Multiquadric Radial Basis Function Research Articles

A Comprehensive Price Prediction System Based on Inverse Multiquadrics Radial Basis Function for Portfolio Selection

Price prediction plays a crucial role in portfolio selection (PS). However, most price prediction strategies only make a single prediction and do not have efficient mechanisms to make a comprehensive price prediction. Here, we propose a comprehensive price prediction (CPP) system based on inverse multiquadrics (IMQ) radial basis function. First, the novel radial basis function (RBF) system based on IMQ function rather than traditional Gaussian (GA) function is proposed and centers on multiple price prediction strategies, aiming at improving the efficiency and robustness of price prediction. Under the novel RBF system, we then create a portfolio update strategy based on kernel and trace operator. To assess the system performance, extensive experiments are performed based on 4 data sets from different real-world financial markets. Interestingly, the experimental results reveal that the novel RBF system effectively realizes the integration of different strategies and CPP system outperforms other systems in investing performance and risk control, even considering a certain degree of transaction costs. Besides, CPP can calculate quickly, making it applicable for large-scale and time-limited financial market.

A Numerical computation for solving the fractional delay differential equations via the MQ-RBF collocation method

In the paper, a numerical technique based on the multiquadric radial basis function (MQ-RBF) collocation method is presented for the fractional delay differential equations. The convergence analysis and the unique fixed point of the collocation methods are propounded in the norm. Numerical results illustrate the credibility and applicability of the numerical technique. The obtained results are compared with those obtained by other methods.

NUMERICAL SOLUTION OF MAGNETOHYDRODYNAMIC FLOW PROBLEM USING RADIAL BASIS FUNCTION BASED FINITE DIFFERENCE METHOD

This paper focuses on the application of Radial Basis Function generated Finite Difference Method (RBF-FD) to solve Magnetohydrodynamic (MHD) flow equation in a rectangular duct in the presence of the transverse external oblique magnetic field. Multiquadric (MQ) Radial Basis Function is used to obtain the numerical solution of the MHD flow problem. Accuracy of the solution can be improved by varying the shape parameter in MQ function. The solution obtained from RBF-FD method is compared with the analytical solution and classical Finite Difference solution. Contours are presented for various Hartmann numbers with different grid sizes and directions of the applied magnetic field. The behaviour of velocity and the magnetic field of the MHD flow have been studied using the contours.

Thermal Accelerometer Simulation by the R‑Functions Method

As well as many modern devices, thermal accelerometers (TAs) need a sophisticated mathematical simulation to find the ways for their performance optimization. In the paper, a novel approach for solving computational fluid dynamics (CFD) problems in the TA’s cavity is proposed (MQ-RFM), which is based on the combined use of Rvachev’s R-functions method (RFM) and the Galerkin technique with multiquadric (MQ) radial basis functions (RBFs). The semi-analytical RFM takes an intermediate position between traditional analytical approaches and numerical methods, such as the finite-element method (FEM), belonging to the family of the so-called meshless techniques which became popular in the last decades in solving various CFD problems in complex-shaped cavities. Mathematical simulation of TA by using the MQ-RFM was carried out with the purpose to simulate the temperature response of the device and to study and improve its performance. The results of numerical experiments were compared with well-known analytical and numerical benchmark solutions for the circular annulus geometry and it demonstrated the effectiveness of the MQ-RFM for solving the convective heat-transfer problem in the TA’s cavity. The use of solution structures allows one to take a relatively small number of expansion terms to achieve an appropriate accuracy of the approximate solution satisfying at the same time the given boundary conditions exactly. The application of the MQ-RFM gives the possibility to obtain semi-analytical solutions to the diffusion-convection problems and to identify the main thermal characteristics of the TA, that allows one to improve the device performance.

Accurate radial basis functions technique for competitive and efficient solutions of non-linear Black-Scholes equations

ABSTRACTObjective: This article aims to solve the non-linear Black Scholes (BS) equation for European call options using Radial Basis Function (RBF) Multi-Quadratic (MQ) Method.Methodology / Approach: This work uses the MQ RBF method applied to the solution of two complex models of nonlinear BS equation for prices of European call options with modified volatility. Linear BS models are also solved to visualize the effects of modified volatility. Additionally, an adaptive scheme is implemented in time based on the Runge-Kutta-Fehlberg (RKF) method.

Using Machine Learning Radial Basis Function (RBF) Method for Predicting Lubricated Friction on Textured and Porous Surfaces

The coefficient of friction (CoF) obtained from tribological tests conducted on textured and porous surfaces was analysed using the machine learning Radial Basis Function (RBF) method. Non-textured and non-porous samples were taken as reference surfaces. Test parameters, such as entrainment velocity and slide-roll ratio (SRR), along with geometric characteristics of surface features (e.g. texture width and depth, coverage area, circularity, spatial distribution and directionality, among others), were selected as training dataset for the machine learning RBF model. The surface features were divided into designed patterns (dimples and grooves) manufactured by laser texturing, and randomised cavities (surface pores) resulted from the sintering process. The principal outcomes of this study are the effective use of the machine learning RBF method for tribological applications, as well as a critical discussion on its feasibility for the experimental dataset selected and the preliminary results obtained. Main results show that the Hardy multiquadric radial basis function provided an overall correlation coefficient of 0.934 for 35 poles. The application of the suggested machine learning technique and methodology can be extended to other experimental results available in the literature to train more robust models for predicting tribological performances of textured and structured surfaces.

Meshless Analysis of Nonlocal Boundary Value Problems in Anisotropic and Inhomogeneous Media

In this work, meshless methods based on a radial basis function (RBF) are applied for the solution of two-dimensional steady-state heat conduction problems with nonlocal multi-point boundary conditions (NMBC). These meshless procedures are based on the multiquadric (MQ) RBF and its modified version (i.e., integrated MQ RBF). The meshless method is extended to the NMBC and compared with the standard collocation method (i.e., Kansa’s method). In extended methods, the interior and the boundary solutions are approximated with a sum of RBF series, while in Kansa’s method, a single series of RBF is considered. Three different sorts of solution domains are considered in which Dirichlet or Neumann boundary conditions are specified on some part of the boundary while the unknown function values of the remaining portion of the boundary are related to a discrete set of interior points. The influences of NMBC on the accuracy and condition number of the system matrix associated with the proposed methods are investigated. The sensitivity of the shape parameter is also analyzed in the proposed methods. The performance of the proposed approaches in terms of accuracy and efficiency is confirmed on the benchmark problems.

Multiquadrics without the Shape Parameter for Solving Partial Differential Equations

In this article, we present multiquadric radial basis functions (RBFs), including multiquadric (MQ) and inverse multiquadric (IMQ) functions, without the shape parameter for solving partial differential equations using the fictitious source collocation scheme. Different from the conventional collocation method that assigns the RBF at each center point coinciding with an interior point, we separated the center points from the interior points, in which the center points were regarded as the fictitious sources collocated outside the domain. The interior, boundary, and source points were therefore collocated within, on, and outside the domain, respectively. Since the radial distance between the interior point and the source point was always greater than zero, the MQ and IMQ RBFs and their derivatives in the governing equation were smooth and globally infinitely differentiable. Accordingly, the shape parameter was no longer required in the MQ and IMQ RBFs. Numerical examples with the domain in symmetry and asymmetry are presented to verify the accuracy and robustness of the proposed method. The results demonstrated that the proposed method using MQ RBFs without the shape parameter acquires more accurate results than the conventional RBF collocation method with the optimum shape parameter. Additionally, it was found that the locations of the fictitious sources were not sensitive to the accuracy.

A methodology based on strain analysis for identifying potential discontinuous deformation zones in sublevel caving mines

Underground mining operations using the sublevel caving method often result in discontinuous ground deformations involving surface cracks. Determining the limit of the discontinuous deformation zone is important for the relocation of neighbouring residential areas, mining infrastructure, and the planning of underground mining operations. The limit of the discontinuous deformation zone is characterised by the outermost surface crack, which is a tensile crack when it occurs and is correlated with the horizontal ground tensile strain. In this study, a mathematical procedure combining the horizontal displacement, as periodically monitored by total station and GPS, multiquadric radial basis function (MQ-RBF) interpolation, and continuum mechanics is provided to calculate the strain state at any point on the ground. Then the relationship between a new tensile crack and horizontal ground tensile strain is studied to determine the criterion representing the presence of a new tensile crack. This means that, when the criterion at any point on the ground is satisfied, the point will have the potential to crack. Based on this, the potential discontinuous deformation zone where surface cracks may occur can be identified, potential break angles at representative sections can then be determined. A case study is presented to illustrate the results given by the above methodology. The result shows that when major principal horizontal ground strain ε1 at any point on the ground reaches the potential cracking major principal strain, PCMPS (6 mm/m in this case study), the point will have the potential to crack. Subsequently, the area within the outermost contour of the PCMPS can be defined as the potential discontinuous deformation zone. Moreover, the approximate matching of the limits of potential discontinuous deformation zones and surface cracks verifies the reliability of the mathematical procedure and the rationality of the PCMPS in identifying the discontinuous deformation zone. Additionally, in this case study, the potential break angles at two respective sections are significantly different, indicating that the break angles should be further studied and care taken in design of works in these two zones, respectively.

Collocation Method for Multiplicative Noise Removal Model

Image denoising is a fundamental problem in both image processing and computer vision with numerous applications. It can be formulated as an inverse problem. Variational methods are commonly used to solve noise removal problems. The Total Variation (TV) regularization has evolved from an image denoising method for images corrupted with multiplicative noise into a more general technique for inverse problems such as denoising, deblurring, blind deconvolution, and inpainting, which also encompasses the Impulse, Poisson, Speckle, and mixed noise models. Multiplicative noise removal based on TV regularization has been widely researched in image science. In multiplicative noise problems, original image is multiplied by a noise rather than added to the original image. This article proposes a novel meshless collocation technique for the solution of a model having multiplicative noise. This technique includes TV and local collocation along with Multiquadric Radial Basis Function (MQ-RBF) for the solution of associated Euler-Lagrange equation for restoring multiplicative noise from digital images. Numerical examples demonstrate that the proposed algorithm is able to preserve small image details while the noise in the homogeneous regions is removed sufficiently. As a consequence, our method yields better denoised results than those of the current state of the art methods with respect to the Peak-Signal to Noise Ratio (PSNR) values.

Surface roughness modeling using response surface methodology and a variant of multiquadric radial basis function

There are several methods for modeling surface roughness in computerized numerical control (CNC) milling process. This work proposes for the first time the use of response surface methodology (RSM) and a variant of the multiquadric radial basis function (MRBF) for the modeling of surface roughness (Ra) of graphite generated by means of the CNC milling process. In this variant, a norm with coefficients (weights) denoting the weighting of the effects of machining parameters is used. The machining parameters that have been most reported in literature were considered, which are cutting speed (S), feed rate (F), and cutting depth (D). Three mathematical models were developed to adjust the values of surface roughness: one model was obtained using RSM and two others were generated with the variant of MRBF. Values were assigned to effects of the weighted norm reducing it to the Euclidean norm, named Euclidean norm model (ENM). For the weighted norm model (WNM), values of effects of the machining parameters calculated by means of a factorial design were considered. Finally, in each model, the mean absolute percentage error (MAPE) was calculated: the lowest value and highest value of MAPE were obtained with response surface methodology model (RSMM) and ENM, respectively. However, lowest surface roughness was estimated and generated experimentally in graphite specimens with the machining parameter values obtained by means of WNM.

Analysing time-fractional exotic options via efficient local meshless method

In this article, we analyse the numerical simulation of the time-fractional Black-Scholes model governing butterfly spread option, digital option and double barrier option. For this purpose, a local meshless collocation method based on multiquadric radial basis function is used. The model equation is discretized in the temporal sense by Liouville-Caputo fractional derivative scheme for 0<β<1, whereas the space derivatives are discretized by the suggested meshless method. Numerical experiments are performed for butterfly spread option, digital option and double barrier option. Efficiency and accuracy of the proposed meshless method are assessed in terms of double mesh procedure since the exact solution of these options are not available. Furthermore, the greeks (delta and gamma) of the options are also computed.

Extending meshless method of approximate particular solutions (MAPS) to two-dimensional convection heat transfer problems.

Natural, forced and mixed convection heat transfer problems are solved by the meshless Method of Approximate Particular Solutions (MAPS). Particular solutions of Poisson and Stokes equations are employed to approximate temperature and velocity, respectively. The latter is used to obtained a closed expression for pressure particular solution. In both cases, the source terms are multiquadric radial basis functions which allow to obtain analytical expressions for these auxiliary problems. In order to couple momentum and energy equations, a relaxation strategy is implemented to avoid convergence problems due to the difference between successive temperature and velocity changes when solving the steady problem from an initial guess. The developed and validated numerical scheme is used to study flow and heat transfer in two two-dimensional problems: natural convection in concentric annulus between a square and a circular cylinder and non-isothermal flow past a staggered tube bundle. Numerical solutions obtained by MAPS are comparable in accuracy to solutions reported by authors who uses denser nodal distributions, showing the capability of the present method to accurately solve heat convection problems with temperature and heat flux boundary conditions as well as curve geometries and internal flow situations.

A Collocation Method Using Radial Polynomials for Solving Partial Differential Equations

In this article, a collocation method using radial polynomials (RPs) based on the multiquadric (MQ) radial basis function (RBF) for solving partial differential equations (PDEs) is proposed. The new global RPs include only even order radial terms formulated from the binomial series using the Taylor series expansion of the MQ RBF. Similar to the MQ RBF, the RPs is infinitely smooth and differentiable. The proposed RPs may be regarded as the equivalent expression of the MQ RBF in series form in which no any extra shape parameter is required. Accordingly, the challenging task for finding the optimal shape parameter in the Kansa method is avoided. Several numerical implementations, including problems in two and three dimensions, are conducted to demonstrate the accuracy and robustness of the proposed method. The results depict that the method may find solutions with high accuracy, while the radial polynomial terms is greater than 6. Finally, our method may obtain more accurate results than the Kansa method.

Analysis of filamentation instability in a current-carrying plasma using meshless method of lines coupled with radial basis functions

The filamentation instability and the nonlinear dynamics of the magnetic field produced in a current-carrying plasma are investigated in the diffusion regime. Starting the coupled fluid equations for electrons and ions and using the Ampere-Maxwell equation, a nonlinear partial differential equation for the magnetic field diffusion is obtained. This paper presents an efficient meshless method of lines using radial basis functions to approximate this equation and consequently to analyze the magnetic field diffusion and the electron density distribution in the plasma. For this purpose, we applied the multi-quadric radial basis functions on some points and found appropriate approximations. Our method provides an expansion which is smooth enough for applying on high order differential equations. This method does not depend on any meshing procedure and has good accuracy, but is seriously parameter dependent. Finally, we have compared our results with two previously well-established finite difference and Adomian decomposition methods.

Solving the Nonlinear Heat Equilibrium Problems Using the Local Multiquadric Radial Basis Function Collocation Method

In this article, the nonlinear heat equilibrium problems are solved by the local multiquadric (MQ) radial basis function (RBF) collocation method. The system of nonlinear algebraic equations is solved by iteration based on the residual norm-based algorithm, in which the direction of evolution is determined by a linear equation. In addition, the role of the collocation point and source point is clearly defined such that in our proposed method the field value of any interested point can be expressed. Six numerical examples are shown to check the performance of the proposed method. As the number of supporting points (mp) increases, the accuracy of numerical solution increases. Among all examples, mp = 50 can perform well. In addition, the selection of shape parameter, c, affects the accuracy. However, as c &lt; 2 the maximum relative absolute error percentage is less than 1%.

ARBF: adaptive radial basis function interpolation algorithm for irregularly scattered point sets

Radial basis functions (RBFs) are isotropic, simple in form, dimensionally independent and mesh-free and are suitable for interpolation and fitting of scattered data. In a scattered point set, the calculation accuracy of multiquadric (MQ) RBF interpolation is strongly related to the selection of the shape factor. There is still no uniform method for determining the shape factor. Many scholars focus on determining the single optimal shape factor and seldom consider the change in the shape factor with the spatial point density in scattered point sets. In this paper, an adaptive radial basis function (ARBF) interpolation algorithm is proposed. The shape factors of MQ functions are determined adaptively by the local point densities of the points to be interpolated. To evaluate the computational performance of the ARBF interpolation algorithm, twelve groups of benchmark tests are conducted in this paper. We found that (1) the numerical error of ARBF interpolation is approximately 10% less than that of commonly used RBF interpolation with the shape factor recommended by Hardy. (2) The computational efficiency of ARBF interpolation is 1–2.5% lower than that of commonly used RBF interpolation with the shape factor recommended by Hardy.

Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method

Fractional differential equations depict nature sufficiently in light of the symmetry properties which describe biological and physical processes. This article is concerned with the numerical treatment of three-term time fractional-order multi-dimensional diffusion equations by using an efficient local meshless method. The space derivative of the models is discretized by the proposed meshless procedure based on the multiquadric radial basis function though the time-fractional part is discretized by Liouville–Caputo fractional derivative. The numerical results are obtained for one-, two- and three-dimensional cases on rectangular and non-rectangular computational domains which verify the validity, efficiency and accuracy of the method.

Nonlinear bending analysis of functionally graded porous beams using the multiquadric radial basis functions and a Taylor series-based continuation procedure

In this paper, the nonlinear bending analysis of Functionally Graded Porous (FGP) beams is investigated using an efficient numerical algorithm associating a meshless collocation technique uses the Multiquadric Radial Basis Function (MQRBF) approximation method and a higher-order Taylor series-based continuation procedure. Material properties of the FGP beams are described by adopting a modified power-law function taking into account the effect of porosities. Based on the First Order Shear Deformation Theory (FSDT) of beams with the von-Kármán kinematic hypothesis, the strong form of nonlinear equations is established. For an efficient application of the proposed numerical approach, a quadratic matrix strong form of the problem is presented. The resulting nonlinear equations are solved numerically with the proposed algorithm which leaned on the following three steps: a higher-order Taylor series expansion to transform the quadratic nonlinear equations into a sequence of linear ones, a meshless collocation technique based on MQRBF approximation method to solve numerically the resulting linear equations and a continuation procedure to get the whole solution branch. To demonstrate the robustness of the developed algorithm, convergence and validation studies have been carried out. Furthermore, the effects of power-law index, porosity volume fraction, Young’s modulus ratio, loads and boundary conditions are investigated.

A leap-frog meshless method with radial basis functions for simulating electromagnetic wave splitter and rotator

In this paper, we develop a meshless method by using radial basis functions (RBFs) to solve time domain Maxwell’s equations resulting from simulating wave propagation in electromagnetic wave splitter and rotator devices. To simulate wave propagation in these devices, we introduce a perfectly matched layer (PML) to reduce the unbounded physical domain problem to a bounded domain problem. Using PML leads to a multi-physics problem with different governing equations in different subdomains, which makes the simulation quite challenging. By following the idea of leap-frog FDTD scheme, we develop the leap-frog RBF meshless method to solve these complicated coupled modeling equations. Extensive numerical results by using multiquadric RBF and Gaussian RBF are presented to demonstrate the effectiveness of our meshless method.