RBF Approximation Research Articles

Training RBF neural networks for solving nonlinear and inverse boundary value problems

Radial basis function neural networks (RBFNN) have been increasingly employed to solve boundary value problems (BVPs). In the current study, we propose such a technique for nonlinear (apparently for the first time) elliptic BVPs of orders two and four in 2-D and 3-D. The method is also extended, in a natural way, to solving 2-D and 3-D inverse BVPs. The RBFNN is trained via the least squares minimization of a nonlinear functional using the MATLAB®routine lsqnonlin. In this way, as well as the solution, appropriate values of the RBF approximation parameters are automatically delivered. The efficacy of the proposed RBFNN is demonstrated through several numerical experiments.

Residual life prediction of bearings based on RBF approximation models

Abstract Once the failure of rotating machinery occurs, it may cause the whole system to paralyze and cause great economic losses, or it may cause casualties. Therefore, the prediction of the remaining life of bearings is of great significance. The purpose of this paper is to analyze the approximate modeling technology and develop a framework for combined approximate modeling technology. A multi-strategy radial-based approximate model optimization model is proposed based on the limitations of radial-based approximate model technology. Utilizing the weight coefficient solving technique, the variable confidence RBF model, i.e., RBF-LSTM model, is established. Propose the remaining methods for life prediction using the bearing life prediction process. The RBF-LSTM combined approximation model is used to construct the evaluation index for rolling bearing remaining life prediction. Using the empirical analysis method, the optimization effects of different models and the accuracy of bearing remaining life prediction are analyzed, respectively. Experiments show that the data range of the RBF-LSTM combined approximation model is between [23,52], the overall fluctuation range of the data is not large, and the time taken is only 31 s. After 230 calculations, the model optimization effect is better. In the remaining life validation, the starting values of 132h and 148h are less different from real life, only 1.53h and 1.3h, respectively, and the model prediction accuracy is high.

Position design of the trailing arm bushing on torsion beam suspension accounting for vehicle transient handling performance and uncertainties

The bushing of the trailing arm on torsion beam suspension plays a pivotal role in vehicle dynamic behavior. In this paper, the connection between bushing position and vehicle dynamic response is elucidated. According to the simulation results, the impact of the bushing position on the transient performance of vehicle is more pronounced at low handling frequencies, and different index under the same bushing position are not always optimal. To design the bushing position that is better for evaluation indexes, this paper formulates the design problem of the bushing position as a multi-objective optimization problem. Due to the influence of actual production and processing, inevitable errors in bushing position may result in vehicle performance not meeting design requirements. Therefore, this paper takes into consideration the uncertainties and conducts robust multi-objective optimization to design the bushing position. To address the computational burden associated with robust optimization, the RBF approximation model is developed in this paper. Finally, the optimization problem is solved with the NSGA-II intelligent algorithm. The optimization results show that the bushing position designed by robust multi-objective optimization results in vehicle with stronger anti-roll performance and better robustness for each evaluation index. It is more suitable for practical applications.

Random sampling and unisolvent interpolation by almost everywhere analytic functions

We prove a.s. (almost sure) unisolvency of interpolation by continuous random sampling with respect to any given density, in spaces of multivariate a.e. (almost everywhere) analytic functions. Examples are given concerning polynomial and RBF approximation.

Meshless symplectic and multi-symplectic scheme for the coupled nonlinear Schrödinger system based on local RBF approximation

Meshless symplectic and multi-symplectic scheme for the coupled nonlinear Schrödinger system based on local RBF approximation

RBF approximation of three dimensional PDEs using tensor Krylov subspace methods

In this paper, we propose different algorithms for the solution of a tensor linear discrete ill-posed problem arising in the application of the meshless method for solving PDEs in three-dimensional space using multiquadric radial basis functions. It is well known that the truncated singular value decomposition (TSVD) is the most common effective solver for ill-conditioned systems, but unfortunately the operation count for solving a linear system with the TSVD is computationally expensive for large-scale matrices. In the present work, we propose algorithms based on the use of the well known Einstein product for two tensors to define the tensor global Arnoldi and the tensor Gloub Kahan bidiagonalization algorithms. Using the so-called Tikhonov regularization technique, we will be able to provide computable approximate regularized solutions in a few iterations. The tensor formulation will allow us to develop an RBF approximation for high dimensional PDEs based on Hierarchical tensors and Adaptive Cross Approximation, which in turn reduces significantly the storage and computational costs, while the accuracy of the method is preserved. The performance of the proposed methods is illustrated with a variety of benchmark examples and large-scale industrial applications with high degrees of freedom.

The RBF-PU Method for Solving 2D Nonlocal Diffusion and Peridynamic Equations

The radial basis function partition of unity (RBF-PU) method  was applied to obtain the numerical solution of 2D nonlocal diffusion and peridynamic problems. The main idea is to partition the original domain into several patches, use the RBF approximation on each local domain, and then give weighting to obtain the global approximation of the unknown function. The radial basis function method based on the strong form of the equation has many advantages, such as avoiding an additional layer of integral calculation, no need to deal with intersections of neighborhoods with the mesh, and easiness of implementation. The numerical results show that, this method can solve nonlocal diffusion equations and peridynamic equations accurately and efficiently.

Numerical simulation of a degenerate parabolic problem occurring in the spatial diffusion of biological population

This paper studies a localized meshless algorithm for calculating the solution of a nonlinear biological population model (NBPM). This model describes the dynamics in the biological population and may provide valuable predictions under different scenarios. The solution of the NBPM is approximated by means of local radial basis function based on the partition of unity (LRBF-PU) technique. First, the partial differential equation (PDE) is converted into a system of ordinary differential equations (ODEs) with the help of radial kernels. Afterwards, the system of ODEs is solved through an ODE solver of high-order. The major advantage of this scheme is that it does not requires any linearization. The LRBF-PU approximation helps handling the issue of the matrix ill conditioning that arises in a global RBF approximation. Three examples highlight the efficiency and accuracy of the numerical method. It is verified that the proposed strategy is more efficient in terms of computational time and accuracy in comparison with others available in the literature.

A Three-Level Radial Basis Function Method for Expensive Optimization.

This article proposes a three-level radial basis function (TLRBF)-assisted optimization algorithm for expensive optimization. It consists of three search procedures at each iteration: 1) the global exploration search is to find a solution by optimizing a global RBF approximation function subject to a distance constraint in the whole search space; 2) the subregion search is to generate a solution by minimizing an RBF approximation function in a subregion determined by fuzzy clustering; and 3) the local exploitation search is to generate a solution by solving a local RBF approximation model in the neighborhood of the current best solution. Compared with some other state-of-the-art algorithms on five commonly used scalable benchmark problems, ten CEC2015 computationally expensive problems, and a real-world airfoil design optimization problem, our proposed algorithm performs well for expensive optimization.

RBF보간 및 근사에 관한 수치적 고찰 및 평가

In this paper, RBF interpolation and approximation methods are introduced in detail, and a comparison of those is made for 3D grid and scattered dataset of various sizes and for well-known RBF functions with respect to the numerical accuracy and computation time. We also examine the RBF global functions and local functions, distribution and size of data, stability of memory and memory usage in detail. In addition, we apply the proposed methods to image reconstruction in order to verify the scalability and usability of this methods. From these numerical investigations, we can see that the proposed RBF approximation offers lower memory requirements and better quality of approximation.

RBF approximation by partition of unity for valuation of options under exponential Lévy processes

The prices of European and American-style contracts on assets driven by Markov processes satisfy partial integro-differential equations (PIDEs). In particular, this holds true for assets driven by Lévy processes, which are very popular in mathematical finance. We focus below on Lévy processes whose jump part has infinite (small jumps) activity, in which case the resulting singular kernel is rather difficult to treat numerically [1,23]. We illustrate, using the CGMY example, that this challenge may be successfully met by using a combination of: a) the Asmussen-Rosinsky approximation of the small jumps by Brownian motion; b) localized RBF approximations known as the RBF partition of unity (RBF-PU), which overcome the ill-conditioning inherent in global mesh-free methods; c) Crank-Nicolson Leap-Frog (CNLF) for time discretization. d) treating the local term using an implicit step, and the nonlocal term using an explicit step (to avoid the inversion of the nonsparse matrix)Finally, to deal with free boundary problems associated to American options, we combine the implicit-explicit method with a penalty method.Efficiency and practical performance are demonstrated by several numerical experiments for pricing European and American contracts.

A compact integrated RBF method for time fractional convection–diffusion–reaction equations

In this paper, a local compact integrated radial basis function (CIRBF) method is proposed to solve the time fractional convection–diffusion–reaction equations. The proposed CIRBF scheme derives high accuracy and fast convergence rate with grid refinement. The spatial derivatives are discretized by combining with the integrated RBF approximation and compact approximation in a stencil. The nodal function values and the nodal second derivatives are used to establish the relation between the physical space and the RBF weight space. The time discretization is replaced by the second order shifted Grünwald scheme. Moreover, the unconditional stability and convergence of the proposed method are shown. Finally, some numerical examples on 2D and 3D domains are carried out to verify the accuracy and efficiency of the proposed algorithm for the fractional differential equations.

Quick and robust meshless analysis of cracked body with coupled generalized hyperbolic thermo-elasticity formulation

In this article, the MLPG method is applied to the generalized linear coupled thermoelectricity equations. Lord–Shulman modification with a relaxation time parameter is used in the hyperbolic heat conduction equations. A new linear test function which is zero on the boundaries of local test domains is introduced. The test function and its partial derivatives are determined by an exponential RBF approximation method. The approximation of test function and main variables are similar. For the construction of shape functions, neighbors of every point are determined based on the definition of the closest adjacent point pattern. Consequently, test function space becomes independent of trial function space. Direct interpolation method and penalty parameter are used to impose essential boundary conditions. The selection of appropriate parameters are demonstrated in two numerical examples. The small number of used points is the advantage of this method over the FEM that is shown in several examples. The accuracy of results is compared between the meshless method and different analytical and FE solutions. The effect of the relaxation time on SIF under thermal shock is discussed in a separate example. The comparison of meshless results with various examples shows that employed method is accurate and reliable.

Research on Temperature Field Reconstruction Based on RBF Approximation with Polynomial Reproduction Considering the Refraction Effect of Sound Wave Paths

The temperature field distribution directly reflects the combustion condition in a furnace．In this paper, acoustic thermometry to reconstruct temperature distribution is investigated. A method based on radial basis function... | Find, read and cite all the research you need on Tech Science Press

A line search algorithm for wind field adjustment with incomplete data and RBF approximation

The problem of concern in this work is the construction of free divergence fields given scattered horizontal components. As customary, the problem is formulated as a PDE constrained least squares problem. The novelty of our approach is to construct the so-called adjusted field, as the unique solution along an appropriately chosen descent direction. The latter is obtained by the adjoint equation technique. It is shown that the classical adjusted field of Sasaki’s is a particular case. On choosing descent directions, the underlying mass consistent model leads to the solution of an elliptic problem which is solved by means of a radial basis functions method. Finally, some numerical results for wind field adjustment are presented.

High-order fluid solver based on a combined compact integrated RBF approximation and its fluid structure interaction applications

In this study, we present a high-order numerical method based on a combined compact integrated RBF (IRBF) approximation for viscous flow and fluid structure interaction (FSI) problems. In the method, the fluid variables are locally approximated by using the combined compact IRBF, and the incompressible Navier-Stokes equations are solved by using the velocity-pressure formulation in a direct fully coupled approach. The fluid solver is verified through various problems including heat, Burgers, convection-diffusion equations, Taylor-Green vortex and lid driven cavity flows. It is then applied to simulate some FSI problems in which an elastic structure is immersed in a viscous incompressible fluid. For FSI simulations, we employ the immersed boundary framework using a regular Eulerian computational grid for the fluid mechanics together with a Lagrangian representation of the immersed boundary. For the immersed fibre/membrane FSI problems, although the order of accuracy of the present scheme is generally similar to FDM approaches reported in the literature, the present approach is nonetheless more accurate than FDM approaches at comparable grid spacings. The numerical results obtained by the present scheme are highly accurate or in good agreement with those reported in earlier studies of the same problems.

A high-order coupled compact integrated RBF approximation based domain decomposition algorithm for second-order differential problems

This paper presents a high-order coupled compact integrated RBF (CCIRBF) approximation based domain decomposition (DD) algorithm for the discretisation of second-order differential problems. Several Schwarz DD algorithms, including one-level additive/ multiplicative and two-level additive/ multiplicative/ hybrid, are employed. The CCIRBF based DD algorithms are analysed with different mesh sizes, numbers of subdomains and overlap sizes for Poisson problems. Our convergence analysis shows that the CCIRBF two-level multiplicative version is the most effective algorithm among various schemes employed here. Especially, the present CCIRBF two-level method converges quite rapidly even when the domain is divided into many subdomains, which shows great promise for either serial or parallel computing. For practical tests, we then incorporate the CCIRBF into serial and parallel two-level multiplicative Schwarz. Several numerical examples, including those governed by Poisson and Navier-Stokes equations are analysed to demonstrate the accuracy and efficiency of the serial and parallel algorithms implemented with the CCIRBF. Numerical results show: (i) the CCIRBF-Serial and -Parallel algorithms have the capability to reach almost the same solution accuracy level of the CCIRBF-Single domain, which is ideal in terms of computational calculations; (ii) the CCIRBF-Serial and -Parallel algorithms are highly accurate in comparison with standard finite difference, compact finite difference and some other schemes; (iii) the proposed CCIRBF-Serial and -Parallel algorithms may be used as alternatives to solve large-size problems which the CCIRBF-Single domain may not be able to deal with. The ability of producing stable and highly accurate results of the proposed serial and parallel schemes is believed to be the contribution of the coarse mesh of the two-level domain decomposition and the CCIRBF approximation. It is noted that the focus of this paper is on the derivation of highly accurate serial and parallel algorithms for second-order differential problems. The scope of this work does not cover a thorough analysis of computational time.

A Fully Coupled Scheme for Viscous Flows in Regular and Irregular Domains Using Compact Integrated RBF Approximation

In this study, we present a numerical discretisation scheme, based on a fully coupled approach and compact local integrated radial basis function (CIRBF) approximations, to solve the Navier-Stokes equation in rectangular/non-rectangular domains. The velocity and pressure fields are simulated in a fully coupled manner [1] with Cartesian grids. The field variables are locally approximated in each direction by using CIRBF approximations defined over 3-node stencils, where nodal values of the first-and second-order derivatives of the field variables are also included [2, 3]. The present scheme, whose system matrix is sparse, is verified through the solutions of several test problems including Taylor-Green vortices. Highly accurate solutions are obtained.

A new approach to radial basis function approximation and its application to QSAR.

We describe a novel approach to RBF approximation, which combines two new elements: (1) linear radial basis functions and (2) weighting the model by each descriptor’s contribution. Linear radial basis functions allow one to achieve more accurate predictions for diverse data sets. Taking into account the contribution of each descriptor produces more accurate similarity values used for model development. The method was validated on 14 public data sets comprising nine physicochemical properties and five toxicity endpoints. We also compared the new method with five different QSAR methods implemented in the EPA T.E.S.T. program. Our approach, implemented in the program GUSAR, showed a reasonable accuracy of prediction and high coverage for all external test sets, providing more accurate prediction results than the comparison methods and even the consensus of these methods. Using our new method, we have created models for physicochemical and toxicity endpoints, which we have made freely available in the form of an online service at http://cactus.nci.nih.gov/chemical/apps/cap.

A direct forcing immersed boundary method employed with compact integrated RBF approximations for heat transfer and fluid flow problems

In this paper, we present a numerical scheme, based on the direct forcing immersed boundary (DFIB) approach and compact integrated radial basis function (CIRBF) approximations, for solving the Navier-Stokes equations in two dimensions. The problem domain of complicated shape is embedded in a Cartesian grid containing Eulerian nodes. Non-slip conditions on the inner boundaries, represented by Lagrangian nodes, are imposed by means of the DFIB method, in which a smoothed version of the discrete delta functions is utilised to transfer the physical quantities between two types of nodes. The velocities and pressure variables are approximated locally on Eulerian nodes using 3-node CIRBF stencils, where first- and second-order derivative values of the field variables are also included in the RBF approximations. The present DFIB-CIRBF scheme is verified through the solution of several test problems including Taylor-Green vortices, rotational flow, lid-driven cavity flow with multiple solid bodies, flow between rotating circular and fixed square cylinders, and natural convection in an eccentric annulus between two circular cylinders. Numerical results obtained using relatively coarse grids are in good agreement with available data in the literature.