Subzone scheme for RBF meshless method in solving multi-medium boundary value problems

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.

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

An accurate and stable RBF method for solving partial differential equations

این مطالعه به منظور بررسی پیش بینی شاخص های عملکرد تولیدی در مرغان تخم گذار با استفاده از شبکه های عصبی مصنوعی و رگرسیون غیرخطی چندگانه انجام شد. بررسی بر روی اطلاعات چهار دوره متوالی پرورش در یک واحد پرورش مرغ تخم گذار صورت گرفت. روش های داده-کاوی شامل رگرسیون خطی و غیر خطی، شبکه عصبی پرسپترون سه لایه، شبکه عصبی پرسپترون چهار لایه و شبکه عصبی با تابع پایه ای شعاعی بود. در این مدل ها از متغیرهای سن گله، میزان خوراک مصرفی و فصل تولید به عنوان متغیر پیشگو و شاخص های عملکرد تولیدی شامل درصد تخم-گذاری، وزن توده ای تخم مرغ تولیدی و ضریب تبدیل غذایی به عنوان متغیر پاسخ استفاده شد. نتایج نهایی رگرسیون های خطی نشان داد که برای تمامی متغیرهای وابسته مورد مطالعه متغیر مستقل سن گله معنی دار می باشد. بنابراین رگرسیون غیر خطی شاخص های عملکرد تولیدی در مقابل سن برای مقایسه با شبکه های عصبی مختلف مورد بررسی قرار گرفت و برای مقایسه کلیه مدل ها از ضریب تعیین (R2) و میانگین قدر مطلق خطا (MAE) استفاده شد. نتایج نشان داد بین شبکه های عصبی مصنوعی مختلف مورد مطالعه، شبکه با تابع پایه ای شعاعی بهتر از سایر مدل های در پیش‌بینی شاخص های عملکرد تولیدی مرغان تخم گذار عمل می کند.

This paper presents a radial basis function (RBF) solution to the optimal control of drug administration in renal anemia treatment. Anemia management is formulated as a constrained optimal control problem and solved by an RBF method. RBF method is a numerical approach using RBF parameterization and orthogonal discretization to transcribe the optimal control problem into a nonlinear programming (NLP) problem. Then, a receding horizon control approach based on the RBF method is developed for individualized anemia management. The simulation results are compared with those obtained from a clinical protocol to verify the efficiency of the proposed method.

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

A new learning method of radial basis function (RBF) network based on fuzzy C-means (FCM) clustering and ant colony optimization (ACO) is presented in the paper. Generally, the crucial problem of learning RBF network is the selection of the unit number, the centers and widths of the Gaussian radial basis function in hidden layer, and the weights between hidden layer and output layer. In this work, the centers are determined by FCM clustering, and the weights are determined by least mean squares (LMS) method. In addition, ACO is introduced to optimize two important parameters of FCM including the weighting exponent which impacts its performance, and the clustering number that is equal to the hidden unit number which impacts the generalization of RBF network, by minimizing an objective function integrated the generalization error with the hidden unit number of RBF network. Simulation results of identifying a nonlinear system illustrate the effectiveness of designing the RBF network with smaller structure but stronger generalization ability, comparing with K-means and orthogonal least squares (OLS) based learning methods.

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.

In this chapter a meshless method, based on the method of fundamental solutions (MFS) and radial basis functions (RBF), is developed to solve thin plate bending on an elastic foundation. In the presented algorithm, the analog equation method (AEM) is firstly used to convert the original governing equation to an equivalent thin plate bending equation without elastic foundations, which can be solved by the MFS and RBF interpolation, and then the satisfaction of the original governing equation and boundary conditions can determine all unknown coefficients. In order to fully reflect the practical boundary conditions of plate problems, the fundamental solution of biharmonic operator with augmented fundamental solution of Laplace operator are employed in the computation. Finally, several numerical examples are considered to investigate the accuracy and convergence of the proposed method.

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 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

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.

Radial basis function methods are modern ways to approximate multivariate functions, especially in the absence of grid data. They have been known, tested and analysed for several years now and many positive properties have been identified. This paper gives a selective but up-to-date survey of several recent developments that explains their usefulness from the theoretical point of view and contributes useful new classes of radial basis function. We consider particularly the new results on convergence rates of interpolation with radial basis functions, as well as some of the various achievements on approximation on spheres, and the efficient numerical computation of interpolants for very large sets of data. Several examples of useful applications are stated at the end of the paper.

Unstructured mesh deformation is an effective way to automatically generate mesh after geometric shape changes such as fluid–structure interaction simulation or aerodynamic shape optimization. The radial basis function method is one of the best mesh deformation methods, which takes into account both computational time and deformation ability. However, the current existing methods are confronted by the contradiction between computational efficiency and deformation accuracy. In this paper, a hybrid deformation method combining the radial basis function and distance-weighted function is proposed, which can effectively reduce computing cost and eliminate deformation error. Firstly, based on the radial basis function method with data reduction scheme, an efficient equidistant sampling method for points selection independent of the specific form of deformation is proposed, and a sampling algorithm based on bisection is devised to make the number of sample points quickly approach the expected value. Secondly, a compact distance-weighted function deformation method is developed, which is used to diffuse the deformation errors of boundary mesh points directly to interior mesh points in order to completely eliminate the deformation errors. Finally, two configurations, AGARD 445.6 wing and HIRENASD wing, are used to test the deformation capability of the hybrid method and the computing time of several key processes. The results show that the hybrid method can accurately realize large mesh deformation with a maximum displacement up to 50% span length, and at the same time, the mesh deformation can be completed with a single core in about 100 s for millions of mesh points, which indicates that the hybrid method in this paper has the ability to be applied to complicated configurations in real engineering.