Using Time‐Space Double Radial Basis Function Method to Solve High‐Dimensional PDEs Arising from Multiasset Option Pricing

Parallel radial basis function methods for the global optimization of expensive functions

Radial basis function (RBF) methods have been actively developed in the last decades. RBF methods are global methods which do not require the use of specialized points and that yield high order accuracy if the function is smooth enough. Like other global approximations, the accuracy of RBF approximations of discontinuous problems deteriorates due to the Gibbs phenomenon, even as more points are added. In this paper we show that it is possible to remove the Gibbs phenomenon from RBF approximations of discontinuous functions as well as from RBF solutions of some hyperbolic partial differential equations. Although the theory for the resolution of the Gibbs phenomenon by reprojection in Gegenbauer polynomials relies on the orthogonality of the basis functions, and the RBF basis is not orthogonal, we observe that the Gegenbauer polynomials recover high order convergence from the RBF approximations of discontinuous problems in a variety of numerical examples including the linear and nonlinear hyperbolic partial differential equations. Our numerical examples using multi-quadric RBFs suggest that the Gegenbauer polynomials are Gibbs complementary to the RBF multi-quadric basis.

Purpose – The purpose of this paper is to solve the interface discontinuities in radial basis function (RBF) method for multi-medium boundary value problems (BVPs). The discontinuity of the solution derivatives is not easily handled with RBF method because of infinitely smoothness. Design/methodology/approach – The essence of solving BVP is to construct the continuous potential function surfaces. Hence, from constructing surface aspect, this paper proposed and compared the global and subzone schemes for RBF method. Their implementation schemes and mathematic models can then be derived. Numerical experiments and comparison are carried out for electric and magnetic field calculation. Findings – In the numerical experiments, the subzone scheme has shown its significant advantageous, it can approximate not only the potential function but also its derivative on interface boundary with high accuracy. So the physical characteristics of discontinuities on the interface can be revealed clearly. The overall precision is significantly improved. Originality/value – This paper proposed an effective subzone scheme for RBF method in multi-medium BVP. It is an improvement for RBF method based on its domain decomposition idea. And it is also a candidate for solving complex BVP.

An accurate and stable RBF method for solving partial differential equations

Disease prediction is an important aspect of modern medicine, which aims to diagnose disease early and provide appropriate treatment to patients. This research uses a hybrid approach that combines the RBF (Radial Basis Function) kernel algorithm with logistic regression to predict various diseases in medical datasets. This method is intended to improve prediction performance by exploiting the advantages of each algorithm. This research uses a dataset containing medical information about several diseases collected from the Kaggle dataset. First, the RBF kernel is applied to transform the data features into a more informative, non-linear representation. Then, the logistic regression model is used to make predictions based on the features that have been processed by the RBF kernel. In this research, the hybrid RBF (Radial Basis Function) method was proven to be superior in predicting multiple diseases. This method shows the highest accuracy of 0.9460, as well as excellent precision, recall, and F1-score values of 0.8680, 0.8097, and 0.8294, respectively. The advantage of the hybrid RBF method lies in its ability to capture complex patterns in data that other methods often cannot identify, as well as its ability to handle non-linear decision boundaries, which are a common characteristic in medical datasets. Keywords: Disease prediction, Hybrid approach, RBF kernel algorithm, Logistic regression, medical datasets

The method of radial basis functions (RBFs) is a method of scattered data interpolation that has many applications in different fields. Based on the RBFs approach, two simple and accurate methods are presented in this paper to favor the solution of fractional optimal control problems (FOCPs) which are nominated as indirect and direct methods. In the first one, the considered FOCP is converted into a system of fractional differential equations (FDEs) that is called two-point boundary value problem (TPBVP) of fractional order. Then, both the context of minimization the total error and a joint application of Volterra integral equation will be used to rewrite this problems as as an unconstrained optimization problem that can be solved by using RBFs approximation. In direct one, we rewrite the FOCP as a classical static optimization problem that can be easily solved using known formulas for computing fractional derivatives of RBFs. Numerical example are given to prove the accuracy, effectiveness and feasibility of these methods.

The Global Navigation Satellite System spacecraft signal power received at the Earth's surface is one of the main signal parameters. The navigation signals power measurements have been carried on using a specialized metrological measuring complex based on the large-aperture dish antennas. It is necessary to measure radiation pattern of the spacecraft antenna to estimate signal power at an arbitrary point on the earth's surface. A description of spacecraft antenna amplitude radiation pattern shape reconstructing based on the two large-aperture antennas observation results is given. The least squares method application in estimating the parameters of the navigation spacecraft antenna amplitude radiation pattern is considered. The amplitude radiation pattern is estimated from the ground-based measurements results of the satellite signals energy characteristics using a special complex based on the large-aperture dish antenna. The method of radial basis functions is used for pattern interpolation. Theoretical and experimental results are presented, the interpolation accuracy is estimated. Examples of amplitude radiation patterns reconstruction by piecewise linear approximation method and radial-basis functions (two-dimensional interpolation) are shown.

A new meshless approach to map electromagnetic loads for FEM analysis on DEMO TF coil system

The local radial basis function (RBF) method is a promising solver for variable‐order time fractional diffusion equation (TFDE), as it overcomes the computational burden of the traditional global method. Application of the local RBF method is limited to Fickian diffusion, while real‐world diffusion is usually non‐Fickian in multiple dimensions. This article is the first to extend the application of the local RBF method to two‐dimensional, variable‐order, time fractional diffusion equation in complex shaped domains. One of the main advantages of the local RBF method is that only the nodes located in the subdomain, surrounding the local point, need to be considered when calculating the numerical solution at this point. This approach can perform well with large scale problems and can also mitigate otherwise ill‐conditioned problems. The proposed numerical approach is checked against two examples with curved boundaries and known analytical solutions. Shape parameter and subdomain node number are investigated for their influence on the accuracy of the local RBF solution. Furthermore, quantitative analysis, based on root‐mean‐square error, maximum absolute error, and maximum error of the partial derivative indicates that the local RBF method is accurate and effective in approximating the variable‐order TFDE in two‐dimensional irregular domains.

An artificial neural network is well known as a flexible mathematical tool that has the ability to generalize patterns in imprecise or noisy and ambiguous input and output data sets. The radial basis function (RBF) method is applied to model the relationship between rainfall and runoff for Sungai Bekok Catchment (Johor, Malaysia) and Sungai Ketil catchment (Kedah, Malaysia). The RBF is used to predict the streamflow hydrograph based on storm events. Evaluation on the performance of RBF is demonstrated based on errors (between predicted and actual) and comparison with the results of the Hydrologic Engineering Center hydrologic modeling system model. It is obvious that the RBF method offers an accurate modeling of streamflow hydrograph.

Mesh deformation on 3D complex configurations using multistep radial basis functions interpolation

The inverse heat conduction problem on the heat transfer characteristics of cooled/heated laminar flows through finite length thick-walled circular tubes is studied, using temperature measurements taken at several different locations within the fluid in this paper. The method of radial basis functions is coupled with the boundary control technique to estimate the unknown temperature on the external surface of the circular pipe. The main idea of the proposed method is to solve the direct problem instead of solving the inverse problem directly. The temperature data obtained from the direct problem are used to simulate the temperature measurement for the inverse problem, and during the calculation, the Tikhonov regularization and L-curve methods are employed to solve the inverse problem. Therefore, this study also considers the influence of errors in these measurements upon the precision of the estimated results as well as the influence of the locations and number of sensors used upon the accuracy of the estimated results. The results indicate that the accuracy of the estimated results is improved by taking temperature measurements in locations close to the the unknown boundary. Finally, the results confirm that the proposed method is capable of yielding accurate results even when errors in the temperature measurements are present.

Interface evolution during mixing of a two-phase fluid in rectangular cavity is studied numerically by using radial basis function (RBF) method combined with domain decomposition method in the form of primitive variables. The mathematical model is governed by modified Navier-Stokes equation. The Cahn-Hilliard equation is implemented into the governing equations to capture the interface evolution. The fractional step method is applied to solve the momentum equations. The numerical method is based on RBF method for spatial discretization and Euler implicit method for time discretization. The effects of viscosity ratio on the interface evolution are investigated. Furthermore, the interface evolutions obtained from the present method agree well with another numerical methods from the literatures.

An efficient parallelized mesh deformation algorithm has been implemented in an unstructured CFD solver based on the Radial Basis Functions (RBFs) method. This method requires no grid connectivity information, which allows the method to be easily implemented in unstructured CFD solvers in parallel fashion. To reduce the computational cost of the RBF method, a Greedy Algorithm is employed to vastly reduce the number of surface basis points. The test cases show that this method can effectively handle vary large mesh deformations of three-dimensional complex geometries due to their translations, rotations and deformations while maintaining a good mesh quality.

The combination of Sinc quadrature method and double exponential transformation (DE) is a powerful tool to approximate the singular integrals, and radial basis functions (RBFs) are useful for the higher-dimensional space problem. In this study, we develop a numerical method base on Gaussian-RBF combined with QR-factorization of arising matrix and DE-quadrature Sinc method to approximate the solution of two-dimensional space-fractional diffusion equations. When the number of central nodes increases, the ill-conditioning of resultant matrix can be eliminated by using GRBF-QR method. Two numerical examples have been presented to test the efficiency and accuracy of the method.