Compactly supported radial basis functions: How and why?

Compactly supported basis functions are widely required and used in many applications. We explain why radial basis functions are preferred to multi-variate polynomials for scattered data approximation in high-dimensional space and give a brief description on how to construct the most commonly used compactly supported radial basis functions - the Wendland functions and the new found missing Wendland functions. One can construct a compactly supported radial basis function with required smoothness according to the procedure described here without sophisticated mathematics. Very short programs and extended tables for compactly supported radial basis functions are supplied

On the spectral distribution of kernel matrices related to\ud radial basis functions

This paper focuses on the spectral distribution of kernel matrices related to radial basis functions. The asymptotic behaviour of eigenvalues of kernel matrices related to radial basis functions with different smoothness are studied. These results are obtained by estimated the coefficients of an orthogonal expansion of the underlying kernel function. Beside many other results, we prove that there are exactly (k+d−1/d-1) eigenvalues in the same order for analytic separable kernel functions like the Gaussian in Rd. This gives theoretical support for how to choose the diagonal scaling matrix in the RBF-QR method (Fornberg et al, SIAM J. Sci. Comput. (33), 2011) which can stably compute Gaussian radial basis function interpolants

Data Interpolants -- That's What Discriminators in Higher-order Gradient-regularized GANs Are

We consider the problem of optimizing the discriminator in generative adversarial networks (GANs) subject to higher-order gradient regularization. We show analytically, via the least-squares (LSGAN) and Wasserstein (WGAN) GAN variants, that the discriminator optimization problem is one of interpolation in $n$-dimensions. The optimal discriminator, derived using variational Calculus, turns out to be the solution to a partial differential equation involving the iterated Laplacian or the polyharmonic operator. The solution is implementable in closed-form via polyharmonic radial basis function (RBF) interpolation. In view of the polyharmonic connection, we refer to the corresponding GANs as Poly-LSGAN and Poly-WGAN. Through experimental validation on multivariate Gaussians, we show that implementing the optimal RBF discriminator in closed-form, with penalty orders $m \approx\lceil \frac{n}{2} \rceil$, results in superior performance, compared to training GAN with arbitrarily chosen discriminator architectures. We employ the Poly-WGAN discriminator to model the latent space distribution of the data with encoder-decoder-based GAN flavors such as Wasserstein autoencoders

Solving high-order partial differential equations with indirect radial basis function networks

This paper reports a new numerical method based on radial basis function networks (RBFNs) for solving high-order partial differential equations (PDEs). The variables and their derivatives in the governing equations are represented by integrated RBFNs. The use of integration in constructing neural networks allows the straightforward implementation of multiple boundary conditions and the accurate approximation of high-order derivatives. The proposed RBFN method is verified successfully through the solution of thin-plate bending and viscous flow problems which are governed by biharmonic equations. For thermally driven cavity flows, the solutions are obtained up to a high Rayleigh number

A trivariate interpolation algorithm using a cube-partition searching procedure

In this paper we propose a fast algorithm for trivariate interpolation, which is based on the partition of unity method for constructing a global interpolant by blending local radial basis function interpolants and using locally supported weight functions. The partition of unity algorithm is efficiently implemented and optimized by connecting the method with an effective cube-partition searching procedure. More precisely, we construct a cube structure, which partitions the domain and strictly depends on the size of its subdomains, so that the new searching procedure and, accordingly, the resulting algorithm enable us to efficiently deal with a large number of nodes. Complexity analysis and numerical experiments show high efficiency and accuracy of the proposed interpolation algorithm

The collocation and meshless methods for differential equations in R(2)

In recent years, meshless methods have become popular ones to solve differential equations. In this thesis, we aim at solving differential equations by using Radial Basis Functions, collocation methods and fundamental solutions (MFS). These methods are meshless, easy to understand, and even easier to implement