On Quasi-interpolation with Radial Basis Functions

ABSTRACTThis paper analyses the performance of the main radial basis functions in the formulation of the Boundary Element Method (DIBEM). This is an alternative for solving problems modeled by non-adjoint differential operators, since it transforms domain integrals in boundary integrals using radial basis functions. The solution of eigenvalue problem was chosen to performance evaluation. Natural frequencies are calculated numerically using several radial functions and their accuracy is evaluated by comparison with the available analytical solutions and with the Finite Element Method as well. The standard radial basis functions have presented similar performance to compact radial functions, being even slightly superior.

The value algorithms of classical function approximation theory have a common drawback: the compute-intensive, poor adaptability, high model and data demanding and the limitation in practical applications. Neural network can calculate the complex relationship between input and output, therefore, neural network has a strong function approximation capability. This paper describes the application of RBFNN in function approximation and interpolation of scattered data. RBF neural network uses Gaussian function as transfer function widespreadly. Using it to train data set, it needs to determine the extension of radial basis function constant SPEAD. SPEAD setting is too small, there will be an over eligibility for function approximation, while SPREAD is too large, there will be no eligibility for function approximation. This paper examines the usage of different radial functions as transferinf functions to design the neural network, and analyzes their numerical applications. Simulations show that, for the same data set, Gaussian radial basis function may not be the best.

In this paper, we study the convergence behavior of interpolants by smooth radial basis functions to polynomial interpolants in $\mathbb{R}^d$ as the radial basis functions are scaled to be increasingly flat. Larsson and Fornberg [Comput. Math. Appl., 49 (2005), pp. 103–130] conjectured a sufficient property for this convergence, and they also conjectured that Bessel radial functions do not satisfy this property. First, in the case of positive definite radial functions, we prove both conjectures by Larsson and Fornberg for the convergence of increasingly flat radial function interpolants. Next, we extend the results to the case of conditionally positive definite radial functions of order $m>0$.

This chapter revisits critical aspects of pseudoatom interpretation of X-ray Bragg diffraction data to derive the crystalline electron density experimentally. The main focus is on the radial basis functions in the nucleus-centered multipole expansion of the model density. Based on the direct-space projection of the stockholder-partitioned quantum-chemical molecular density onto nucleus-centered spherical harmonics, “bonded-atom” radial functions are derived, in terms of which the original density can be reconstructed. The method allows for a quantitative access of errors introduced by restrictions on the pseudoatom radial functions and for designing more adequate basis sets than those routinely used in the standard model. The applicability of the upgraded multipole formalism to X-ray charge density analysis is demonstrated through pseudoatom fitting of simulated and experimental data. The chemical transferability of stockholder pseudoatoms is analyzed through small-molecule model studies. A protocol to build a stockholder-atom library and the feasibility of simple data-refinement strategies using the databank radial functions is also discussed.

If additional smoothness requirements and boundary conditions are met, the well–known approximation orders of scattered data interpolants by radial functions can roughly be doubled.

In this study, the prediction errors of various spatial interpolation methods used to model values at unmeasured locations was compared and the accuracy of these predictions was evaluated. The root mean square (RMS) was calculated by processing different parameters associated with spatial interpolation by using techniques such as inverse distance weighting, kriging, local polynomial interpolation and radial basis function to known elevation data of the east coastal area under the same condition. As a result, a circular model of simple kriging reached the smallest RMS value. Prediction map using the multiquadric method of a radial basis function was coincident with the spatial distribution obtained by constructing a triangulated irregular network of the study area through the raster mathematics. In addition, better interpolation results can be obtained by setting the optimal power value provided under the selected condition.

این مطالعه به منظور بررسی پیش بینی شاخص های عملکرد تولیدی در مرغان تخم گذار با استفاده از شبکه های عصبی مصنوعی و رگرسیون غیرخطی چندگانه انجام شد. بررسی بر روی اطلاعات چهار دوره متوالی پرورش در یک واحد پرورش مرغ تخم گذار صورت گرفت. روش های داده-کاوی شامل رگرسیون خطی و غیر خطی، شبکه عصبی پرسپترون سه لایه، شبکه عصبی پرسپترون چهار لایه و شبکه عصبی با تابع پایه ای شعاعی بود. در این مدل ها از متغیرهای سن گله، میزان خوراک مصرفی و فصل تولید به عنوان متغیر پیشگو و شاخص های عملکرد تولیدی شامل درصد تخم-گذاری، وزن توده ای تخم مرغ تولیدی و ضریب تبدیل غذایی به عنوان متغیر پاسخ استفاده شد. نتایج نهایی رگرسیون های خطی نشان داد که برای تمامی متغیرهای وابسته مورد مطالعه متغیر مستقل سن گله معنی دار می باشد. بنابراین رگرسیون غیر خطی شاخص های عملکرد تولیدی در مقابل سن برای مقایسه با شبکه های عصبی مختلف مورد بررسی قرار گرفت و برای مقایسه کلیه مدل ها از ضریب تعیین (R2) و میانگین قدر مطلق خطا (MAE) استفاده شد. نتایج نشان داد بین شبکه های عصبی مصنوعی مختلف مورد مطالعه، شبکه با تابع پایه ای شعاعی بهتر از سایر مدل های در پیش‌بینی شاخص های عملکرد تولیدی مرغان تخم گذار عمل می کند.

In this paper, a numerically stable method of calculating atomic integrals is suggested. The commutation relations among the components of the angular momentum and the Runge-Lenz vector are used to deduce recurrence relations for the Sturmian radial functions. The radial part of the one- and two-electron integrals is evaluated by means of these recurrence relations. The product of two radial functions is written as a linear combination of the radial functions. This enables us to write the integrals over four radial functions as a linear combination of the integrals over two radial functions. The recurrence relations for the functions are used to derive the recursion relations for the coefficients of the linear combination and for the integrals over two functions.

In this paper we use asymptotic expansions of certain integral transforms in order to derive conditions on a radial basis function o: R≥0→R that imply the existence of a cardinal function $$x\left( x \right)=\sum\limits_{k\in {{Z}^{n}}}{{{c}_{k}}\phi \left( \parallel x-k\parallel \right)},x\in {{R}^{n}},$$ which satisfies x(l) = δol for all l ∈ Zn. We also study the rate of decay of |X(x)| for large ‖x‖ and the polynomial recovery of interpolation on Zn using this cardinal function. The conditions hold for many important examples of radial basis functions, such as the multiquadrics and related radial functions, and in contrast to some earlier work by the author they are expressed in terms of asymptotic properties of o rather than in terms of its Fourier transform.

The human face has a unique shape and size, as well as a 3D character face model. During this process of animated facial expression of 3D virtual characters are mostly still done manually by moving the rig in each frame. The more characters used, the more production costs incurred. The absence of a cheap facial motion transfer system is also one of the reasons why not many studios are using motion capture technology in Indonesia. This study will evaluate the implementation of radial basis function (RBF) as a method of marker-transfer used as a reference to rig movement in the facial animation system. Testing is done by performing variations of radial function, namely: Gaussian, Inverse Quadratic, Inverse Multiquadric, and Multiquadric. The value of epsilon used is 0.01. The experimental results show that the range of feature point shifts generated by RBF Gaussian, Inverse Multiquadric, Inverse Quadratic, and Multiquadric have the same pattern with the difference in the distance of the marker rigging point shift on the target 3D model. The farthest range of differences is generated by RBF Gaussian. The resulting maximum range difference can reach 92.37% and a minimum of 2.47% compared to other methods. This appears on the Gaussian chart to have a sharper pattern. As with the concept of exaggeration, the Gaussian method which has the maximum range is the right method to apply the exaggeration principle in 3D face-expression animation.

Fast computation of RBF coefficients using FFT

A Meshless Radial Basis Function Method for Fluid Flow with Heat Transfer K Nagamani Devi1, D.W. Pepper2 Summary Over the past few years, efforts have been made to solve fluid flow and heat transfer problems using radial basis functions. This approach is meshless, easy to understand, and simple to implement. Preliminary results indicate accuracies on the order of finely meshed conventional techniques, but with considerably less computational effort. In this study, a projection-based technique is used to solve the primitive equations of motion and energy using radial basis functions. Three benchmark test cases are examined: (1) lid-driven cavity flow, (2) natural convection in a square enclosure, and (3) flow with forced convection over backward facing step. Results are compared with COMSOL and FLUENT – two popular commercial CFD packages.

In this paper, radial basis functions are utilised for numerical prediction of the bulk properties of particulate suspensions under simple shear conditions. The suspending fluid is Newtonian and the suspended particles are rigid. Results obtained are compared well with those based on finite elements in the literature.

Radial basis functions can be used as an alternative method for mineral resource estimation when ordinary kriging cannot be done because it is impossible to calculate an experimental variogram. Interpolation based on radial based functions is a method that is similar to ordinary kriging. In this context, this article presents the results of gold resource estimation and modeling of the União Luis and Morro do Carrapato gold deposits located in Alta Floresta Gold Province, Mato Grosso. Experimental variograms in the main direction (90-degree strike) resulted in a pure nugget effect, whereas in the orthogonal direction (180/85), they resulted in a structured variogram. However, the structured variogram for 180/85 is useless because three orthogonal variograms are needed to define the anisotropy ellipsoid for deriving a correlation model used in ordinary kriging. Thus, mineral resource estimation was done using multiquadric equations, a very popular radial based function kernel. The 3D model for gold grades showed a vertical distribution suggesting a structural conditioning for gold mineralization. The grade-tonnage curve with a simulated cutoff of 5.0 g/ton resulted in a gold mineral resource of 2.122 tons. Through the above, the gold estimation and multiquadric equation-based 3D model for the studied area can be considered effective in its objective to estimate ore resources from the available data.

We study cubature formulas on relatively small scattered samples in the unit square, obtained by integrating radial basis function (RBF) interpolants. Numerical tests show that, due to the small size of the corresponding weights (which are not all positive in general), thin-plate splines and Wendland's compactly supported radial functions give the most reliable RBF cubature methods.