Inverse Multiquadric Function Research Articles

Spherical interpolation of scattered data using least squares thin-plate spline and inverse multiquadric functions

Spherical interpolation of scattered data using least squares thin-plate spline and inverse multiquadric functions

An efficient radial basis function generated finite difference meshfree scheme to price multi-dimensional PDEs in financial options

In this paper, the purpose is to focus on proposing a new solver for pricing financial option at the presence of several assets. Toward this goal, first we introduce the weighting coefficients of the radial basis function-finite difference (RBF-FD) procedure for approximating the derivatives of the functions under the inverse multi-quadric function (IMQ). The weights are employed on a nonuniform mesh of points when tackling multi-dimensional partial differential equations (PDEs). A new solver based on RBF-FD methodology is then constructed employing matrix notations for solving option pricing problems. The theoretical discussions will be confirmed via numerical experiments on several test problems consisting of four to five underlying assets.

Inverse Multiquadric Function to Price Financial Options under the Fractional Black–Scholes Model

The inverse multiquadric radial basis function (RBF), which is one of the most important functions in the theory of RBFs, is employed on an adaptive mesh of points for pricing a fractional Black–Scholes partial differential equation (PDE) based on the modified RL derivative. To solve this problem, discretization along space is carried out on a non-uniform grid in order to focus on the hot area, at which the initial condition of the pricing model, i.e., the payoff, has discontinuity. The L1 scheme having the convergence order 2−α is used along the time fractional variable. Then, our proposed numerical method is built by matrices of differentiations to be as efficient as possible. Computational pieces of evidence are brought forward to uphold the theoretical discussions and show how the presented method is efficient in contrast to the exiting solvers.

An ensemble machine learning approach for enhanced path loss predictions for 4G LTE wireless networks

SummaryAccurate path loss prediction models are indispensable in modern wireless communication systems. In recent times, several path loss prediction models have been proposed to improve network performance. However, most of these models have not addressed the fundamental issues. The problem of deploying a single path loss prediction model that fits well in all wireless propagation environments remains. To address this problem, we present machine learning‐based ensemble methods to path loss predictions. Specifically, ensemble methods have been introduced to improve signal prediction accuracy and performance. Additionally, the radial basis function (RBF) and multilayer perceptron (MLP) neural network models were deployed and improved by adding more network parameters to their respective input layers. Results revealed that the RBF model with an increase in the number of centroids reduces the mean square error (MSE). Also, the Gaussian kernel function gives lower MSE compared to the multiquadric and inverse multiquadric functions. The bagging and blending ensemble path loss prediction models were introduced for optimal performance. An RBF neural network of different clusters was incorporated into the bagging algorithm as base learners. The RBF, MLP, blending, and bagging were examined using standard metrics for the training, testing, and validation dataset for model validation. The developed bagging ensemble path loss prediction model gave the lowest errors (MSE = 0.0011 dB, SSE = 0.6069 dB, MAE = 0.0245 dB, &R = 0.7484 dB) for the datasets. The bagging ensemble method acts as a variance and error reduction mechanism because it predicted path loss closest to measured data and is suitable for near precise path loss predictions.

An RBF-FD sparse scheme to simulate high-dimensional Black–Scholes partial differential equations

The objective of this research is to investigate the numerical solution of high-dimensional Black–Scholes partial differential equations (PDEs). To do this, the weights of the radial basis function-finite difference (RBF-FD) scheme using inverse multi-quadric function are used with an emphasis on the hotzone. The proposed RBF-FD technique is capable to price multi-asset options efficiently.

The application of radial basis function interpolation in reactor core power distribution on-line monitoring

Reactor core power mapping can be regarded as an interpolation problem of scattered data which is common in many scientific and engineering fields, and the Radial Basis Function (RBF) method provides an excellent tool for interpolating multidimensional scattered data and thus an appropriate approach for power mapping. By using the dual form of the RBF interpolation expression, this study shows that the Ordinary Kriging (OK) method authors utilized before can be regarded as an instance of RBF framework. Moreover, Gaussian function, multiquadric function, inverse quadratic function and inverse multiquadric function have been implemented in RBF interpolation framework to carry out core power mapping. Measurements from DayaBay Unit 1 PWR are used to verify the accuracies of these four RBF functions, and comparisons are made among these four methods and the OK method. The reconstructed assembly power distribution results and the calculation speeds show that the RBF interpolation framework is suitable for on-line core power distribution monitoring.

Computer modeling system for the numerical solution of the one-dimensional non-stationary Burgers’ equation

The computer modeling system for numerical solution of the nonlinear one-dimensional non-stationary Burgers’ equation is described. The numerical solution of the Burgers’ equation is obtained by a meshless scheme using the method of partial solutions and radial basis functions. Time discretization of the one-dimensional Burgers’ equation is obtained by the generalized trapezoidal method (θ-scheme). The inverse multiquadric function is used as radial basis functions in the computer modeling system. The computer modeling system allows setting the initial conditions and boundary conditions as well as setting the source function as a coordinate- and time-dependent function for solving partial differential equation. A computer modeling system allows setting such parameters as the domain of the boundary-value problem, number of interpolation nodes, the time interval of non-stationary boundary-value problem, the time step size, the shape parameter of the radial basis function, and coefficients in the Burgers’ equation. The solution of the nonlinear one-dimensional non-stationary Burgers’ equation is visualized as a three-dimensional surface plot in the computer modeling system. The computer modeling system allows visualizing the solution of the boundary-value problem at chosen time steps as three-dimensional plots. The computational effectiveness of the computer modeling system is demonstrated by solving two benchmark problems. For solved benchmark problems, the average relative error, the average absolute error, and the maximum error have been calculated.

Constructing radial kernels with higher-order generalized Strang-Fix conditions

The paper provides an approach for constructing multivariate radial kernels satisfying higher-order generalized Strang-Fix conditions from a given univariate generator. There are three key features of the approach. First, the kernels are explicitly expressed only by the derivatives of the f-form of the generator without computing any Fourier transforms. Second, it includes the radial kernels with the highest-order generalized Strang-Fix conditions. Finally, it requires only computing univariate derivatives of the f-form. Therefore, the approach is simple, efficient and easy to implement. As examples, the paper constructs radial kernels from some commonly used generators, including the Gaussian functions, the inverse multiquadric functions and compactly supported positive definite functions.

Health risks zonation in megacities vis-à-vis PM using GIS-based model

In megacities, exposure to high concentrations of air pollution, as a major concern on public health, is being felt worldwide problem. The issue of particulate matter especially the PM 2.5 has become extremely crucial in Tehran, due to industrialization and population growth. Therefore, it is crucial to estimate the PM 2.5 concentrations for the purpose of health risk evaluation. Since the air quality, considering PM 2.5 varies over space and time, in this paper, RBF method in a based GIS model was utilized to zone air quality and its health risks upon PM 2.5 concentrations dispersion over Tehran, during one year, from 21 March 2013 to 20 March 2014. The RBF method was applied to obtain the zoning maps and determine the highest concentration of PM 2.5 in the 22 Tehran’s regions for each season. The RMSE min values according to the types of functions in the RBF method, viz.: Completely regularized spline, Spline with tension, Multiquadric function, Inverse multiquadric function, and Thin-plate spline, for each month have been assessed. The numbers of neighbors in the model for each function were varied from 3 to 15. The results indicate that the models with 4 neighbors have the best performance with the lowest RMSE values by using RBF method. In accordance with health risks assessment, the most and less favorable health conditions happen during spring and winter respectively. However, the most critical zone is city center. Keywords: Air pollution; PM 2.5 ; Tehran; Health risks; GIS method; RBF model; Zonation; RMSE

The solitary wave solution of coupled Klein–Gordon–Zakharov equations via two different numerical methods

In this research, we propose two different methods to solve the coupled Klein–Gordon–Zakharov (KGZ) equations: the Differential Quadrature (DQ) and Globally Radial Basis Functions (GRBFs) methods. In the DQ method, the derivative value of a function with respect to a point is directly approximated by a linear combination of all functional values in the global domain. The principal work in this method is the determination of weight coefficients. We use two ways for obtaining these coefficients: cosine expansion (CDQ) and radial basis functions (RBFs-DQ), the former is a mesh-based method and the latter categorizes in the set of meshless methods. Unlike the DQ method, the GRBF method directly substitutes the expression of the function approximation by RBFs into the partial differential equation. The main problem in the GRBFs method is ill-conditioning of the interpolation matrix. Avoiding this problem, we study the bases introduced in Pazouki and Schaback (2011) [44]. Some examples are presented to compare the accuracy and easy implementation of the proposed methods. In numerical examples, we concentrate on Inverse Multiquadric (IMQ) and second-order Thin Plate Spline (TPS) radial basis functions. The variable shape parameter (exponentially and random) strategies are applied in the IMQ function and the results are compared with the constant shape parameter.