Univariate interpolation by exponential functions and gaussian RBFs for generic sets of nodes

We consider interpolation of univariate functions on arbitrary sets of nodes by Gaussian radial basis functions or by exponential functions. We derive closed-form expressions for the interpolation error based on the Harish-Chandra-Itzykson-Zuber formula. We then prove the exponential convergence of interpolation for functions analytic in a sufficiently large domain. As an application, we prove the global exponential convergence of optimization by expected improvement for such functions.Comment: Some stylistic improvements and added references following feedback from the reviewer

Adaptive multiquadric collocation for boundary layer problems

AbstractAn adaptive collocation method based upon radial basis functions is presented for the solution of singularly perturbed two-point boundary value problems. Using a multiquadric integral formulation, the second derivative of the solution is approximated by multiquadric radial basis functions. This approach is combined with a coordinate stretching technique. The required variable transformation is accomplished by a conformal mapping, an iterated sine-transformation. A new error indicator function accurately captures the regions of the interval with insufficient resolution. This indicator is used to adaptively add data centres and collocation points. The method resolves extremely thin layers accurately with fairly few basis functions. The proposed adaptive scheme is very robust, and reaches high accuracy even when parameters in our coordinate stretching technique are not chosen optimally. The effectiveness of our new method is demonstrated on two examples with boundary layers, and one example featuring an interior layer. It is shown in detail how the adaptive method refines the resolution

On spherical averages of radial basis functions

A radial basis function (RBF) has the general form $$s(x)=\sum_{k=1}^{n}a_{k}\phi(x-b_{k}),\quad x\in\mathbb{R}^{d},$$ where the coefficients a 1,…,a n are real numbers, the points, or centres, b 1,…,b n lie in ℝ d , and φ:ℝ d →ℝ is a radially symmetric function. Such approximants are highly useful and enjoy rich theoretical properties; see, for instance (Buhmann, Radial Basis Functions: Theory and Implementations, [2003]; Fasshauer, Meshfree Approximation Methods with Matlab, [2007]; Light and Cheney, A Course in Approximation Theory, [2000]; or Wendland, Scattered Data Approximation, [2004]). The important special case of polyharmonic splines results when φ is the fundamental solution of the iterated Laplacian operator, and this class includes the Euclidean norm φ(x)=‖x‖ when d is an odd positive integer, the thin plate spline φ(x)=‖x‖2log  ‖x‖ when d is an even positive integer, and univariate splines. Now B-splines generate a compactly supported basis for univariate spline spaces, but an analyticity argument implies that a nontrivial polyharmonic spline generated by (1.1) cannot be compactly supported when d>1. However, a pioneering paper of Jackson (Constr. Approx. 4:243–264, [1988]) established that the spherical average of a radial basis function generated by the Euclidean norm can be compactly supported when the centres and coefficients satisfy certain moment conditions; Jackson then used this compactly supported spherical average to construct approximate identities, with which he was then able to derive some of the earliest uniform convergence results for a class of radial basis functions. Our work extends this earlier analysis, but our technique is entirely novel, and applies to all polyharmonic splines. Furthermore, we observe that the technique provides yet another way to generate compactly supported, radially symmetric, positive definite functions. Specifically, we find that the spherical averaging operator commutes with the Fourier transform operator, and we are then able to identify Fourier transforms of compactly supported functions using the Paley–Wiener theorem. Furthermore, the use of Haar measure on compact Lie groups would not have occurred without frequent exposure to Iserles’s study of geometric integration

Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators

In this paper we introduce a generalized Sobolev space by defining a semi-inner product formulated in terms of a vector distributional operator $\mathbf{P}$ consisting of finitely or countably many distributional operators $P_n$, which are defined on the dual space of the Schwartz space. The types of operators we consider include not only differential operators, but also more general distributional operators such as pseudo-differential operators. We deduce that a certain appropriate full-space Green function $G$ with respect to $L:=\mathbf{P}^{\ast T}\mathbf{P}$ now becomes a conditionally positive definite function. In order to support this claim we ensure that the distributional adjoint operator $\mathbf{P}^{\ast}$ of $\mathbf{P}$ is well-defined in the distributional sense. Under sufficient conditions, the native space (reproducing-kernel Hilbert space) associated with the Green function $G$ can be isometrically embedded into or even be isometrically equivalent to a generalized Sobolev space. As an application, we take linear combinations of translates of the Green function with possibly added polynomial terms and construct a multivariate minimum-norm interpolant $s_{f,X}$ to data values sampled from an unknown generalized Sobolev function $f$ at data sites located in some set $X \subset \mathbb{R}^d$. We provide several examples, such as Mat\'ern kernels or Gaussian kernels, that illustrate how many reproducing-kernel Hilbert spaces of well-known reproducing kernels are isometrically equivalent to a generalized Sobolev space. These examples further illustrate how we can rescale the Sobolev spaces by the vector distributional operator $\mathbf{P}$. Introducing the notion of scale as part of the definition of a generalized Sobolev space may help us to choose the "best" kernel function for kernel-based approximation methods.Comment: Update version of the publish at Num. Math. closed to Qi Ye's Ph.D. thesis (\url{http://mypages.iit.edu/~qye3/PhdThesis-2012-AMS-QiYe-IIT.pdf}

Reconstruction from Radon projections and orthogonal expansion on a ball

The relation between Radon transform and orthogonal expansions of a function on the unit ball in \RR^d is exploited. A compact formula for the partial sums of the expansion is given in terms of the Radon transform, which leads to algorithms for image reconstruction from Radon data. The relation between orthogonal expansion and the singular value decomposition of the Radon transform is also exploited.Comment: 15 page

ОСОБЛИВОСТІ САРКОЇДОЗУ В ТЕРНОПІЛЬСЬСКІЙ ОБЛАСТІ

Purpose: to evaluate the incidence and prevalence of sarcoidosis in the Ternopil region and to study the gender, age characteristics of the patients.Materials and Methods. We analyzed 341 cards of ambulatory patient with sarcoidosis, who were at the dispensary observation in the Ternopil Regional TB Dispensary during 1985–2017.Results. The prevalence of sarcoidosis in Ternopil region is 32.3 per 100 thousand population, this indicator for the urban population is 3 times higher than for rural residents. The incidence of sarcoidosis in the region for 10 years (2005–2015) has increased almost in 4 times, and in 2016 it was 4.0 per 100 thousand population, which is 1.5 times higher than the maximum inUkraine.Conclusions. The prevalence and morbidity of sarcoidosis in the Ternopil region exceeds the maximum inUkraine by 4 and 1.5 times respectively. Sarcoidosis is more likely to be detected in city dwellers, especially in stage I, which is probably due to better access of people to specialized medical facilities, while residents of the oblast find more complicated for diagnostic ІІ and ІV stages.Мета: оцінити захворюваність та поширеність саркоїдозу в Тернопільській області та вивчити гендерні, вікові особливості пацієнтів.Матеріали і методи. Опрацьовано 341 карту амбулаторного хворого на саркоїдоз, всі вони перебували на диспансерному спостереженні в Тернопільському обласному протитуберкульозному диспансері протягом 1985–2017рр.Результати. Поширеність саркоїдозу в Тернопільській області становить 32,3 на 100 тис. населення, цей показник для міського населення в 3 рази вище, ніж для сільських жителів. Захворюваність на саркоїдоз в області за 10 років (2005–2015 рр.) зросла майже в 4 рази, а в 2016 р. становила 4,0 на 100 тис. населення, що в 1,5 раза перевищує максимальний показник по Україні.Висновки. Поширеність та захворюваність на саркоїдоз у Тернопільській області перевищують максимальні по Україні у 4 та 1,5 раза відповідно. Саркоїдоз частіше виявляють у жителів міст, особливо І стадії, що, імовірно, пов’язано із кращим доступом населення до спеціалізованих медичних закладів, у той час як у мешканців області виявляють більш складні для діагностики ІІ та ІV стадій

A symmetric integrated radial basis function method for solving differential equations

In this article, integrated radial basis functions (IRBFs) are used for Hermite interpolation in the solution of differential equations, resulting in a new meshless symmetric RBF method. Both global and local approximation-based schemes are derived. For the latter, the focus is on the construction of compact approximation stencils, where a sparse system matrix and a high-order accuracy can be achieved together. Cartesian-grid-based stencils are possible for problems defined on nonrectangular domains. Furthermore, the effects of the RBF width on the solution accuracy for a given grid size are fully explored with a reasonable computational cost. The proposed schemes are numerically verified in some elliptic boundary-value problems governed by the Poisson and convection-diffusion equations. High levels of the solution accuracy are obtained using relatively coarse discretisations

A generalised finite difference scheme based on compact integrated radial basis function for flow in heterogeneous soils

In the present paper, we develop a generalised finite difference approach based on compact integrated radial basis function (CIRBF) stencils for solving highly nonlinear Richards equation governing fluid movement in heterogeneous soils. The proposed CIRBF scheme enjoys a high level of accuracy and a fast convergence rate with grid refinement owing to the combination of the integrated RBF approximation and compact approximation where the spatial derivatives are discretised in terms of the information of neighbouring nodes in a stencil. The CIRBF method is first verified through the solution of ordinary differential equations, 2-D Poisson equations and a Taylor-Green vortex. Numerical comparisons show that the CIRBF method outperforms some other methods in the literature. The CIRBF method in conjunction with a rational function transformation method and an adaptive time-stepping scheme is then applied to simulate 1-D and 2-D soil infiltrations effectively. The proposed solutions are more accurate and converge faster than those of the finite different method employed with a second-order central difference scheme. Additionally, the present scheme also takes less time to achieve target accuracy in comparison with the 1D-IRBF and HOC schemes

A novel dual reciprocity boundary element formulation for two-dimensional transient convection–diffusion–reaction problems with variable velocity

This paper describes a new formulation of the dual reciprocity boundary element method (DRBEM) for two-dimensional transient convection-diffusion-reaction problems with variable velocity. The formulation decomposes the velocity field into an average and a perturbation part, with the latter being treated using a dual reciprocity approximation to convert the domain integrals arising in the boundary element formulation into equivalent boundary integrals. The integral representation formula for the convection-diffusion-reaction problem with variable velocity is obtained from the Green’s second identity, using the fundamental solution of the corresponding steady-state equation with constant coefficients. A finite difference method (FDM) is used to simulate the time evolution procedure for solving the resulting system of equations. Numerical applications are included for three different benchmark examples for which analytical solutions are available, to establish the validity of the proposed approach and to demonstrate its efficiency. Finally, results obtained show that the DRBEM results are in excellent agreement with the analytical solutions and do not present oscillations or damping of the wave front, as it appears in other numerical techniques