B-spline Functions Research Articles

A reduced-order model based on cubic B-spline basis function and SSP Runge–Kutta procedure to investigate option pricing under jump-diffusion models

The purpose of this research is to develop a numerical method that can be used to deal with option pricing in jump-diffusion models. The proposed model is made up of a backward partial integro-differential equation with diffusion and advection factors. For the first and second order derivatives, the pseudo-spectral technique is used in conjunction with cubic B-spline functions. As a consequence, the first- and second-order derivative matrices are constructed, and an ODE system is created. Second-order Strong Stability Preserved Runge–Kutta (SSP) procedure is used to solve the ODE problem. According to what has been stated, the main model falls under the advection–diffusion classification. As a result, in order to reach the ultimate time, we need to increase both the number of collocations and the number of time steps in order to improve our results. That is why the final algebraic system of equations has been condensed using a technique known as proper orthogonal decomposition (POD). The POD-cubic B-Spline (POD-CB-S) function collocation approach can be used to describe this numerical procedure. Finally, a number of test cases are conducted to demonstrate the proposed method’s efficiency and accuracy.

IG-LSPIA: Least Squares Progressive Iterative Approximation for Isogeometric Collocation Method

The isogeometric collocation method (IGA-C), which is a promising branch of isogeometric analysis (IGA), can be considered fitting the load function with the combination of the numerical solution and its derivatives. In this study, we develop an iterative method, isogeometric least-squares progressive-iterative approximation (IG-LSPIA), to solve the fitting problem in the collocation method. IG-LSPIA starts with an initial blending function, where the control coefficients are combined with the B-spline basis functions and their derivatives. A new blending function is generated by constructing the differences for collocation points (DCP) and control coefficients (DCC), and then adding the DCC to the corresponding control coefficients. The procedure is performed iteratively until the stop criterion is reached. We prove the convergence of IG-LSPIA and show that the computation complexity in each iteration of IG-LSPIA is related only to the number of collocation points and unrelated to the number of control coefficients. Moreover, an incremental algorithm is designed; it alternates with knot refinement until the desired precision is achieved. After each knot refinement, the result of the last round of IG-LSPIA iterations is used to generate the initial blending function of the new round of iteration, thereby saving great computation. Experiments show that the proposed method is stable and efficient. In the three-dimensional case, the total computation time is saved twice compared to the traditional method.

An algebraic characterization of B-splines

B-splines of order k can be viewed as a mapping N taking a (k+1)-tuple of increasing real numbers a0<⋯<ak and giving as a result a certain piecewise polynomial function. Looking at this mapping N as a whole, basic properties of B-spline functions imply that it has the following algebraic properties: (1) N(a0,…,ak) has local support contained in the interval [a0,ak]; (2) N(a0,…,ak) allows refinement, i.e. for every a∈∪j=0k−1(aj,aj+1) we have that if (α0,…,αk+1) is the increasing rearrangement of the points {a0,…,ak,a}, the ‘old’ function N(a0,…,ak) is a linear combination of the ‘new’ functions N(α0,…,αk) and N(α1,…,αk+1); (3) N is translation and dilation invariant. It is easy to see that derivatives of N(a0,…,ak) satisfy properties (1)–(3) as well.Let F be a mapping taking (k+1)-tuples of increasing real numbers to some generalized function. In this paper we show that under some additional mild condition on the size of the supports of F(a0,…,ak) relative to the interval [a0,ak], properties (1)–(3) are already sufficient to deduce that F(a0,…,ak) is a non-zero multiple of (some derivative of) a B-spline function. However, and somewhat surprisingly, we explicitly give examples of choices of F satisfying (1)–(3) but are not of this form.

Airfoil Shape Morphing through a Novel Parameterization and Fitting Optimization Method Based on Uniform Non-Rational B-Spline Functions

The aim of this work is to implement an innovative parameterization and fitting procedure for the definition of a mathematical model useful to describe a wide range of airfoils. They are partitioned into three sections: central box, leading edge, and trailing edge. Each section is mathematically represented by two opposed, uniform, non-rational B-spline curves, describing the upper and lower airfoil segments’ perimeter. A novel approach is used to ensure both the desired continuity between two adjacent segments (up to 2nd derivatives) and sufficient model versatility and flexibility while managing a limited number of parameters, defining tangent and curvature vectors as scale factor variables. These parameters allow for a variable separation approach during the geometric fitting procedure that can be carried out considering two nested optimization processes, one based on a genetic algorithm and the other on a numerical gradient evaluation of the objective function. The representation method has been verified against different airfoils, comparing the geometric and aerodynamic properties of the input and model-based generated profile. To show the mathematical model’s capabilities and possible applications, a comparison between existing and proposed airfoil approximation methods has been provided together with examples of “global” and “local” morphing and CFD analyses of the resulting airfoils.

Improvement of FFD parametric approach in the application of a lifting body

AbstractFFD (free-form deformation method) is one of the most commonly used parameterisation methods at present. It places the parameterised objects inside the control volume through coordinate system transformation, and controls the control volume through control points, thus realising the deformation control of its internal objects. Firstly, this paper systematically analyses and compares the characteristics and technical requirements of Bernstein, B-spline and NURBS (non-uniform rational b-splines) basic functions that can be adopted by FFD, and uses the minimum number of control points required to achieve the specified control effect threshold to express the control capability. Aiming at the problem of discontinuity at the right end in the actual calculation of B-spline basis function, a method of adding a small epsilon is proposed to solve it. Then, three basic functions are applied to the FFD parameterisation method, respectively, and the differences are compared from two aspects of the accurate expression of the model and the ability of deformation control. It is found that the BFFD (b-spline free-form deformation) approach owns better comprehensive performance when the control points are distributed correctly. In this paper, the BFFD method is improved, and a p-BFFD (reverse solution points based BFFD) method based on inverse solution is proposed to realise the free distribution of control points under the specified topology. Further, for the lifting body configuration, the control points of the p-BFFD method are brought closer to the airframe forming the EDGE-p-BFFD (edge constraints based p-BFFD) method. For the case in this paper, the proposed EDGE-p-BFFD method not only has fairly high parameterisation accuracy, but also reduces the expression error from 1.01E-3 to 1.25E-4, which is nearly ten times. It can also achieve effective lifting body guideline constraints, and has the ability of local deformation adapting to the configuration characteristics. In terms of the proportion of effective control points, the EDGE-p-BFFD method increases the proportion of effective control points from 36.7% to 50%, and the more control points, the more obvious the proportion increase effect. The new method also has better effect on the continuity of geometric deformation. At the same time, this paper introduces the independent deformation method of the upper and lower surfaces based on the double control body frames, which effectively avoids the deformation coupling problem of the simultaneous change of the upper and lower surfaces caused by the movement of control points in the traditional single control framework.

Numerical and dynamical behaviors of nonlinear traveling wave solutions of the Kudryashov–Sinelshchikov equation

In this article, a mathematical model symbolizing numerical solutions of the nonlinear Kudryashov–Sinelshchikov (KS) equation has been investigated. For this aim, we use collocation finite element method with septic B-spline functions and appropriate initial and boundary conditions on uniform mesh points. The proficiency and effectiveness of the presented method are examined in four test examples, consisting; behavior of a single soliton, collision of two solitons, Maxwellian initial condition and growth of a wavy hole. L2 and L∞ error norms are used to validate practicality and reliability of our numerical algorithm. In addition, two invariants are calculated to determine the conservation properties of the presented algorithm and the numerical results have been shown to be unconditionally stable. Obtained numerical results have been illustrated with tables and graphics for easy visualization of properties of the problem modeled. Numerical calculations to solve such problems prove the accuracy and effectiveness of the finite element method. In collocation method, using B-spline functions gives rise to a technique that needs only the unknown parameters at certain node points to create the solution. Collocation method has two excellent advantages: establishing method does not include integrations and the resulting matrix system is banded with small band width. Consequently, B-splines associate with the collocation provide a simple solution procedure of linear and nonlinear partial differential equations (PDEs). After that, we have analyzed error estimation for the Galerkin finite element method. Furthermore, dynamical properties of traveling waves of the KS equation are investigated in absence and presence of external periodic perturbation. In absence of perturbation, periodic traveling wave solution of the KS equation are studied. On the other hand, a collection of various quasiperiodic wave features of the KS equation is presented in presence of periodic perturbation. Therefore, it can be said that the numerical technique is effective and valid in obtaining numerical solutions of such PDEs. In addition, the study on dynamical properties of the KS equation provides qualitative behavior of the corresponding wave features.

An interval spline finite point method for size-dependent mechanical behaviors of defective functionally graded material nanobeam

In this paper, a novel method, i.e., interval spline finite point method is presented to investigate the mechanical behavior of functionally graded material nanobeam with defects, which can provide some guidance for the design of nanoscale devices. The size-dependent behavior of the functionally graded material nanobeam is considered, and the defects of nanobeam is taken into account and simulated by weakening/strengthening the stiffness of the nanobeam, where the material properties are considering as interval parameters. Combing the interval mathematic and the Timoshenko beam theory, the governing equation of the defective functionally graded material nanobeam is derived based on principle of minimum potential energy and variational principle. To solve the high order equilibrium equation with interval parameters, the interval spline finite point method is proposed, where the B-spline function is adopted as basic function to guarantee the accuracy. Moreover, to obtain the mechanical response of the nanobeam efficiently, a gradient-based vertex strategy is presented. Finally, two numerical examples are given and the effectiveness of the proposed method are validated by comparison with the existing methods.

Non-Ellipsoidal Infrared Group/Extended Target Tracking Based on Poisson Multi-Bernoulli Mixture Filter and B-Spline

This study provides a solution for multiple group/extended target tracking with an arbitrary shape. Many tracking approaches for extended/group targets have been proposed. However, these approaches make assumptions about the target shape, which have limitations in practical applications. To address this problem, in this work, an extended/group target tracking algorithm based on B-spline is proposed. Specifically, the extension of an extended or a group target was modeled as a spatial probability distribution characterized by the control points of a B-spline function that was then jointly propagated with the measurement rate model and kinematic component model over time using the Poisson multi-Bernoulli mixture (PMBM) filter framework. In addition, an amplitude-aided measurement partitioning approach is proposed to improve the accuracy caused by distance-based approaches. The simulation results demonstrate that the extension, shape and orientation of targets can be estimated better by the proposed algorithm, even if the shape changes. The tracking performance is also improved by about 10% and 13% compared to the other two algorithms.

Functional Data Analysis on Global COVID-19 Data

No one can deny that COVID-19 has spread around the world and is still emerging as a new variation in some areas. Since the time of its breakout, three coronavirus waves have emerged. Daily cases of COVID-19 from 79 countries around the world were selected for the study. The main objective of this research was to model and analyze the behavior of the disease's first wave in the world and on the Asian continent using functional data analysis methods. Functional models were fitted to the data using B-spline basis functions at different orders, and the best-fit curves were further analyzed with respect to their functional behavior and the rate of change. These curves were visualized during the preliminary analysis and later clustered within each continent using functional cluster analysis. The results indicated that all continents, apart from Asia, had two clusters based on their functional behavior, whereas the world data had three clusters. All the continents except the Asian continent had different functional forms and numbers of peaks, but they all had the same number of clusters. The world's 18 countries were divided into two categories, with the remaining 61 countries clustered into a single group. The identified cluster indices were further modeled using multinomial logistic regression models with six popular health index variables. In the world, people over the age of 70 made a significant contribution to selecting cluster 2 over cluster 1 and cluster 3 over cluster 2. On the Asian continent, the female smoker variable preferred cluster 2 over cluster 1, while cluster 3 over cluster 1 could be determined by the median age variable. The findings of the overall study would be helpful for the researchers to understand the spread of the disease and the impact of the health indices on its functional behavior.

Isogeometric indirect BEM solution based on virtual continuous sources placed directly on the boundary of 2D Helmholtz acoustic problems

An indirect boundary element method (BEM) based on isogeometric analysis (IGA) is proposed for 2D Helmholtz acoustic problems using virtual continuous sources placed directly on the problem boundary. The virtual sources form a virtual boundary identical to the main problem boundary. The proposed solution couples the CAD model with the analysis model by approximating both the CAD model and the virtual continuous sources with the same non-uniform rational B-spline functions (NURBS). Moreover, neither domain discretization nor truncation boundaries at the far-field are required. This solution creates only one coefficient matrix by directly arranging the linear system of equations.The solution follows a collocation scheme on the boundaries based on Greville abscissae with offsets wherever C0 continuity is encountered to permit an easy prediction for the normal directions and the free-terms at the collocation points required in the cases of Neumann and Robin boundary conditions. It allows us also to treat all integrals with standard Gauss quadrature points.Several numerical examples for exterior and interior acoustic problems are discussed to verify the proposed solution with comparisons to analytical solutions and previously published numerical results. These examples prove the robustness of the proposed solution even for high wavenumbers, in contrast to the previous attempts which implemented extensive investigations to find out the optimum place of the sources outside the domain producing minimum errors. Furthermore, no fictitious eigenfrequency problem is observed for exterior acoustic problems.

A strategy improving stiffness to resist local vibration of sandwich plates by extended Legendre higher-order model

Sandwich structures have been increasingly used in aerospace engineering, while local vibration modes of sandwich plates might occur earlier than global vibration modes attributing to the weak stiffness of the core layer, and it is difficult to predict such vibration modes. Therefore, local vibration modes are often discarded in the process of structural design, so that potential dangers might be encountered when the frequency of external load is close to the natural frequency of local vibration. To avoid such an issue, an extended Legendre higher-order theory (ELHT) is developed to accurately forecast the local dynamic behaviors of sandwich plates for the first time. Subsequently, an effective multi-objective optimization platform will be constructed to optimize the distribution of graphene-nanoplatelets (GPLs) in the core layer, so that natural frequencies of local vibration modes will be improved and largely more than the fundamental frequency. The B-spline basis functions are employed to describe various GPLs distributions, which is more convenient to produce original data for the machine learning method in the optimization process. Then, extreme gradient boosting optimized by Sparrow Search Algorithm (SSA-XGBoost) is utilized as the high-fidelity surrogate model to construct the objective functions to accelerate the interaction of the ELHT and non-dominated sorting sparrow search algorithm (NSSSA). By optimizing for minimum global and local fundamental frequency ratio and total mass of GPLs reinforced sandwich plates, the Pareto-optimal solutions are obtained from the present algorithm which can give an alternative suggestion for the future designer to ensure the structural dynamic safety of thick sandwich plates.

Linear-Time Algorithm for Computing the Bernstein–Bézier Coefficients of B-spline Basis Functions

A new differential-recurrence relation for the B-spline functions of the same degree is proved. From this relation, a recursive method of computing the coefficients of B-spline functions of degree m in the Bernstein-Bézier form is derived. Its complexity is proportional to the number of coefficients in the case of coincident boundary knots. This means that, asymptotically, the algorithm is optimal. In other cases, the complexity is increased by at most O ( m 3 ) . When the Bernstein-Bézier coefficients of B-spline basis functions are known, it is possible to compute any B-spline function in linear time with respect to its degree by performing the geometric algorithm proposed recently by the authors. This algorithm scales well when evaluating the B-spline curve at multiple points, e.g., in order to render it, since one only needs to find the coefficients for each knot span once. When evaluating many B-spline curves at multiple points (as is the case when rendering tensor product B-spline surfaces), such approach has lower computational complexity than using the de Boor-Cox algorithm. The numerical tests show that the new method is efficient. The problem of finding the coefficients of the B-spline functions in the power basis can be solved similarly. • A new differential-recurrence relation for the B-spline functions of the same degree is proved. • From this relation, a recursive method of computing the coefficients of B-spline functions of degree m in the Bernstein-Bézier form is derived. • Its complexity is proportional to the number of coefficients in the case of coincident boundary knots. This means that, asymptotically, the algorithm is optimal. • When the Bernstein-Bézier coefficients of B-spline basis functions are known, it is possible to compute any B-spline function in linear time with respect to its degree by performing the geometric algorithm proposed recently by the authors. • This algorithm scales well when evaluating many B-spline curves with the same knot sequences at multiple points, e.g., when rendering tensor product B-spline surfaces. • When evaluating many B-spline curves at multiple points, such approach has lower computational complexity than using the de Boor-Cox algorithm. • The numerical tests show that the new method is efficient. • The problem of finding the coefficients of the B-spline functions in the power basis can be solved similarly.

Application of trigonometric B-spline functions for solving Caputo time fractional gas dynamics equation

&lt;abstract&gt;&lt;p&gt;In this study, we present a numerical method that utilizes trigonometric cubic B-spline functions to solve the time fractional gas dynamics equation, which is a key component in the study of physical phenomena such as explosions, combustion, detonation and condensation in a moving flow. The Caputo formula is used to define the fractional time derivative, which generalizes the framework for both singular and non-singular kernels. To discretize the unknown function and its derivatives in the spatial direction, we employ trigonometric cubic B-spline functions, while the usual finite difference formulation is used to approximate the Caputo time fractional derivative. A stability analysis of the scheme is provided to ensure that errors do not propagate over time, and a convergence analysis is conducted to measure the accuracy of the solution. To demonstrate the effectiveness of the proposed methodology, we solve various relevant examples and present graphical and tabular results to evaluate the outcomes of the strategy.&lt;/p&gt;&lt;/abstract&gt;

Patient Spontaneous Effort Estimation in Digital Twin Model with B-spline Function

Patient work of breathing is of significant clinical interest for several decades. It is particularly relevant when gauging patient-ventilator interaction, patient-specific level of mechanical ventilation (MV) support, and weaning from full support to spontaneous breathing MV modes or off of MV entirely. Current monitoring approaches require additional equipment (usually expensive), well-trained clinicians (to collect and interpret these signals), or/and extra clinical interventions (increases difficulty and cost). This study extends a digital twin model to estimate patient spontaneous breathing effort (Pp curve) with a previously proposed model-based estimation method using b-spline functions. Data from 22 patients for two assisted MV modes, NAVA (neurally adjusted ventilatory assist) and PSV (pressure support ventilation), are employed. Estimation results are compared to breathing effort reflected by electrical activity of the diaphragm (EAdi) signals. Physiologically-relevant correlations in identified Pp curve area (negative and positive) and EAdi signal can be found in both NAVA and PSV data analysis. While Pp curves yield more negative area (larger PRCTneg), the corresponding breaths tend to have lower peak EAdi values and area under curve of EAdi signal (AUC[EAdi]) during inspiration. R2 values for NAVA data yield an interquartile range (IQR) from 0.31 to 0.68 for peak EAdi versus PRCTneg and 0.40 to 0.61 for AUC[EAdi] versus PRCTneg, respectively. Results differ between NAVA and PSV modes based on poorer patient-ventilator interaction observed in PSV, while the same level of expected physiological relevance is still observed. Overall, the extended digital twin model with b-spline functions to quantify patient-specific inspiratory effort shows promising application in helping guide weaning or changes in MV settings at bedside for assisted breathing modes of MV. In future, the identified Pp curves could also potentially be used to replace the need for costly measurement of EAdi signals.

Model-based Estimation of Neonatal Pleural Pressure

Spontaneous breathing during mechanical ventilation is common in the neonatal intensive care unit. The breathing efforts in spontaneous breaths generate negative pressures in the pleural space, complicating model-based analysis due to unknown parameter tradeoff. This paper estimates the pleural pressure and elastance in a single compartment model applied to data from a neonate undergoing synchronous intermittent mandatory ventilation. Two sets of 60 breaths were analysed. Resistance is empirically modelled based on endotracheal tube dimensions. Breath-to-breath elastance is estimated from expiration data, assuming passive expiration, and inspiratory pleural pressure is modelled using B-spline functions. Typical breathing trends were observed, patient-triggered inspiration was apparent, and the work of breathing was significantly decreased in ventilator supported breaths compared to unsupported breaths. Additionally, examples of ventilator-patient asynchrony were observed when comparing pleural pressure to airway pressure. This study contributes to the development of computational models for respiration mechanics. The fundamental aim is to provide clinicians with more physiological insight and tools to improve neonatal ventilation support.

Numerical study of stagnation point flow of Casson-Carreau fluid over a continuous moving sheet

&lt;abstract&gt; &lt;p&gt;This paper is devoted to analysis the behavior of heat transfer of Casson-Carreau fluid at the magnetohydrodynamic (MHD) stagnation point with thermal radiation over a continuous moving sheet. The suitable similarity transform is utilized to transfer the governing differential equations into a system of differential equations and then solve the converted non-linear system by the collocation technique based on the B-spline function (CTBS) and Runge-Kutta method (RK). The quasi-linearization technique is utilized to approach the non-linear equations of the model to a system of linear equations and used CTBS to acquire the solution of the system of linear equations. The obtained results are investigated with the present literature by direct comparison. It is found that an increment in the value of the Weissenberg number decreases the velocity profile and enhances the temperature profile for Casson and Carreau fluids. Conversely, increasing the values of the magnetic parameter, shrinking parameter, and Casson fluid parameter improve the velocity profile and depreciate the thermal distribution. Further, the temperature profile declines with an improvement in radiation parameter and Prandtl number for Casson and Carreau fluids. The influence of distinct physical parameters on the velocity and temperature profiles are depicted via tables and illustrative graphs.&lt;/p&gt; &lt;/abstract&gt;

Parallel ray tracing through freeform lenses with NURBS surfaces

We implement Monte Carlo-based parallel ray tracing to achieve quick irradiance evaluation for freeform lenses with non-uniform rational B-splines (NURBS) surfaces. We employ the inverse transform sampling method to sample rays uniformly from the Lambertian light source and adopt the analytical form of the B-spline basis function to achieve fast surface interpolation. When performing parallel calculations for the intersections between the rays and the NURBS surfaces, we propose a parameter transformation method to avoid the parameters escaping from the defined range in the iteration process. Simulation results of two complex picture-generating freeform lenses show that our method is fast and effective.

A weighted online regularization for a fully nonparametric model with heteroscedasticity

&lt;abstract&gt;&lt;p&gt;In this paper, combining B-spline function and Tikhonov regularization, we propose an online identification approach for reconstructing a smooth function and its derivative from scattered data with heteroscedasticity. Our methodology offers the unique advantage of enabling real-time updates based on new input data, eliminating the reliance on historical information. First, to address the challenge of heteroscedasticity and computation cost, we employ weight coefficients along with a judiciously chosen set of knots for interpolation. Second, a reasonable approach is provided to select weight coefficients and the regularization parameter in objective functional. Finally, We substantiate the efficacy of our approach through a numerical example and demonstrate its applicability in solving inverse problems. It is worth mentioning that the algorithm not only ensures the calculation efficiency, but also trades the data accuracy through the data volume.&lt;/p&gt;&lt;/abstract&gt;

Numerical solution for the system of Lane-Emden type equations using cubic B-spline method arising in engineering

&lt;abstract&gt;&lt;p&gt;In this study, we develop a collocation method based on cubic B-spline functions for effectively solving the system of Lane-Emden type equations arising in physics, star structure, and astrophysics. To overcome the singularity behavior of the considered system at τ = 0, we apply the L'Hôpital rule. Furthermore, we have carried out a convergence analysis of the proposed method and have demonstrated that it has a second-order convergence. To demonstrate the effectiveness, accuracy, simplicity, and practicality of the method, five test problems are solved numerically and the maximum absolute errors of the proposed method are compared with those of some existing methods.&lt;/p&gt;&lt;/abstract&gt;

B-spline approximation of discontinuous functions defined on a closed contour in the complex plane

In this paper we propose an efficient algorithm for approximating piecewise continuous functions, defined on a closed contour $\Gamma $ in the complex plane. The function, defined numerically on a finite set of points of $\Gamma $, is approximated by a linear combination of B-spline functions and Heaviside step functions, defined on $\Gamma $. Theoretical and practical aspects of the convergence of the algorithm are presented, including the vicinity of the discontinuity points.