Radial Basis Function Approximation Research Articles

Solving 2D Reaction-Diffusion Equations with Nonlocal Boundary Conditions by the RBF-MLPG Method

This paper is concerned with the development of a new approach for the numerical solution of linear and nonlinear reaction-diffusion equations in two spatial dimensions with Bitsadze-Samarskii type nonlocal boundary conditions. Proper finite-difference approximations are utilized to discretize the time variable. Then, the weak equations of resultant elliptic type PDEs are constructed on local subdomains. These local weak equations are discretized by using the multiquadric (MQ) radial basis function (RBF) approximation where an iterative procedure is proposed to treat the nonlinear terms in each time step. Numerical test problems are given to verify the accuracy of the obtained numerical approximations and stability of the proposed method versus the parameters of the nonlocal boundary conditions.

A numerical study of Asian option with radial basis functions based finite differences method

The purpose of this paper is to design and describe the valuation of Asian option by radial basis function approximation. A one state variable partial differential equation which characterizes the price of European type Asian option is discussed. The governing equation is discretized by the θ-method and the option price is approximated by radial basis function based finite difference method. Numerical experiments are performed with European option and Asian option and results are compared with theoretical and numerical results available in the literature. We show numerically that the scheme is second order accurate. Stability of the scheme is also discussed.

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.

Analysis of Transient Bioheat Transfer in the Human Eye Using Hybrid Finite Element Model

Simulation of transient bioheat transfer in a two dimensional (2D) human eye model is conducted using a newly developed hybrid fundamental solution-finite element method (HFS-FEM) coupling with the radial basis function (RBF) approximation. Firstly, a time stepping scheme based on the finite difference method (FDM) is used to handle time variable in the transient Pennes bioheat equation. Secondly, the particular solution of the governing equation is approximated by a RBF approach. Then, the homogeneous solution is calculated by means of HFS-FEM. The obtained results are compared with those from ABAQUS and a good agreement between them is observed.

A Numerical Solution Based on the Fokker-Planck Equation for Dilute Polymer Solutions Using High Order RBF Methods

This paper presents a numerical method for the Fokker-Planck Equation (FPE) based on mesoscopic modelling of dilute polymer solutions using Radial Basis Function (RBF) approaches. The stress is determined by the solution of a FPE while the velocity field is locally calculated via the solution of conservation Differential Equations (DEs) [1,2]. The FPE and PDEs are approximated separately by two different Integrated RBF methods. The time implicit discretisation of both FPE and PDEs is carried out using collocation methods where the high order RBF approximants improve significantly the accuracy of the numerical solutions and the convergence rate. As an illustration of the method, the time evolution of a start-up flow is studied for the Finitely Extensible Nonlinear Elastic (FENE) dumbbell model.

Robust parameter design optimization using Kriging, RBF and RBFNN with gradient-based and evolutionary optimization techniques

The dual response surface methodology is one of the most commonly used approaches in robust parameter design to simultaneously optimize the mean value and keep the variance minimum. The commonly used meta-model is the quadratic polynomial regression. For highly nonlinear input/output relationship, the accuracy of the fitted model is limited. Many researchers recommended to use more complicated surrogate models. In this study, three surrogate models will replace the second order polynomial regression, namely, ordinary Kriging, radial basis function approximation (RBF) and radial basis function artificial neural network (RBFNN). The results show that the three surrogate model present superior accuracy in comparison with the quadratic polynomial regression.The mean squared error (MSE) approach is widely used to link the mean and variance in one cost function. In this study, a new approach has been proposed using multi-objective optimization. The new approach has two main advantages over the classical method. First, the conflicting nature of the two objectives can be efficiently handled. Second, the decision maker will have a set of Pareto-front design points to select from.

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.

Reducing solid particle erosion of an axial fan with sweep and lean using multidisciplinary design optimization

A multidisciplinary design optimization (MDO) system is established to reduce solid particle erosion of an axial induced draft fan with sweep and lean. The method improves the erosion resistance of the fan blade in the aerodynamic design stage through a change of blade sweep and lean. The multidisciplinary design optimization approach takes the place of the traditional time-consuming design method by automatic calculation of the flow field, stress distribution, dynamic frequencies, and erosion distribution for blade, controlled by an optimization strategy. A multi-objective particle swarm optimization (MOPSO) algorithm combined with radial basis function approximation model is employed for finding a compromise between the conflicting demands of high efficiency and low average erosion rate with constraints on the pressure ratio and structural responses for the blade. The Navier–Stokes solver, finite element method (FEM) is used to predict the aerodynamic performance and mechanical performance of the blade, respectively. Particle paths in a viscous flow are calculated using the Lagrangian method and Tabakoff rebound model. And then Tabakoff erosion model is used to predict erosion of the blade surface. Several representative designs are selected along the Pareto front to verify using computer aided engineering tools. A compromise solution is used to analyze in detail. Compared with the reference design, the optimal design increases the η/ η0 slightly by 0.53%, while decreases the ɛavg / ɛavg0 markedly by 13.7%. The result shows that the optimized blade favors a reduced total pressure due to its forward sweep. The decrease of the ɛavg/ ɛavg0 is attributed to a reduced impact velocity and impact angle. The analysis of variance technique indicates that the blade lean has a direct impact on performances with respect to efficiency, erosion, and von Mises stress of the blade, and the blade sweep law near hub has an immeasurable influence on blade von Mises stress. As a conclusion, it can be drawn that the proposed approach may open a new opportunity for the design of axial fan to reduce erosion damage of blade taking other disciplines into consideration. Meanwhile, the multidisciplinary design optimization system can be extended to other turbomachinery and erosion-resistant design fields.

Acceleration Technique for Extended Boundary-Node Method

The acceleration technique for the extended boundary-node method (X-BNM) has been proposed and its performance has been numerically investigated. The results of computations show the X-BNM with the radial basis function (RBF) approximation is always faster than that with the moving least-squares (MLS) approximation. Moreover, the accuracy of the X-BNM with the RBF approximation is almost equivalent to that with the MLS approximation. From these results, it is found that the RBF approximation is more useful than the MLS approximation as the acceleration technique of the X-BNM.

A Fast Semidirect Least Squares Algorithm for Hierarchically Block Separable Matrices

We present a fast algorithm for linear least squares problems governed by hierarchically block separable (HBS) matrices. Such matrices are generally dense but data-sparse and can describe many important operators including those derived from asymptotically smooth radial kernels that are not too oscillatory. The algorithm is based on a recursive skeletonization procedure that exposes this sparsity and solves the dense least squares problem as a larger, equality-constrained, sparse one. It relies on a sparse QR factorization coupled with iterative weighted least squares methods. In essence, our scheme consists of a direct component, comprised of matrix compression and factorization, followed by an iterative component to enforce certain equality constraints. At most two iterations are typically required for problems that are not too ill-conditioned. For an $M \times N$ HBS matrix with $M \geq N$ having bounded off-diagonal block rank, the algorithm has optimal $\mathcal{O} (M + N)$ complexity. If the rank increases with the spatial dimension as is common for operators that are singular at the origin, then this becomes $\mathcal{O} (M + N)$ in 1D, $\mathcal{O} (M + N^{3/2})$ in 2D, and $\mathcal{O} (M + N^{2})$ in 3D. We illustrate the performance of the method on both over- and underdetermined systems in a variety of settings, with an emphasis on radial basis function approximation and efficient updating and downdating.

Computing Deformable Image Registration by Inverse Multiquadric Demons Algorithm

Radiation therapy is the mainstay of cancer treatment. However, significant weight loss and tumor shrinkage during treatment could affect the medical outcome. Advanced forms of adaptive radiotherapy use deformable image registration (DIR) accounting for such anatomical changes. Accurately defining the exact location and extent of the target with respect to surrounding organs at risk are difficult tasks. DIR model is commonly used as an alternative to measure how the tumor changes in size and location. In this paper, a reciprocal Multiquadric radial basis function approximation method is incorporated with the Demons algorithm for solving the DIR model. The proposed scheme could enhance the accuracy and efficiency of the deformable image registration. The numerical performance is compared with the classical DIR Demons algorithms. The result demonstrated that the combination of the reciprocal Multiquadric approximation and the Demons algorithm is useful in image registration.

Temperature Distribution Monitoring Using Ultrasonic Thermometry Based on Markov Radial Basis Function Approximation and Singular Values Decomposition

Information about temperature distribution is complex but of critical importance for the control of various microwave applications. In this paper, an innovative way of temperature distribution monitoring using ultrasonic thermometry in microwave field is investigated. The principle of ultrasonic thermometry in the situation of ideal gas is elaborated, and reconstruction algorithm based on Markov radial basis function approximation and singular values decomposition is presented and described in detail. In order to validate the performance of temperature distribution reconstruction of our presented algorithm, four two-dimensional temperature distribution models with different complexities are utilized in simulation experiments. Especially, simulation experiments taking error of measurement into account are studied to verify the robustness. Figure profiles show remarkable correspondence between the reconstructed ones and their models, while quantitative analysis, including the overall temperature error analysis and the hotspot positioning analysis, shows that different kinds of errors calculated are all within the limit ranges. In addition, the time analysis of simulation experiments also demonstrates its well real-time capability.

A numerical algorithm for multidimensional modeling of scattered data points

In this paper we propose an N-dimensional (Nd) algorithm for surface modeling of multivariate scattered data points. This code is implemented in MATLAB environment to numerically approximate (usually) large data point sets in $$\mathbb {R}^N$$ , for any $$N \in \mathbb {N}$$ . Since we need to organize the points in a Nd space, we build a kd-tree space-partitioning data structure, which is used to efficiently apply a partition of unity interpolant. This global method is combined with local radial basis function approximants and compactly supported weight functions. A detailed design of the partition of unity algorithm and a complexity analysis of the computational procedures are also considered. Finally, in several numerical experiments we show the performances, i.e., accuracy, efficiency and stability, of the Nd interpolation algorithm, considering various sets of Halton data points for $$N \le 5$$ .

A Fourier error analysis for radial basis functions on an infinite uniform grid. Part 2: Spectral-plus is special

A frequent claim in the radial basis function (RBF) literature is that with a wise choice of the “shape parameter” α, spectrally accurate RBFs can be “spectral-plus” in the sense that the RBF approximation is orders of magnitude more accurate than the corresponding spectral method. This is surprising because in the so-called “flat limit”, α→0, RBF methods provably reduce to a standard spectral method. On an infinite uniform grid, for example, Gaussian, hyperbolic secant, inverse quadratic, and multiquadric and inverse multiquadric RBFs reduce to the classical sinc basis. Using the error theorem proved in Part 1, we show that indeed it is possible for RBFs to be greatly superior to the sinc basis. However, rather special conditions are required for this superiority. The generic case is that RBFs are less accurate than the sinc method for nonzero values of the shape parameter.

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.

Reconstruction of dynamically changing boundary of multilayer heat conduction composite walls

Abstract This paper presents an inverse reconstruction procedure to determine the inner boundary location of heat conduction composite walls from the measurement data of temperature and heat flux on the exterior boundary. Our procedure uses a meshless forward solver that was developed for solving inhomogeneous heat transfer problems across a multilayer composite wall with Cauchy conditions. The forward solver uses the radial basis functions (RBFs) approximation in both time and space in a unified fashion, and hence is well suited for inverse problems. In this work, we consider that the length of the inner layer of the composite wall may vary caused by the material erosion at very high temperature such as in iron-making blast furnaces. In order to mitigate the ill-posed inverse problem, we use the Tikhonov regularization technique to obtain a stable and accurate numerical approximation of the moving boundary. Numerical experiments for a number of examples are presented to demonstrate the effectiveness of our inverse procedure. It can be observed that the error of the inverse solution is smaller or at the same level of noises in the simulated measurement data, demonstrating that our inverse procedure is effective and stable with respect to noisy data for moving boundary problems.

A compact 9 point stencil based on integrated RBFs for the convection–diffusion equation

In this paper, a high-order compact stencil for solving the convection–diffusion equation in two dimensions is proposed. The convection and diffusion terms are both approximated by means of radial basis functions (RBFs) that are constructed over 3×3 rectangular stencils. Salient features here are that (i) integration is employed to construct local RBF approximations; and (ii) through the constants of integration, values of the convection–diffusion equation at several selected nodes on the stencil are also enforced. Numerical results indicate that (i) the inclusion of the governing equation into the stencil leads to a significant improvement in accuracy; (ii) when the convection dominates, accurate solutions are obtained at a regime of the RBF width which makes the RBFs peaked; and (iii) high levels of accuracy are achieved using relatively coarse grids.

Proper orthogonal decomposition‐based model order reduction via radial basis functions for molecular dynamics systems

SUMMARYModel order reduction for molecular dynamics (MD) systems exhibits intrinsic complexities because of the highly nonlinear and nonlocal multi‐atomic interactions in high dimensions. In the present work, we introduce a proper orthogonal decomposition‐based method in conjunction with the radial basis function (RBF) approximation of the nonlinear and nonlocal potential energies and inter‐atomic forces for MD systems. This approach avoids coordinate transformation between the physical and reduced‐order coordinates, and allows the potentials and inter‐atomic forces to be calculated directly in the reduced‐order space. The RBF‐approximated potential energies and inter‐atomic forces in the reduced‐order space are discretized on the basis of the Smolyak sparse grid algorithm to further enhance the effectiveness of the proposed method. The good approximation properties of RBFs in interpolating scattered data make them ideal candidates for the reduced‐order approximation of MD inter‐atomic force calculations. The proposed approach is validated by performing the reduced‐order simulations of DNA molecules under various external loadings. Copyright © 2013 John Wiley & Sons, Ltd.

Finite integration method for partial differential equations

A finite integration method is proposed in this paper to deal with partial differential equations in which the finite integration matrices of the first order are constructed by using both standard integral algorithm and radial basis functions interpolation respectively. These matrices of first order can directly be used to obtain finite integration matrices of higher order. Combining with the Laplace transform technique, the finite integration method is extended to solve time dependent partial differential equations. The accuracy of both the finite integration method and finite difference method are demonstrated with several examples. It has been observed that the finite integration method using either radial basis function or simple linear approximation gives a much higher degree of accuracy than the traditional finite difference method.

The uncertain optimisation of buffering characteristics of landing airbag in manned airdrop

In this paper, the approximation non-linear interval number programming (A-NINP) method is suggested to obtain the optimal buffering characteristics of a landing airbag in manned airdrop. The landing velocities of airdrop equipment and ground friction coefficient are regarded as the uncertain parameters, and an uncertain objective function that represents the human neck injury index is created. Based on an order relation of interval number, the uncertain objective function is transformed into two deterministic objective functions. Through a modified possibility degree, the uncertain inequality constraints are changed to deterministic inequality constraints. In order to reduce the computational cost, radial basis function approximation models are used, and a local-densifying method is suggested to guarantee the accuracy of the approximation models. The intergeneration projection genetic algorithm is employed to solve the finally obtained deterministic and non-constraint optimisation problem. The results indicate that using the presented method, better buffering characteristics of the landing airbag can be obtained to ensure the safety of astronauts.