B-spline Method Research Articles

Registration and estimation for functional generalized linear model

. Functional data analysis has recently concentrated on the challenges of registration and estimation. Our research team has developed an innovative model that enables the simultaneous registration and estimation of multivariate functional data. This approach eliminates the need for pre-smoothing, thereby improving computational efficiency. We utilize the inverse warping function in conjunction with generalized functional principal component analysis (GFPCA) to approximate the covariance structure effectively. The inverse warping function, population mean function, and orthonormal eigenfunctions are estimated using B-spline methods. We have derived the joint probability distribution and established its asymptotic normality. Through extensive simulation studies and real data analyses, we compare our method with existing approaches and demonstrate its exceptional performance, showcasing superior fitting capabilities.

Comparing Solutions to the Linear Space-Fractional Telegraph Equation

The applications of the fractional differentiation for the mathematical modelling of real-world physical problems such as the earthquake modelling, and allometric scaling laws in biology, etc., have been widespread in this modern era. Fractional differential theory has gained much more attention as the fractional order system response ultimately converges to the integer order equations. Differential equations of fractional order have been successfully employed for modelling the so-called anomalous phenomena during last two decades. One of these nonlinear partial differential equations, the linear space-fractional telegraph equation. That applied into signal analysis for transmission, propagation of electrical signals, and so on. The aim of this article is to compare were the fractional Sumudu decomposition method (SDM), a double Sumudu matching transformation method, a finite difference scheme, a finite difference scheme based on a combination of the extended cubic B-splines (ExCuBs) method, and a quadratic spline functions with the solution of the linear space-fractional telegraph equation. We will conduct a comparison of the stability of the methods and convergence. In addition, numerical examples will be presented to illustrate the accuracy of these methods.

Natural Cubic Spline Approximation of Risk-Neutral Density

The risk-neutral density is a fundamental concept in pricing financial derivatives, risk management, and assessing financial markets’ perceptions over significant political or economic events. In this paper, we propose a new nonparametric method for estimating the risk-neutral density using natural cubic splines (NCS). The estimated density is twice continuously differentiable with linear tails at both ends. Our method targets the logarithm of the underlying asset price, releasing the restriction to the positive domain. We theoretically prove the consistency of our NCS method. We conduct a comprehensive empirical study comparing the proposed NCS method with a piecewise constant method, a uniform quartic B-spline method, and a cubic spline method from the literature using 20 years of S&P 500 index option data. The empirical results show that our NCS method is more robust than the piecewise constant method, which can only produce a discontinuous density, especially for options with maturities longer than six months. Moreover, our NCS method outperforms other historical continuous methods in terms of optimization feasibility and option price estimation.

B-Spline Solution of the Two-Center Dirac Equation in the Electronic Continuum for Relativistic Molecular Photoionization.

In this work, the two-center Dirac equation is solved numerically using an extension of an adapted B-spline basis set method previously implemented in relativistic atomic calculations (Fischer, C. F.; Zatsarinny, O. Comput. Phys. Commun. 2009, 180, 879). The robustness of the chosen numerical method, which avoids the appearance of spurious states common in other approaches, allows us to investigate molecular photoionization within a relativistic framework by simply adapting those methods already available in the nonrelativistic case (Brosolo, M.; Decleva, P. Chem. Phys. 1992, 159, 185; Brosolo, M.; Decleva, P.; Lisini, A. Mol. Opt. Phys. 1992, 25, 3345). First, light diatomic molecules (i.e., H2+ and HeH2+) are investigated with the purpose of testing the validity and efficiency of the method. Then, a series of one-electron molecular hydrides (i.e., HF9+, HCl17+ and HI53+) is explored by computing the total photoionization cross sections, asymmetry β-parameters and partial phase shifts. The present methodology can be easily extended to treat N-electron molecules following previous approaches in nonrelativistic calculations (Plesiat, E.; Decleva, P.; Martin, F. Phys. Chem. Chem. Phys. 2012, 14, 10853). The inclusion of a second photon can be also accomplished just like in atomic investigations aiming at reproducing pump-probe experiments capable to extract the photoionization time-delays (Vinbladh, J.; Dahlstrom, J. M.; Lindroth, E. Phys. Rev A 2019, 100, 043424; Vinblach, J.; Dahlstrom, J. M.; Lindroth, E. Atoms 2022, 10, 80).

Numerical study of the time-fractional partial differential equations by using quartic B-spline method

This paper utilizes the quartic B-spline method for the numerical resolution of time-fractional partial differential equations. The fractional Caputo derivative is employed to depict anomalous diffusion processes influenced by memory effects. The proposed numerical method utilizes quartic B-spline functions for spatial discretization and employs a finite difference method to address the time-fractional derivative. Based on the Fourier method, its stability has been evaluated to demonstrate its efficacy in addressing fractional-order models. Numerical experiments, encompassing both linear and nonlinear scenarios, are performed to illustrate the method’s effectiveness and accuracy. The results obtained are compared with exact solutions and alternative numerical methods, demonstrating improved performance in convergence and computational efficiency.

Fast and universal single-molecule localization using multi-dimensional point spread functions

The recent development of single-molecule imaging techniques has enabled not only high accuracy spatial resolution imaging but also information rich functional imaging. Abundant information about single molecules can be encoded in its diffraction pattern and be extracted precisely (e.g. 3D position, wavelength, dipole orientation). However, sophisticated high dimensional point spread function (PSF) modeling and analyzing methods have greatly impeded the broad accessibility of these techniques. Here, we present a graphics processing unit (GPU) -based B-spline PSF modeling method that could flexibly model high dimensional PSFs with arbitrary shape without greatly increasing the model parameters. Our B-spline fitter achieves 100 times speed improvement and minimal uncertainty for each dimension, enabling efficient high dimensional single-molecule analysis. We demonstrated, both in simulations and experiments, the universality and flexibility of our B-spline fitter to accurately extract the abundant information from different types of high dimensional single-molecule data, including multicolor PSF (3D + color), multi-channel four-dimensional 4Pi-PSF (3D + interference phase) and five-dimensional vortex PSF (3D + dipole orientation).

Numerical solution and analysis of extended Fisher–Kolmogorov equation using an improved collocation algorithm

The paper investigated the approximate solution of the extended Fisher–Kolmogorov. For this, an advanced approach: the improved quintic B-spline collocation technique has been employed, which is an enhancement over the conventional B-spline collocation method. The B-spline interpolant of degree five has been refined through the posteriori corrections, leading to the development of the improved B-spline solution. This proposed technique demonstrates superior convergence compared to the standard B-spline method, as assessed both theoretically and numerically. In tackling the extended Fisher–Kolmogorov equation, the space discretization is conducted using the improved quintic B-spline collocation methodology (IQSCM), and for the temporal domain discretization, the Crank-Nicolson scheme is applied. The error bounds and convergence analysis is established using the Green's functions. Von-Neumann stability analysis is carried out to discuss the stability of the technique. A few examples are solved and are represented graphically which helps in determining the nature of the solution. Also, L 2 , L ∞ error norms, and order of convergence are calculated to demonstrate the contribution of the new improved technique over the standard spline collocation technique.

Investigating novel optical soliton solutions for a generalized (3+1)-dimensional q-deformed equation

In this study, we investigate the (3+1) q-deformed tanh-Gordon equation due to its importance in the context of mathematical physics. It describes solitonic solutions in quantum field theory; it can sometimes be used in condensed matter physics to describe interactions between particles in magnetic materials or superconductors; it can model light propagation in nonlinear optical fibers or photonic crystals where the refractive index has a q-deformed structure; and it also can be applied in studying shock waves, turbulence and rogue waves where the deformation introduces corrections to classical wave phenomena. Utilizing the (G′ωG′+G+r)-expansion technique, we derive novel analytical solutions that enhance our understanding of the underlying dynamics. Additionally, we employ a finite element method (extended cubic B-spline method) to validate our analytical findings and explore the behavior of the q-deformed equation under different parameter regimes. Our results demonstrate the versatility of the q-deformed framework in generating rich optical phenomena, paving the way for future research in both theoretical and applied physics.

A Sixth-Order Cubic B-Spline Approach for Solving Linear Boundary Value Problems: An In-Depth Analysis and Comparative Study

This research presents an efficient and highly accurate cubic B-spline method (CBSM) for solving second-order linear boundary value problems (BVPs). The method achieves sixth-order convergence, supported by rigorous error analysis, ensuring rapid error reduction with mesh refinement. The effectiveness of the CBSM is validated through four numerical examples, showcasing its accuracy, reliability, and computational efficiency, making it well-suited for large-scale problems. A comparative analysis with existing methods confirms the superior performance of the CBSM, positioning it as a practical and powerful tool for solving second-order BVPs.

Comparative Analysis of NURBS and Finite Element Method in Computational Fluid Dynamics Applications: Case Study on NACA 2412 Airfoil Aerodynamics

In this research, an attempt was made to employ the Non-Uniform Rational B-Splines (NURBS) method for a challenging computational fluid dynamics (CFD) problem of aerodynamics around NACA 2412 airfoils. The comparison was carried out thoroughly by using the same boundary conditions and geometry, comparing NURBS to standard FEM implementations. Our study was interested in demonstrating the foreseeable functionalities of NURBS for solving complex CFD problems and conducting a comparative effectiveness performance evaluation between them with traditional FEM methodologies.

Extended B-Spline Collocation Method for Singularly Perturbed Time Delay Parabolic Differential-Difference Problems

The paper presents a robust collocation B-spline numerical scheme for solving singularly perturbed time lag parabolic differential–difference problems. The method uses Taylor’s series expansion for the approximation of retarded terms, the [Formula: see text]-method (when [Formula: see text]) for time discretization, and the extended third-degree B-spline collocation method for spatial discretization in a piecewise uniform Shishkin mesh. The advantage of using an extended cubic B-spline rather than the classical third-degree B-spline method is that it introduces one additional free parameter to control the global shape parameter. The stability and uniform convergence of the proposed scheme are investigated. The method is first-order accurate in [Formula: see text] and almost second-order accurate in [Formula: see text], surpassing existing literature methods.

Adaptive mesh extended cubic B-spline method for singularly perturbed delay Sobolev problems

The purpose of this paper is to develop a robust numerical scheme for a class of singularly perturbed delay Sobolev (pseudo-parabolic) problems that have wide application in various branches of mathematical physics and fluid mechanics.For the small perturbation parameter, the standard numerical schemes for the solution of these problems fail to resolve the boundary layer(s) and the oscillations occur near the boundary layer. Thus, in this paper to resolve the boundary layer(s), im-plicit Euler scheme for the time derivatives on uniform mesh and extended B-spline basis functions consisting of free parameter λ are presented for spatial variable on Bakhvalov type mesh. The stability and uniform convergence analyisis of the proposed method are established. The error estimation of the developed method is shown to be firts order accurate in time and second order accurate in space. Numerical exprementation is carried out to validate the applicability of the developednumerical method. The numerical results reveals that the computational result is in agreement with the theoretical estimations

Cubic B-spline based elastic and viscoelastic wave propagation method

The requirement for high computational efficiency in inversion imaging poses challenges to further enhancing the accuracy of imaging large elastic and viscoelastic models. Considering that forward modeling serves as a prerequisite for inversion imaging, it is essential to introduce a wave propagation simulation method that can enhance simulation accuracy without significantly compromising computational speed. In our paper, cubic B-spline is introduced into the forward modeling of the wave equation. The B-spline collocation method offers a linear complexity for computation and exhibits fourth-order convergence in terms of computational errors. We also find that it can perfectly adapt to the boundary conditions of the perfectly matched layer in the forward modeling solution of the wave simulation. Several typical numerical examples, including 2-D acoustic wave equation, 2-D elastic wave propagation, and 2-D viscoelastic propagation in homogeneous media, are provided to validate its convergence, accuracy, and computational efficiency.

Comparative analysis of parametric B-spline and Hermite cubic spline based methods for accurate ECG signal modeling

Analyzing Electrocardiogram (ECG) signals is imperative for diagnosing cardiovascular diseases. However, evaluating ECG analysis techniques faces challenges due to noise and artifacts in actual signals. Machine learning for automatic diagnosis encounters data acquisition hurdles due to medical data privacy constraints. Addressing these issues, ECG modeling assumes a crucial role in biomedical and parametric spline-based methods have garnered significant attention for their ability to accurately represent the complex temporal dynamics of ECG signals. This study conducts a comparative analysis of two parametric spline-based methods—B-spline and Hermite cubic spline—for ECG modeling, aiming to identify the most effective approach for accurate and reliable ECG representation. The Hermite cubic spline serves as one of the most effective interpolation methods, while B-spline is an approximation method. The comparative analysis includes both qualitative and quantitative evaluations. Qualitative assessment involves visually inspecting the generated spline-based models, comparing their resemblance to the original ECG signals, and employing power spectrum analysis. Quantitative analysis incorporates metrics such as root mean square error (RMSE), Percentage Root Mean Square Difference (PRD) and cross correlation, offering a more objective measure of the model's performance. Preliminary results indicate promising capabilities for both spline-based methods in representing ECG signals. However, the analysis unveils specific strengths and weaknesses for each method. The B-spline method offers greater flexibility and smoothness, while the cubic spline method demonstrates superior waveform capturing abilities with the preservation of control points, a critical aspect in the medical field. Presented research provides valuable insights for researchers and practitioners in selecting the most appropriate method for their specific ECG modeling requirements. Adjustments to control points and parameterization enable the generation of diverse ECG waveforms, enhancing the versatility of this modeling technique. This approach has the potential to extend its utility to other medical signals, presenting a promising avenue for advancing biomedical research.

A modified basis of cubic B-spline with free parameter for linear second order boundary value problems: Application to engineering problems

The traditional cubic B-spline method offers limited local control over the curve solution. Adjusting the position of a control point affects the entire curve, making it challenging to make localized changes, e.g., smoothness. Moreover, the basis functions vanish on one side by the cubic B-spline method near the end conditions where the initial and boundary conditions are applied. To address these limitations, this research proposes a new basis by including a free parameter γ with the purpose of modifying the weights of nearby control points. This free parameter γ can influence the curve’s behavior in specific regions as well as the entire curve. This modification of the cubic B-spline method was used to approximate the second-order derivative at each collocation point. The convergence test showed that the proposed method was second-order convergent. Numerical examples of ordinary differential equations were used with different step values to evaluate the accuracy of the proposed method. The findings persistently indicated that the proposed technique provided better error estimates as compared to the other methods discussed in the literatures.

A robust B-spline method for two parameter singularly perturbed parabolic differential equations with discontinuous initial condition

A robust B-spline method for two parameter singularly perturbed parabolic differential equations with discontinuous initial condition

Exponential B-Spline Method for Second Order Singularly Perturbed Boundary Value Problem with Negative Shift

Exponential B-Spline Method for Second Order Singularly Perturbed Boundary Value Problem with Negative Shift

Hawking radiation of 5D Ricci-flat black hole

In this paper, we investigate the properties of a five-dimensional Ricci-flat black hole within the framework of the Space–Time–Matter theory. By employing the Klein–Gordon equation, we derive the wave function equations for both the mass dimensional coordinate and the radial dimensional coordinate. We present the analytical solutions for the wave function on the mass dimensional coordinate under two distinct scenarios. Remarkably, we observe the absence of quantum effects on the mass dimensional coordinate, which aligns with the absence of a barrier in the wave function equation. For the radial component, we employ the B-spline method to numerically solve its wave function. Our findings reveal the existence of Hawking radiation between the event horizon and the cosmological horizon, with the transmission and reflection probabilities being determined by the presence of a barrier. Furthermore, we discover a significantly higher probability of particle detection near the cosmological horizon, while the probability near the event horizon approaches zero.

Optimal section design of surface piercing propeller using artificial neural networks algorithm

ABSTRACT Section optimisation of a surface piercing propeller (SPP) with the combination of artificial neural networks and NSGA II algorithm is investigated in this research. The blade section have a sharp leading edge and thick trailing edge. The geometry of this propeller is mostly designed experimentally, and comprehensive studies have not been performed to examine its parameters. In this regard, the propeller's parametric geometry is designed based on the cubic B-spline method. The SPP was first fabricated and tested, and the simulated data of the tested propeller were verified. Then, limited data were extracted according to a verified CFD model based on which the ANN was trained. The ANN results indicated the mean square error of the trained network is 3e-9, 1e-7, and 1e-7 for training, validation, and test data, respectively. comparing the optimisation and experimental results showed a relative error amounting to 3.5% in thrust and 4.24% in torque.

Stochastic occupancy modeling for spaces with irregular occupancy patterns using adaptive B-Spline-based inhomogeneous Markov Chains

This paper presents a discrete time, discrete state-space in-homogeneous Markov Chains model for stochastic occupancy modeling in spaces with irregular occupancy patterns. The goal of the model is to provide accurate predictions of occupancy numbers, enabling appropriate actions to be taken for HVAC system to maintain optimal indoor environment. The proposed Markov Chain model incorporates time in-homogeneity by coupling the time-varying model parameters using a Periodic B-Spline expansion with adaptive knots, which effectively captures patterns in occupancy activity. This method optimizes the distribution of knots based on specific occupancy characteristics observed in different types of rooms. To evaluate the effectiveness of the proposed method, six months of occupancy data collected from a meeting room are utilized. A comprehensive comparison is conducted between the proposed adaptive B-Spline method and other approaches, including the counting method and uniform B-Spline method. The comparison considers both model accuracy and complexity, using metrics such as the Akaike Information Criterion and Bayesian Information Criterion. Results indicate that the proposed model achieves more accurate predictions with fewer model parameters compared to other methods. These forecasts are particularly useful in optimizing the control of HVAC systems, where accurate predictions of future occupancy numbers are essential.