Global Radial Basis Functions Research Articles

Calibration Method of Orthogonally Splitting Imaging Pose Sensor Based on KDFcmPUM.

This paper proposes a partition of unity method (PUM) based on KDFCM (KDFcmPUM) that can be implemented to solve the dense matrix problem that occurs when the radial basis function (RBF) interpolation method deals with a large amount of scattered data. This method introduces a kernel fuzzy clustering algorithm to improve clustering accuracy and achieve the partition of unity. The local compact support RBF is used to construct the weight function, and local expression is obtained from the interpolation of the global RBF. Finally, the global expression is constructed by the weight function and local expression. In this paper, the method is applied to the orthogonally splitting imaging pose sensor to establish the mathematical model and the calibration and test experiments are carried out. The calibration and test accuracy both reached ±0.1 mm, and the number of operations was reduced by 4% at least. The experimental results show that KDFcmPUM is effective.

Radial basis function methods for optimal control of the convection–diffusion equation: A numerical study

In this paper, we perform a numerical study for the solution of optimal constrained optimization problems for linear convection–diffusion PDEs by local and global radial basis function techniques. To the best of our knowledge, these control problems have not been treated in the literature by RBFs methods. It is well-known that the algebraic system of RBFs methods presents a larger condition number and a higher numerical complexity as the number of nodes (or shape parameter), increases. In this work, and in the context of optimal constrained optimization problems, we explore a possible answer to both problems. Specifically, we introduce a local RBF method (denoted as LAM-DQ), based on the combination of an asymmetric RBFs local method (LAM), inspired in local Hermite interpolation (LHI), combined with the differential quadrature method (DQ). We also propose a preconditioning technique that in combination with extended arithmetic precision let us treat the ill-conditioning problem. We numerically prove that as the number of nodes increases, then for errors of the same order, the condition number remains tractable, in quad-precision, and the numerical complexity of the local method remains bounded.

A meshless procedure for analysis of fluid flow and heat transfer in an internally finned square duct

Application of the method of fundamental solutions in combination with the global radial basis function collocation method for analysis of fluid flow and heat transfer in an internally finned square duct is presented in the paper. Fluid flow problem is solved using the modified method of fundamental solutions. After that, the average fluid velocity and product of friction factor and Reynolds number can be determined. Heat transfer problem in the fluid is governed by a nonlinear equation with linear boundary conditions. The Picard iteration method is employed in the paper in order to transform the nonlinear problem into a sequence of inhomogeneous problems. At each iteration step, the general solution is obtained using the modified method of fundamental solutions and the particular solution is obtained using the global radial basis function collocation method. When the iteration process is stopped, the Nusselt number can be determined.

An accurate and stable RBF method for solving partial differential equations

Most standard radial basis function (RBF) methods strongly depend on a shape parameter in accuracy and stability. However, the selection of optimal shape parameter in the RBF methods has long been a challenging task. In this paper, a new global RBF, known as coupled radial basis function (CRBF), is presented based on coupling the infinitely smooth RBFs with the conical spline. It is found that the Kansa’s method with the CRBF inherits the strengths of the aforementioned two types of RBFs, and achieves an accurate and stable numerical solution almost independent of the shape parameter values. Therefore, the proposed methodology completely overcome the troublesome issue of the optimal shape parameter that is a formidable obstacle to global RBF schemes. Two test examples verify the accuracy and robustness of the proposed CRBF method.

Numerical solution of time-dependent stochastic partial differential equations using RBF partition of unity collocation method based on finite difference

Meshfree methods based on radial basis functions (RBFs) are popular tools for the numerical solution of stochastic partial differential equations (SPDEs) due to their nice properties. However, the RBF collocation methods in global view have some disadvantages for the numerical solution of time-dependent SPDEs. Calculation of matrix condition number in the resulting dense linear systems indicates that the meshless method using global RBFs may be unstable at each realization to solve SPDEs. In order to avoid numerical instabilities in global RBF methods, we are interested in the use of RBF methods in local view for the numerical solution of time-dependent SPDEs. In this paper, the RBF partition of unity collocation method based on a finite difference scheme for the Gaussian random field (RBF-PU-FD) as a localized RBF approximation presented to deal with these issues. For this purpose, we simulate the Gaussian field with spatial covariance structure at a finite collection of predetermined collocation points. The matrices formed during the RBF-PU-FD method will be sparse and, hence, will not suffer from ill-conditioning and high computational cost. We will show that the method is viable through analyzing its numerical accuracy, CPU time, stability and sparsity structure. For the test problems, we perform 1000 realizations and statistical criterions such as mean, standard deviation, lower bound and upper bound of prediction are computed and evaluated using the Monte-Carlo method.

A Local Radial Basis Function Method for Pricing Options Under the Regime Switching Model

This paper is devoted to develop an efficient meshfree method based on radial basis functions (RBFs) to solve a system of partial differential equations arising from pricing options under the regime switching model. For global RBF methods, one of the major disadvantages is the computational cost and ill-conditioning associated with the dense linear systems that arise. So, we employ one of the local meshfree methods known as radial basis function based finite difference method. Then with an operator splitting method, sparse and well-conditioned system of complementarity problems are solved very fast for the American option. Also, the uniqueness of solution is proved for the discretized system of equations. Numerical examples presented in the last section illustrate the robustness and practical performance of the proposed algorithm for pricing European and American options.

Local and global characteristics-based kernel hybridization to increase optimal support vector machine performance for stock market prediction

In this paper, a novel multi-kernel support vector machine (MKSVM) combining global and local characteristics of the input data is proposed. Along with, a parameter tuning approach is developed using the fruit fly optimization (FFO), which is applied to stock market movement direction prediction problem. At first, factor analysis is used for identifying reduced key features called as factor scores from the raw stock index data which when applied to the model contributes to improvement in prediction performance. Subsequently, a hybrid kernel method combining local and global characteristics of input data is proposed, where polynomial is used for global kernel and radial basis function is utilized for local kernel. Additionally, FFO-based parameter tuning scheme is proposed to enhance the prediction performance further. Lastly, the evolving MKSVM with best feature subset and optimal parameters is used to predict stock market movement direction based upon historical data series. For evaluation and illustration purposes, three significant stock databases, NYSE, DJI and S&P 500 are used as testing targets. The effectiveness of this proposed approach is proved by three different stock market datasets, which demonstrate that the proposed approach outperforms the MKSVM with default parameter, MKSVM with PSO, MKSVM with GA and other methods. In addition, our findings reveal that the optimization strategy proposed here may be used as a promising choice forecasting tool for better generalization ability higher forecasting accuracy.

A computationally efficient numerical approach for multi-asset option pricing

ABSTRACTNumerical solution of the multi-dimensional partial differential equations arising in the modelling of option pricing is a challenging problem. Mesh-free methods using global radial basis functions (RBFs) have been successfully applied to several types of such problems. However, due to the dense linear systems that need to be solved, the computational cost grows rapidly with dimension. In this paper, we propose a numerical scheme to solve the Black–Scholes equation for valuation of options prices on several underlying assets. We use the derivatives of linear combinations of multiquadric RBFs to approximate the spatial derivatives and a straightforward finite difference to approximate the time derivative. The advantages of the scheme are that it does not require solving a full matrix at each time step and the algorithm is easy to implement. The accuracy of our scheme is demonstrated on a test problem.

Mesh-free semi-Lagrangian methods for transport on a sphere using radial basis functions

We present three new semi-Lagrangian methods based on radial basis function (RBF) interpolation for numerically simulating transport on a sphere. The methods are mesh-free and are formulated entirely in Cartesian coordinates, thus avoiding any irregular clustering of nodes at artificial boundaries on the sphere and naturally bypassing any apparent artificial singularities associated with surface-based coordinate systems. For problems involving tracer transport in a given velocity field, the semi-Lagrangian framework allows these new methods to avoid the use of any stabilization terms (such as hyperviscosity) during time-integration, thus reducing the number of parameters that have to be tuned. The three new methods are based on interpolation using 1) global RBFs, 2) local RBF stencils, and 3) RBF partition of unity. For the latter two of these methods, we find that it is crucial to include some low degree spherical harmonics in the interpolants. Standard test cases consisting of solid body rotation and deformational flow are used to compare and contrast the methods in terms of their accuracy, efficiency, conservation properties, and dissipation/dispersion errors. For global RBFs, spectral spatial convergence is observed for smooth solutions on quasi-uniform nodes, while high-order accuracy is observed for the local RBF stencil and partition of unity approaches.

RBF-PU method for pricing options under the jump–diffusion model with local volatility

Meshfree methods based on radial basis functions (RBFs) are of general interest for solving partial differential equations (PDEs) because they can provide high order or spectral convergence for smooth solutions in complex geometries. For global RBF methods, one of the major disadvantages is the computational cost associated with the dense linear systems that arise. Therefore, this paper is currently directed toward localized RBF approximations known as the RBF partition of unity (RBF-PU) method for partial integro-differential equation (PIDE) arisen in option pricing problems in jump–diffusion model. RBF-PU method produces algebraic systems with sparse matrices which have small condition number. Also, for comparison, some stable time discretization schemes are combined with the operator splitting method to get a fully discrete problem. Numerical examples are presented to illustrate the convergence and stability of the proposed algorithms for pricing European and American options with Merton and Kou models.

Improved localized radial basis functions with fitting factor for dominated convection-diffusion differential equations

Ill-conditioning problem of Global Radial Basis Functions (GRBFs) is a fundamental limitation for approximation of differential equation using this method. Most recently, some researchers have applied Local Radial Basis Functions (LRBFs) to approximate singularly perturbed convention diffusion problems. In these kinds of problems, mostly appear boundary and (or) interior layers. Existence of these regions causes instability and oscillation solutions in numerical methods. To avoid these problems, upwind LRBFs are used. In other words, convective part and diffusion part of equations discretize through upwind LRBFs and central LRBFs method, respectively. Although this technique stabilizes the scheme but reduces the accuracy, by the way, the selection of upwind direction is more complicated. To overcome these problems, in this paper, we multiply an artificial diffusion to diffusion part and apply central scheme to discretize convective part. Since central method for these kinds of problems are unstable, adding artificial diffusion causes the central method is stabilized. This strategy significantly improve the accuracy. For verification, several numerical examples are considered and compared with other schemes. The comparisons show the superiority of the method to tradition approaches. Results of this study reveal that the proposed method can be successfully applied for singularly perturbed differential equations.

Novel two-level discretization method for high dimensional semilinear elliptic problems base on RBF-FD scheme

ABSTRACTIn this article, we propose a new two-level method for solving 2D and 3D semilinear elliptic boundary value problem base on radial basis function (RBF). In the first step, we use the global RBF method to solve a semilinear problem on the coarse mesh or a small number of collocation points, in the second step, finite difference (FD), finite element (FE), and radial basis function-generated finite difference (RBF-FD) methods are used to solve the linearized problem on the fine mesh or a large number of collocation points, respectively. Numerical examples are provided to verify the feasibility and efficiency of the two-level RBF method. Moreover, compared to the FD and FE methods, RBF-FD method used in the second step further improve the accuracy of numerical solution.

Topology analysis of global and local RBF transformations for image registration

For elastic registration, topology preservation is a necessary condition to be satisfied, especially for landmark-based image registration. In this paper, we focus on the topology preservation properties of two different families of radial basis functions (RBFs), known as Gneiting and Matérn functions. Firstly, we consider a small number of landmarks, dealing with the cases of one, two and four landmark matching; in all these situations we analyze topology preservation and compare numerical results with those obtained by Wendland functions. Secondly, we discuss the registration properties of these two families of functions, when we have a larger number of landmarks. Finally, we analyze the behavior of Gneiting and Matérn functions, considering some test examples known in the literature and a real application.

An Investigation of Radial Basis Function-Finite Difference (RBF-FD) Method for Numerical Solution of Elliptic Partial Differential Equations

The Radial Basis Function (RBF) method has been considered an important meshfree tool for numerical solutions of Partial Differential Equations (PDEs). For various situations, RBF with infinitely differentiable functions can provide accurate results and more flexibility in the geometry of computation domains than traditional methods such as finite difference and finite element methods. However, RBF does not suit large scale problems, and, therefore, a combination of RBF and the finite difference (RBF-FD) method was proposed because of its own strengths not only on feasibility and computational cost, but also on solution accuracy. In this study, we try the RBF-FD method on elliptic PDEs and study the effect of it on such equations with different shape parameters. Most importantly, we study the solution accuracy after additional ghost node strategy, preconditioning strategy, regularization strategy, and floating point arithmetic strategy. We have found more satisfactory accurate solutions in most situations than those from global RBF, except in the preconditioning and regularization strategies.

Meshless collocation procedures for time-dependent inverse heat problems

Inverse heat problems are known to be ill-posed, which make them difficult to be solved numerically. In this paper we use meshless methods based on local and global radial basis functions for numerical solution of one- and two-dimensional inverse heat problems. In one-dimensional case, the unknown heat source and the right boundary condition are recovered, while in two-dimensional case, only the unknown heat source is recovered. Comparison of the global and the local meshless numerical schemes based on MQ radial basis function is performed for one- and two-dimensional cases. Several test problems related to inverse heat parabolic PDEs are numerically solved to verify accuracy and efficiency of the local meshless method. Without application of any regularization technique, accurate and stable solution has been obtained in the case of local meshless method for the input data, which is contaminated with a comparatively large level of noise.

A new method for evaluating options based on multiquadric RBF-FD method

In this paper, a new local meshless approach based on radial basis functions (RBFs) is presented to price the options under the Black–Scholes model. The global RBF approximations derived from the conventional global collocation method usually lead to ill-conditioned matrices. Employing the idea of local approximants of the finite difference (FD) method and combining it with the radial basis function (RBF) method can result in a local meshless approach such as RBF-FD. It removes the difficulty of ill-conditionness of the original method. The new proposed approach is unconditionally stable as it is shown by Von-Neumann stability analysis. It is fast and produces high accurate results as shown in numerical experiments. Moreover, we took into account the variation of shape parameter and analyzed numerically the behavior of the RBF-FD method.

Estimation of the non-linear diffusion coefficient with Marcov Chain Monte Carlo method based on the integral information

PurposeThis paper aims to present development and application of the Bayesian inverse approach for retrieving parameters of non-linear diffusion coefficient based on the integral information.Design/methodology/approachThe Bayes formula was used to construct posterior distribution of the unknown parameters of non-linear diffusion coefficient. The resulting aposteriori distribution of sought parameters was integrated using Markov Chain Monte Carlo method to obtain expected values of estimated diffusivity parameters as well as their confidence intervals. Unsteady non-linear diffusion equation was discretised with the Global Radial Basis Function Collocation method and solved in time using Crank–Nicholson technique.FindingsA number of manufactured analytical solutions of the non-linear diffusion problem was used to verify accuracy of the developed inverse approach. Reasonably good agreement, even for highly correlated parameters, was obtained. Therefore, the technique was used to compute concentration dependent diffusion coefficient of water in paper.Originality/valueAn original inverse technique, which couples efficiently meshless solution of the diffusion problem with the Bayesian inverse methodology, is presented in the paper. This methodology was extensively verified and applied to the real-life problem.

Zooming from global to local: a multiscale RBF approach

Because physical phenomena on Earth's surface occur on many different length scales, it makes sense when seeking an efficient approximation to start with a crude global approximation, and then make a sequence of corrections on finer and finer scales. It also makes sense eventually to seek fine scale features locally, rather than globally. In the present work, we start with a global multiscale radial basis function (RBF) approximation, based on a sequence of point sets with decreasing mesh norm, and a sequence of (spherical) radial basis functions with proportionally decreasing scale centered at the points. We then prove that we can zoom on a region of particular interest, by carrying out further stages of multiscale refinement on a local region. The proof combines multiscale techniques for the sphere from Le Gia, Sloan and Wendland, SIAM J. Numer. Anal. 48 (2010) and Applied Comp. Harm. Anal. 32 (2012), with those for a bounded region in ?d from Wendland, Numer. Math. 116 (2010). The zooming in process can be continued indefinitely, since the condition numbers of matrices at the different scales remain bounded. A numerical example illustrates the process.

An RBF collocation method for solving optimal control problems

A direct solution to optimal control problems is introduced based on interpolating global radial basis functions (RBFs) on arbitrary collocation points. In the proposed approach, called the RBF collocation method, states and controls are parameterized with any arbitrary global RBF and the continuous-time optimal control problem is discretized using arbitrary collocation points to transcribe it into a nonlinear programming problem. Numerical examples including a multiple UAV navigation problem are provided to evaluate the performance of the proposed method. It is shown that the RBF collocation method is more accurate and also computationally more efficient than a local polynomial (B-spline) method and a global polynomial (Legendre pseudospectral) method for the UAV navigation problem. The fact that RBF method can employ a variety of RBF functions for the interpolation and also a wide range of arbitrary nodes for the discretization means that the proposed method offers a very flexible RBF framework for direct trajectory optimization. The numerical results confirm the efficiency of the proposed method for solving optimal control problems.

A meshfree approach to non-Newtonian free surface ice flow: Application to the Haut Glacier d'Arolla

Numerical models of glacier and ice sheet dynamics traditionally employ finite difference or finite element methods. Although these are highly developed and mature methods, they suffer from some drawbacks, such as inability to handle complex geometries (finite differences) or a costly assembly procedure for nonlinear problems (finite elements). Additionally, they are mesh-based, and therefore moving domains become a challenge. In this paper, we introduce a novel meshfree approach based on a radial basis function (RBF) method. The meshfree nature of RBF methods enables efficient handling of moving margins and free ice surface. RBF methods are also accurate, easy to implement, and allow for reduction the computational cost associated with the linear system assembly, since stated in strong form. To demonstrate the global RBF method we model the velocity field of ice flow in the Haut Glacier d'Arolla, which is governed by the nonlinear Stokes equations. We test the method for different basal conditions and for a free moving surface. We also compare the global RBF method with its localized counterpart—the RBF partition of unity method (RBF-PUM)—that allows for a significant gain in the computational efficiency. Both RBF methods are compared with the classical finite element method in terms of accuracy and efficiency. We find that the RBF methods are more efficient than the finite element method and well suited for ice dynamics modeling, especially the partition of unity approach.