4 citations · 9 across the 11 of their papers we have counts for
11 papers
On the Numerical Approximation of the Karhunen-Loève Expansion for Random Fields with Random Discrete Data
Michael Griebel, Guanglian Li, Christian Rieger
In many applications, random fields reflect uncertain parameters, and often their moments are part of the modeling process and thus well known. However, there are practical situati…
Highly Localized RBF Lagrange Functions for Finite Difference Methods on Spheres
Wolfgang Erb, Thomas Hangelbroek, Francis J. Narcowich +2
The aim of this paper is to show how rapidly decaying RBF Lagrange functions on the spheres can be used to create effective, stable finite difference methods based on radial basis…
Kernel Multi-Grid on Manifolds
Thomas Hangelbroek, Christian Rieger
Kernel methods for solving partial differential equations on surfaces have the advantage that those methods work intrinsically on the surface and yield high approximation rates if…
Extending error bounds for radial basis function interpolation to measuring the error in higher order Sobolev norms
Thomas Hangelbroek, Christian Rieger
Radial basis functions (RBFs) are prominent examples for reproducing kernels with associated reproducing kernel Hilbert spaces (RKHSs). The convergence theory for the kernel-based…
On the Numerical Approximation of the Karhunen-Loève Expansion for Random Fields with Random Discrete Data
Michael Griebel, Guanglian Li, Christian Rieger
Many physical and mathematical models involve random fields in their input data. Examples are ordinary differential equations, partial differential equations and integro--different…
A Particle Method without Remeshing
Matthias Kirchhart, Christian Rieger
We propose a simple tweak to a recently developed regularisation scheme for particle methods. This allows us to chose the particle spacing proportional to the regularisation le…
On Particles and Splines in Bounded Domains
Matthias Kirchhart
We propose numerical schemes that enable the application of particle methods for advection problems in general bounded domains. These schemes combine particle fields with Cartesian…
Kernel-based stochastic collocation for the random two-phase Navier-Stokes equations
Michael Griebel, Christian Rieger, Peter Zaspel
In this work, we apply stochastic collocation methods with radial kernel basis functions for an uncertainty quantification of the random incompressible two-phase Navier-Stokes equa…
A representer theorem for deep kernel learning
Bastian Bohn, Michael Griebel, Christian Rieger
In this paper we provide a finite-sample and an infinite-sample representer theorem for the concatenation of (linear combinations of) kernel functions of reproducing kernel Hilbert…
An inverse theorem for compact Lipschitz regions in using localized kernel bases
Thomas Hangelbroek, Francis J. Narcowich, Christian Rieger +1
While inverse estimates in the context of radial basis function approximation on boundary-free domains have been known for at least ten years, such theorems for the more important…
Direct and Inverse Results on Bounded Domains for Meshless Methods via Localized Bases on Manifolds
Thomas Hangelbroek, Francis J. Narcowich, Christian Rieger +1
This article develops direct and inverse estimates for certain finite dimensional spaces arising in kernel approximation. Both the direct and inverse estimates are based on approxi…