Related papers: Stable computations with flat radial basis functio…
Scattered data fitting is a frequently encountered problem for reconstructing an unknown function from given scattered data. Radial basis function (RBF) methods have proven to be highly useful to deal with this problem. We describe two…
A radial basis function (RBF) method based on matrix-valued kernels is presented and analyzed for computing two types of vector decompositions on bounded domains: one where the normal component of the divergence-free part of the field is…
Approximation of scattered data is often a task in many engineering problems. The Radial Basis Function (RBF) approximation is appropriate for large scattered (unordered) datasets in d-dimensional space. This approach is useful for a higher…
Derivative boundary conditions introduce challenges for mesh-free discretizations of PDEs on surfaces, especially when the domain is represented by randomly sampled point clouds. The recently developed two-step tangent-space RBF-generated…
In this paper, we present a spectral method based on Radial Basis Functions (RBFs) for numerically solving the fully nonlinear 1D Serre Green-Naghdi equations. The approximation uses an RBF discretization in space and finite differences in…
In this paper, we deal with the challenging computational issue of interpolating large data sets, with eventually non-homogeneous densities. To such scope, the Radial Basis Function Partition of Unity (RBF-PU) method has been proved to be a…
The Reduced Basis Method (RBM) is a rigorous model reduction approach for solving parametrized partial differential equations. It identifies a low-dimensional subspace for approximation of the parametric solution manifold that is embedded…
We propose two localized Radial Basis Function (RBF) methods, the Radial Basis Function Partition of Unity method (RBF-PUM) and the Radial Basis Function generated Finite Differences method (RBF-FD), for solving financial derivative pricing…
A rational approximation by a ratio of polynomial functions is a flexible alternative to polynomial approximation. In particular, rational functions exhibit accurate estimations to nonsmooth and non- Lipschitz functions, where polynomial…
This work is concerned with the kernel-based approximation of a complex-valued function from data, where the frequency response function of a partial differential equation in the frequency domain is of particular interest. In this setting,…
We have developed a parallel algorithm for radial basis function (RBF) interpolation that exhibits O(N) complexity,requires O(N) storage, and scales excellently up to a thousand processes. The algorithm uses a GMRES iterative solver with a…
Radial Basis Function (RBF), or Gaussian, kernels are among the most widely used parametric kernels in machine learning, particularly in methods such as Support Vector Machines (SVM) and kernel-based subspace approaches. The kernel…
This paper introduces a novel parametric activation function based on Wendland radial basis functions (RBFs) for deep neural networks. Wendland RBFs, known for their compact support, smoothness, and positive definiteness in approximation…
In this paper we propose a new efficient interpolation tool, extremely suitable for large scattered data sets. The partition of unity method is used and performed by blending Radial Basis Functions (RBFs) as local approximants and using…
Reconstruction of high-fidelity 3D objects or scenes is a fundamental research problem. Recent advances in RGB-D fusion have demonstrated the potential of producing 3D models from consumer-level RGB-D cameras. However, due to the discrete…
A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for…
Cubature formulas (CFs) based on radial basis functions (RBFs) have become an important tool for multivariate numerical integration of scattered data. Although numerous works have been published on such RBF-CFs, their stability theory can…
We consider the problem of computing univariate polynomial matrices over a field that represent minimal solution bases for a general interpolation problem, some forms of which are the vector M-Pad\'e approximation problem in [Van Barel and…
Radial basis functions (RBFs) are prominent examples for reproducing kernels with associated reproducing kernel Hilbert spaces (RKHSs). The convergence theory for the kernel-based interpolation in that space is well understood and optimal…
The paper is concerned with classic kernel interpolation methods, in addition to approximation methods that are augmented by gradient measurements. To apply kernel interpolation using radial basis functions (RBFs) in a stable way, we…