Radial Basis Function Approximation Research Articles

Computing the survival probability density function in jump-diffusion models: A new approach based on radial basis functions

We propose a numerical method to compute the survival (first-passage) probability density function in jump-diffusion models. This function is obtained by numerical approximation of the associated Fokker–Planck partial integro-differential equation, with suitable boundary conditions and delta initial condition. In order to obtain an accurate numerical solution, the singularity of the Dirac delta function is removed using a change of variables based on the fundamental solution of the pure diffusion model. This approach allows to transform the original problem to a regular problem, which is solved using a radial basis functions (RBFs) meshless collocation method. In particular the RBFs approximation is carried out in conjunction with a suitable change of variables, which allows to use radial basis functions with equally spaced centers and at the same time to obtain a sharp resolution of the gradients of the survival probability density function near the barrier. Numerical experiments are presented in which several different kinds of radial basis functions are employed. The results obtained reveal that the numerical method proposed is extremely accurate and fast, and performs significantly better than a conventional finite difference approach.

On-line 3-D system for the inspection of deformable parts

In spite of the development of automated tolerance inspection systems for manufactured parts over the years, there are still processes that inevitably require manual intervention making full automation impossible in most cases; in particular when dealing with deformable parts. In most current industrial in- spection systems, a deformable part under inspection must first be mechanically constrained on a rigid sup- port or jig so as to be able to compare it with its nominal shape. This paper presents a new system to perform real-time surface inspection of deformable parts that does not require fixturing. Instead, the proposed system applies virtual forces to the part's CAD model as if the part was installed in the fixturing device. Normally, a precise finite element method (FEM) simulation should be used to approximate the deformation that appends when the part is installed in the device. Even with a fast parallel computer, FEM is far from being real-time and cannot be used for on-line inspection. In the proposed system, a radial basis function approximation of the FEM simulation is trained off-line and used to speed- up the simulation by an order of magnitude. Experi- mental evaluation of the proposed system is presented for three plastic parts. Using the proposed scheme, an approximation of 0.25 mm compared with the real de- formation was obtained. In this paper, statistical results are presented such as the average deviation, standard deviation, and processing time between the approxima- tions obtained with the proposed method and with the finite element method applied to the full CAD model.

RBFs approximation method for time fractional partial differential equations

In this paper, radial basis functions (RBFs) approximation method is implemented for time fractional advection–diffusion equation on a bounded domain. In this method the first order time derivative is replaced by the Caputo fractional derivative of order α ∈ (0, 1], and spatial derivatives are approximated by the derivative of interpolation in the Kansa method. Stability and convergence of the method is discussed. Several numerical examples are include to demonstrate effectiveness and accuracy of the method.

Analysis and optimization of externally stiffened crush tubes

Nonlinear finite element analysis is used to investigate the quasi-static axial collapse response of cylindrical tubes which are externally stiffened by multiple identical rings. The rings divide the long tube into a series of short thin-walled tubes. It is assumed that the size and shape of integral stiffeners are controlled through a machining process. The effects of various geometric parameters such as wall thickness, ring spacing, ring thickness and width on the collapse response, crush force and energy absorption of monolithic, integrally stiffened steel tubes are studied and used as a general framework for a design optimization study. Through design and analysis of computer experiments, global metamodels are developed for the mean crush force and energy absorption, using the radial basis function approximation technique. Using both single- and multi-objective design optimization formulations, optimum designs for different response characteristics are found. The crush mode in the form of progressive collapse or buckling is found to heavily depend on the ratio of stiffener spacing to stiffener height as well as the ratio of wall thickness to stiffener thickness. The optimization results show the viability of externally stiffened tubes as efficient energy absorbers.

Stable Computations with Gaussian Radial Basis Functions

Radial basis function (RBF) approximation is an extremely powerful tool for representing smooth functions in nontrivial geometries since the method is mesh-free and can be spectrally accurate. A perceived practical obstacle is that the interpolation matrix becomes increasingly ill-conditioned as the RBF shape parameter becomes small, corresponding to flat RBFs. Two stable approaches that overcome this problem exist: the Contour-Padé method and the RBF-QR method. However, the former is limited to small node sets, and the latter has until now been formulated only for the surface of the sphere. This paper focuses on an RBF-QR formulation for node sets in one, two, and three dimensions. The algorithm is stable for arbitrarily small shape parameters. It can be used for thousands of node points in two dimensions and still more in three dimensions. A sample MATLAB code for the two-dimensional case is provided.

Proper Orthogonal Decomposition and Radial Basis Functions in material characterization based on instrumented indentation

For the mechanical characterization of structural materials non-destructive tests combined with computer simulations and inverse analyses are more and more frequently and advantageously employed in various engineering fields. The contribution to such development presented in this paper can be outlined as follows. With reference to isotropic elastic–plastic material models, indentation test simulations are done preliminarily, once-for-all, by a conventional finite element forward operator. Results of these simulations are employed in a procedure which is centered on Proper Orthogonal Decomposition and Radial Basis Functions approximation and is used for fast interpolations which replace further finite element analyses in the parameter identification process. Comparative computational exercises are presented in order to point out the consequent significant reduction of computing times in test simulations and, hence, in the minimization of the discrepancy function by the Trust Region Algorithm, namely by a traditional first-order mathematical programming method. Such a parameter identification procedure may be carried out routinely and economically on small computers for in situ structural diagnoses. Both the force–penetration relationship (provided by an instrumented indenter) and the average imprint profile (achievable by laser profilometer) are considered as sources of measurable response quantities or experimental data.

Radial basis functions method for parabolic inverse problem

The radial basis functions (RBFs) method is employed to handle a class of multi-dimensional parabolic inverse problems. Because they are not modelled by classical parabolic initial-boundary value problems, theoretical behaviour and numerical approximation of these problems have been active areas of research. Based on the idea of RBF approximation, a fast and highly accurate meshless method is developed for solving an inverse problem with a control parameter. Moreover, with the meshless property, it can be used to handle multi-dimensional parabolic inverse problems defined on very complicated geometries.

Six strategies for defeating the Runge Phenomenon in Gaussian radial basis functions on a finite interval

Radial basis function (RBF) interpolation is a “meshless” strategy with great promise for adaptive approximation. Because it is meshless, there is no canonical grid to act as a starting point for function-adaptive grid modification. Uniform grids are therefore common with RBFs. Like polynomial interpolation, which is the limit of RBF interpolation when the RBF functions are very wide (the “flat limit”), RBF interpolants are vulnerable to the Runge Phenomenon, whether the grid is uniform or irregular: the N-point interpolant may diverge as N→∞ on the interval spanned by the interpolation points even for a function f(x) that is analytic everywhere on and near this interval. In this work, we discuss six strategies for defeating the Runge Phenomenon, specializing to a uniform grid on a finite interval and one particular species of RBFs in one space dimension: ϕ(x)=exp(−[α2/h2]x2) where h is the grid spacing. Three strategies fail, but three are successful. One good strategy is to vary α with the number of interpolation points N as N−1/4. Unfortunately this gives both a subgeometric rate of convergence for the approximation (error falling as exp(−qN) for some constant q) and a matrix condition number that grows as exp(pN) for some p>0. (In order to explain why fixed α, independent of N, is not a convergent strategy, we digress to discuss error saturation, and show experimentally that the saturation error on the finite interval is about 0.6exp(−0.47/α2)‖f‖∞; if a user-chosen error tolerance δ is acceptable, then the optimum choice is αoptimum(δ)=1/−2log(δ/0.06).) The second good strategy is RBF Extension, which uses RBF functions with centers on a interval slightly larger than the target interval to approximate f(x). A third strategy is to split the interval into two “boundary layers” and a large middle interval and apply separate RBF approximations on each. The slow-decrease-of-α strategy is much cheaper, but RBF Extension is much more resistant to ill-conditioning and therefore can achieve much lower errors. The three-interval method is the least accurate, but is robust and inexpensive.

On kernel engineering via Paley–Wiener

A radial basis function approximation takes the form $$s(x)=\sum_{k=1}^na_k\phi(x-b_k),\quad x\in {\mathbb{R}}^d,$$ where the coefficients a 1,?,a n are real numbers, the centres b 1,?,b n are distinct points in ? d , and the function ?:? d ?? is radially symmetric. Such functions are highly useful in practice and enjoy many beautiful theoretical properties. In particular, much work has been devoted to the polyharmonic radial basis functions, for which ? is the fundamental solution of some iterate of the Laplacian. In this note, we consider the construction of a rotation-invariant signed (Borel) measure μ for which the convolution ?=μ ? is a function of compact support, and when ? is polyharmonic. The novelty of this construction is its use of the Paley---Wiener theorem to identify compact support via analysis of the Fourier transform of the new kernel ?, so providing a new form of kernel engineering.

The combination of GNSS-levelling data and gravimetric (quasi-) geoid heights in the presence of noise

We propose a methodology for the combination of a gravimetric (quasi-) geoid with GNSS-levelling data in the presence of noise with correlations and/or spatially varying noise variances. It comprises two steps: first, a gravimetric (quasi-) geoid is computed using the available gravity data, which, in a second step, is improved using ellipsoidal heights at benchmarks provided by GNSS once they have become available. The methodology is an alternative to the integrated processing of all available data using least-squares techniques or least-squares collocation. Unlike the corrector-surface approach, the pursued approach guarantees that the corrections applied to the gravimetric (quasi-) geoid are consistent with the gravity anomaly data set. The methodology is applied to a data set comprising 109 gravimetric quasi-geoid heights, ellipsoidal heights and normal heights at benchmarks in Switzerland. Each data set is complemented by a full noise covariance matrix. We show that when neglecting noise correlations and/or spatially varying noise variances, errors up to 10% of the differences between geometric and gravimetric quasi-geoid heights are introduced. This suggests that if high-quality ellipsoidal heights at benchmarks are available and are used to compute an improved (quasi-) geoid, noise covariance matrices referring to the same datum should be used in the data processing whenever they are available. We compare the methodology with the corrector-surface approach using various corrector surface models. We show that the commonly used corrector surfaces fail to model the more complicated spatial patterns of differences between geometric and gravimetric quasi-geoid heights present in the data set. More flexible parametric models such as radial basis function approximations or minimum-curvature harmonic splines perform better. We also compare the proposed method with generalized least-squares collocation, which comprises a deterministic trend model, a random signal component and a random correlated noise component. Trend model parameters and signal covariance function parameters are estimated iteratively from the data using non-linear least-squares techniques. We show that the performance of generalized least-squares collocation is better than the performance of corrector surfaces, but the differences with respect to the proposed method are still significant.

RBFs approximation method for Kawahara equation

In this paper, we apply radial basis function (RBF) method for the numerical solution of Kawahara equation. Derivatives of interpolation are used to approximate the spatial derivatives and the temporal derivative approximated by a low order forward difference scheme. Computational experiments are performed for single soliton, two solitons, and three solitons interaction to examine accuracy of the method. The results obtained by the present method are in good agreement with exact solutions and some earlier work in literature. Accuracy of the method is estimated in terms of the error norms L 2, L ∞ , the three invariants, number of nodes in the domain of influence, time step size, parameter dependent, and parameter independent RBFs.

Iterative adaptive RBF methods for detection of edges in two-dimensional functions

Radial basis functions have gained popularity for many applications including numerical solution of partial differential equations, image processing, and machine learning. For these applications it is useful to have an algorithm which detects edges or sharp gradients and is based on the underlying basis functions. In our previous research, we proposed an iterative adaptive multiquadric radial basis function method for the detection of local jump discontinuities in one-dimensional problems. The iterative edge detection method is based on the observation that the absolute values of the expansion coefficients of multiquadric radial basis function approximation grow exponentially in the presence of a local jump discontinuity with fixed shape parameters but grow only linearly with vanishing shape parameters. The different growth rate allows us to accurately detect edges in the radial basis function approximation. In this work, we extend the one-dimensional iterative edge detection method to two-dimensional problems. We consider two approaches: the dimension-by-dimension technique and the global extension approach. In both cases, we use a rescaling method to avoid ill-conditioning of the interpolation matrix. The global extension approach is less efficient than the dimension-by-dimension approach, but is applicable to truly scattered two-dimensional points, whereas the dimension-by-dimension approach requires tensor product grids. Numerical examples using both approaches demonstrate that the two-dimensional iterative adaptive radial basis function method yields accurate results.

Radial basis function approximation methods with extended precision floating point arithmetic

Radial basis function (RBF) methods that employ infinitely differentiable basis functions featuring a shape parameter are theoretically spectrally accurate methods for scattered data interpolation and for solving partial differential equations. It is also theoretically known that RBF methods are most accurate when the linear systems associated with the methods are extremely ill-conditioned. This often prevents the RBF methods from realizing spectral accuracy in applications. In this work we examine how extended precision floating point arithmetic can be used to improve the accuracy of RBF methods in an efficient manner. RBF methods using extended precision are compared to algorithms that evaluate RBF methods by bypassing the solution of the ill-conditioned linear systems.

Study on Chloride Diffusion in Concrete with Non-Homogenous Coefficient Using Meshless Boundary Element Method

A new robust numerical technique was proposed for analyzing chloride transient diffusion in concrete with non-homogenous coefficient. The method was based on a meshless boundary element method which results in an integral equation for explicitly evaluating field chloride quantities. Weighted residual method and Green’s function were adopted to derive domain and boundary integral equations. A radial integration method coupling with radial basis function approximation technology was used to convert domain integral into equivalent boundary integral. With central finite difference method, an explicit time iteration scheme was established for solving transient diffusion equation. Two numerical examples for 2D diffusion problem were given to demonstrate the robustness of the proposed method. Numerical results show that the non-homogenous diffusion coefficient causes the chloride distribution non-uniform, and the diffusion process is nonlinear with respect to time.

Recovery of High Order Accuracy in Radial Basis Function Approximations of Discontinuous 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.

Nonlinear approximation using Gaussian kernels

It is well known that nonlinear approximation has an advantage over linear schemes in the sense that it provides comparable approximation rates to those of the linear schemes, but to a larger class of approximands. This was established for spline approximations and for wavelet approximations, and more recently by DeVore and Ron (in press) [2] for homogeneous radial basis function (surface spline) approximations. However, no such results are known for the Gaussian function, the preferred kernel in machine learning and several engineering problems. We introduce and analyze in this paper a new algorithm for approximating functions using translates of Gaussian functions with varying tension parameters. At heart it employs the strategy for nonlinear approximation of DeVore–Ron, but it selects kernels by a method that is not straightforward. The crux of the difficulty lies in the necessity to vary the tension parameter in the Gaussian function spatially according to local information about the approximand: error analysis of Gaussian approximation schemes with varying tension are, by and large, an elusive target for approximators. We show that our algorithm is suitably optimal in the sense that it provides approximation rates similar to other established nonlinear methodologies like spline and wavelet approximations. As expected and desired, the approximation rates can be as high as needed and are essentially saturated only by the smoothness of the approximand.

A new variational radial basis function approximation for inference in multivariate diffusions

In this paper we derive and present a new radial basis function framework that extends a recently proposed variational Bayesian algorithm for approximate inference in diffusion processes. Inference, for the state and in particular for the (hyper-) parameters, in such systems is a challenging and crucial task. We show that the new radial basis function approximation based algorithm not only converges to the original variational algorithm but also has beneficial characteristics when estimating (hyper-) parameters. We validate our new approach on three highly non-linear dynamical systems, namely the univariate stochastic double well, and the multivariate Lorenz 3D and Lorenz 40D systems. We show that we are able to recover good estimates of the system and noise parameters in the multivariate case, even for chaotic systems.

An analytic approximation to the cardinal functions of Gaussian radial basis functions on an infinite lattice

Gaussian radial basis functions (RBFs) have been very useful in computer graphics and for numerical solutions of partial differential equations where these RBFs are defined, on a grid with uniform spacing h, as translates of the “master” function ϕ ( x ; α , h ) ≡ exp ( - [ α 2 / h 2 ] x 2 ) where α is a user-choosable constant. Unfortunately, computing the coefficients of ϕ ( x - jh ; α , h ) requires solving a linear system with a dense matrix. It would be much more efficient to rearrange the basis functions into the equivalent “Lagrangian” or “cardinal” basis because the interpolation matrix in the new basis is the identity matrix; the cardinal basis C j ( x ; α , h ) is defined by the set of linear combinations of the Gaussians such that C j ( kh ) = 1 when k = j and C j ( kh ) = 0 for all integers k ≠ j . We show that the cardinal functions for the uniform grid are C j ( x ; h , α ) = C ( x / h - j ; α ) where C ( X ; α ) ≈ ( α 2 / π ) sin ( π X ) / sinh ( α 2 X ) . The relative error is only about 4 exp ( - 2 π 2 / α 2 ) as demonstrated by the explicit second order approximation. It has long been known that the error in a series of Gaussian RBFs does not converge to zero for fixed α as h → 0 , but only to an “error saturation” proportional to exp ( - π 2 / α 2 ) . Because the error in our approximation to the master cardinal function C ( X ; α ) is the square of the error saturation, there is no penalty for using our new approximations to obtain matrix-free interpolating RBF approximations to an arbitrary function f ( x ) . The master cardinal function on a uniform grid in d dimensions is just the direct product of the one-dimensional cardinal functions. Thus in two dimensions C ( X , Y ; α ) ∼ ( α 4 / π 2 ) sin ( π X ) sin ( π Y ) / [ sinh ( α 2 X ) sinh ( α 2 Y ) ] . We show that the matrix-free interpolation can be extended to non-uniform grids by a smooth change of coordinates.

Kernel based approximation in Sobolev spaces with radial basis functions

In this paper, we study several radial basis function approximation schemes in Sobolev spaces. We obtain an optional error estimate by using a class of smoothing operators. We also discussed sufficient conditions for the smoothing operators to attain the desired approximation order. We then construct the smoothing operators by some compactly supported radial kernels, and use them to approximate Sobolev space functions with optimal convergence order. These kernels can be simply constructed and readily applied to practical problems. The results show that the approximation power depends on the precision of the sampling instrument and the density of the available data.

A random variable shape parameter strategy for radial basis function approximation methods

Several variable shape parameter methods have been successfully used in radial basis function approximation methods. In many cases variable shape parameter strategies produced more accurate results than if a constant shape parameter had been used. We introduce a new random variable shape parameter strategy and give numerical results showing that the new random strategy often outperforms both existing variable shape and constant shape strategies.