Related papers: Kernel-based collocation methods for Zakai equatio…
Matrices resulting from the discretization of a kernel function, e.g., in the context of integral equations or sampling probability distributions, can frequently be approximated by interpolation. In order to improve the efficiency, a…
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 construct approximate Fekete point sets for kernel-based interpolation by maximising the determinant of a kernel Gram matrix obtained via truncation of an orthonormal expansion of the kernel. Uniform error estimates are proved for kernel…
Data sites selected from modeling high-dimensional problems often appear scattered in non-paternalistic ways. Except for sporadic clustering at some spots, they become relatively far apart as the dimension of the ambient space grows. These…
Several kernel-based methods for the numerical solution of fractional differential equations have been developed in the recent past; however, these techniques exclusively relied on the use of radial basis function approximations. In the…
We consider scattered data approximation on product regions of equal and different dimensionality. On each of these regions, we assume quasi-uniform but unstructured data sites and construct optimal sparse grids for scattered data…
We introduce a novel kernel learning framework toward efficiently solving nonlinear partial differential equations (PDEs). In contrast to the state-of-the-art kernel solver that embeds differential operators within kernels, posing…
Kernel based regularized interpolation is a well known technique to approximate a continuous multivariate function using a set of scattered data points and the corresponding function evaluations, or data values. This method has some…
This paper deals with the kernel-based approximation of a multivariate periodic function by interpolation at the points of an integration lattice -- a setting that, as pointed out by Zeng, Leung, Hickernell (MCQMC2004, 2006) and Zeng,…
We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the…
Interpolation and approximation of functionals with conditionally positive definite kernels is considered on sets of centers that are not determining for polynomials. It is shown that polynomial consistency is sufficient in order to define…
Consider a linear operator equation $x - Kx = f$, where $f$ is given and $K$ is a Fredholm integral operator with a Green's function type kernel defined on $C[0, 1]$. For $r \geq 0$, we employ the interpolatory projection at $2r + 1$…
In the recent paper [8], a new method to compute stable kernel-based interpolants has been presented. This \textit{rescaled interpolation} method combines the standard kernel interpolation with a properly defined rescaling operation, which…
The polynomial kernels are widely used in machine learning and they are one of the default choices to develop kernel-based classification and regression models. However, they are rarely used and considered in numerical analysis due to their…
The systems of nonlinear Volterra integral equations of the first kind with jump discontinuous kernels are studied. The iterative numerical method for such nonlinear systems is proposed. Proposed method employs the modified…
Consider a non-linear operator equation $x - K(x) = f$, where $f$ is a given function and $K$ is a Urysohn integral operator with Green's function type kernel defined on $L^\infty [0, 1]$. We apply approximation methods based on…
Can graded meshes yield more accurate numerical solution than uniform meshes? A time-dependent nonlocal diffusion problem with a weakly singular kernel is considered using collocation method. For its steady-state counterpart, under the…
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…
We propose and study a new quasi-interpolation method on spheres featuring the following two-phase construction and analysis. In Phase I, we analyze and characterize a large family of zonal kernels (e.g., the spherical version of Poisson…
This paper proposes and studies a numerical method for approximation of posterior expectations based on interpolation with a Stein reproducing kernel. Finite-sample-size bounds on the approximation error are established for posterior…