditional ones considered in the study.

An approximated exponentially tilted empirical likelihood estimator of moment condition models

. This study proposes an approximated estimator for moment condition models. Our estimator approximates the exponentially tilted empirical likelihood (ETEL) estimator in Schennach (2007) by using a second-order approximation of implied probabilities (IPs) for the exponential tilting (ET). The resulting approximated ETEL estimator generates positive IPs. As the nested optimization of ETEL and EL is avoided, it is computationally simple. It is second-order asymptotically equivalent to the EL estimator. In particular, it has the same O(n −1) bias as EL. It also has the same O(n −2) variance as EL. Like ET and ETEL estimators, it is n convergent under model misspecification, while the EL estimator may not be.

Inversion of fracture weakness parameters based on 3C vertical seismic profile data — Part I: Horizontal transversely isotropic media composed of a single fracture set in the isotropic background media

Natural fractures in oil and gas reservoirs are a crucial factor that cannot be ignored because they significantly influence the reservoir’s petrophysical properties and hydrocarbon development. A horizontal transversely isotropic (HTI) medium composed of a single fracture set in an isotropic background is a typical anisotropic medium. Meanwhile, the S-wave splitting is a sensitive response of such anisotropic media, resulting in the generation of fast and slow S waves. The normal and tangential fracture weaknesses are crucial parameters that characterize the anisotropy of fractured media. We develop an inversion method for fracture weakness based on 3C vertical seismic profiling (3CVSP) data. First, assuming weak anisotropy and an HTI medium containing a single fracture set, we derive a first-order linear approximation of the traveltimes of the converted fast and slow S waves (PS1 and PS2 waves) with respect to fracture weakness parameters in the phase velocity domain. By solving for the horizontal projection of the slowness vector, approximate equations of the traveltimes of the PS1 and PS2 waves are converted from the phase velocity domain to the group velocity domain. Furthermore, we devise an inversion workflow consisting of three primary steps: (1) preprocessing the VSP data to derive the traveltimes and azimuth of the HTI medium, (2) constructing a forward model with undetermined fracture weakness parameters, and (3) following the establishment of the objective function, conducting the inversion for the fracture weakness parameters. We demonstrate the reliability of the method through numerical examples and synthetic 3CVSP data. The inversion errors are primarily influenced by the azimuth angle, with minimal influence from the receiver depth. Furthermore, a collective set of inverted results derived from all geophones is more stable and accurate than individual geophones. The application to actual 3CVSP data further confirms the effectiveness of our approach.

Efficient Optimization of a Support Vector Regression Model with Natural Logarithm of the Hyperbolic Cosine Loss Function for Broader Noise Distribution

While traditional support vector regression (SVR) models rely on loss functions tailored to specific noise distributions, this research explores an alternative approach: ε-ln SVR, which uses a loss function based on the natural logarithm of the hyperbolic cosine function (lncosh). This function exhibits optimality for a broader family of noise distributions known as power-raised hyperbolic secants (PHSs). We derive the dual formulation of the ε-ln SVR model, which reveals a nonsmooth, nonlinear convex optimization problem. To efficiently overcome these complexities, we propose a novel sequential minimal optimization (SMO)-like algorithm with an innovative working set selection (WSS) procedure. This procedure exploits second-order (SO)-like information by minimizing an upper bound on the second-order Taylor polynomial approximation of consecutive loss function values. Experimental results on benchmark datasets demonstrate the effectiveness of both the ε-ln SVR model with its lncosh loss and the proposed SMO-like algorithm with its computationally efficient WSS procedure. This study provides a promising tool for scenarios with different noise distributions, extending beyond the commonly assumed Gaussian to the broader PHS family.

Non-destructive evaluation of the steel fibre content and anisometry in thin UHPFRC elements

The tensile response of ultra-high performance fibre-reinforced composites (UHPFRC) is decisive in many applications and depends on the steel fibre content and orientation. These vary troughout the structural element and may differ from those in the laboratory specimens used to characterize the material behaviour. This work presents the developments on a non-destructive test method based on the measurement of the magnetic inductance, substantiating its use for the determination of the fibre content and orientation in thin UHPFRC elements and allowing the estimation of the directionally dependent post-cracking tensile strength of the material in the structure. Starting from a probabilistic description of the fibre orientation, an existing physical model of the magnetic circuit composed of a U-shaped inductor and the composite is generalized and is used to derive the relations between the magnetic inductance measurements, the fibre volumetric fraction and the fibre orientation factor. A second-order tensor approximation of the relative magnetic permeability of the composite is proposed to determine the in-plane fibre orientation factor along any direction based on any three non-collinear measurements. Experimental evidence is presented supporting the theoretical developments. The factors that may affect the measurements are experimentally quantified. The paper concludes with an application example.

Validity conditions of approximations for a target-mediated drug disposition model: A novel first-order approximation and its comparison to other approximations.

Target-mediated drug disposition (TMDD) is a phenomenon characterized by a drug's high-affinity binding to a target molecule, which significantly influences its pharmacokinetic profile within an organism. The comprehensive TMDD model delineates this interaction, yet it may become overly complex and computationally demanding in the absence of specific concentration data for the target or its complexes. Consequently, simplified TMDD models employing quasi-steady state approximations (QSSAs) have been introduced; however, the precise conditions under which these models yield accurate results require further elucidation. Here, we establish the validity of three simplified TMDD models: the Michaelis-Menten model reduced with the standard QSSA (mTMDD), the QSS model reduced with the total QSSA (qTMDD), and a first-order approximation of the total QSSA (pTMDD). Specifically, we find that mTMDD is applicable only when initial drug concentrations substantially exceed total target concentrations, while qTMDD can be used for all drug concentrations. Notably, pTMDD offers a simpler and faster alternative to qTMDD, with broader applicability than mTMDD. These findings are confirmed with antibody-drug conjugate real-world data. Our findings provide a framework for selecting appropriate simplified TMDD models while ensuring accuracy, potentially enhancing drug development and facilitating safer, more personalized treatments.

Numerical simulations for fractional Hirota–Satsuma coupled Korteweg–de Vries systems

Abstract In this investigation, the fractional Hirota–Satsuma coupled Korteweg–de Vries (KdV) problem is solved using two modern semi-analytic techniques known as the Aboodh residual power series method (ARPSM) and Aboodh transform iteration method (ATIM). The two suggested approaches are briefly explained, along with how to use them to solve the fractional Hirota–Satsuma coupled KdV problem. Some analytical approximate solutions for the current problem are derived using the proposed techniques until the second-order approximation. To ensure high accuracy of the derived approximation, they are analyzed numerically and graphically and compared with the exact solutions of the integer cases. The offered techniques demonstrate more accuracy in their outcomes compared to other alternatives. The numerical results show that ARPSM and ATIM are highly accurate, practical, and beneficial for solving nonlinear equation systems. The current results are expected to help many physics researchers in modeling their different physical problems, especially those interested in plasma physics.

On choosing initial values of iteratively reweighted ℓ1 algorithms for the piece-wise exponential penalty

Computing the proximal operator of the sparsity-promoting piece-wise exponential (PiE) penalty [Formula: see text] with a given shape parameter [Formula: see text], which is treated as a popular nonconvex surrogate of [Formula: see text]-norm, is fundamental in feature selection via support vector machines, image reconstruction, zero-one programming problems, compressed sensing, neural networks, etc. Due to the nonconvexity of PiE, for a long time, its proximal operator is frequently evaluated via an iteratively reweighted [Formula: see text] algorithm, which substitutes PiE with its first-order approximation, however, the obtained solutions only are the critical point. Based on the exact characterization of the proximal operator of PiE, we explore how the iteratively reweighted [Formula: see text] solution deviates from the true proximal operator in certain regions, which can be explicitly identified in terms of [Formula: see text], the initial value and the regularization parameter in the definition of the proximal operator. Moreover, the initial value can be adaptively and simply chosen to ensure that the iteratively reweighted [Formula: see text] solution belongs to the proximal operator of PiE.