41 citations · 108 across the 19 of their papers we have counts for
19 papers
Refined rates of convergence for target-data dependent greedy generalized interpolation with Sobolev kernels
Bernard Haasdonk, Gabriele Santin, Tizian Wenzel +1
Greedy methods have recently been successfully applied to generalized kernel interpolation, or the recovery of a function from data stemming from the evaluation of linear functiona…
On the optimal shape parameter for kernel methods: Sharp direct and inverse statements
Tizian Wenzel, Gabriele Santin
The search for the optimal shape parameter for Radial Basis Function (RBF) kernel approximation has been an outstanding research problem for decades. In this work, we establish a t…
Kernel-based Greedy Approximation of Parametric Elliptic Boundary Value Problems
Bernard Haasdonk, Gabriele Santin, Tizian Wenzel
We recently introduced a scale of kernel-based greedy schemes for approximating the solutions of elliptic boundary value problems. The procedure is based on a generalized interpola…
General superconvergence for kernel-based approximation
Toni Karvonen, Gabriele Santin, Tizian Wenzel
Kernel interpolation is a fundamental technique for approximating functions from scattered data, with a well-understood convergence theory when interpolating elements of a reproduc…
Kernel Methods in the Deep Ritz framework: Theory and practice
Hendrik Kleikamp, Tizian Wenzel
In this contribution, kernel approximations are applied as ansatz functions within the Deep Ritz method. This allows to approximate weak solutions of elliptic partial differential…
Greedy Kernel Methods for Approximating Breakthrough Curves for Reactive Flow from 3D Porous Geometry Data
Robin Herkert, Patrick Buchfink, Tizian Wenzel +3
We address the challenging application of 3D pore scale reactive flow under varying geometry parameters. The task is to predict time-dependent integral quantities, i.e., breakthrou…
On the optimality of target-data-dependent kernel greedy interpolation in Sobolev Reproducing Kernel Hilbert Spaces
Gabriele Santin, Tizian Wenzel, Bernard Haasdonk
Kernel interpolation is a versatile tool for the approximation of functions from data, and it can be proven to have some optimality properties when used with kernels related to cer…
Sharp inverse statements for kernel interpolation
Tizian Wenzel
While direct statements for kernel based interpolation on regions are well researched, far less is known about corresponding inverse statements. The availab…
Application of Deep Kernel Models for Certified and Adaptive RB-ML-ROM Surrogate Modeling
Tizian Wenzel, Bernard Haasdonk, Hendrik Kleikamp +2
In the framework of reduced basis methods, we recently introduced a new certified hierarchical and adaptive surrogate model, which can be used for efficient approximation of input-…
Data-driven kernel designs for optimized greedy schemes: A machine learning perspective
Tizian Wenzel, Francesco Marchetti, Emma Perracchione
Thanks to their easy implementation via Radial Basis Functions (RBFs), meshfree kernel methods have been proved to be an effective tool for e.g. scattered data interpolation, PDE c…
Adaptive meshfree approximation for linear elliptic partial differential equations with PDE-greedy kernel methods
Tizian Wenzel, Daniel Winkle, Gabriele Santin +1
We consider meshless approximation for solutions of boundary value problems (BVPs) of elliptic Partial Differential Equations (PDEs) via symmetric kernel collocation. We discuss th…
A new certified hierarchical and adaptive RB-ML-ROM surrogate model for parametrized PDEs
B. Haasdonk, H. Kleikamp, M. Ohlberger +2
We present a new surrogate modeling technique for efficient approximation of input-output maps governed by parametrized PDEs. The model is hierarchical as it is built on a full ord…
Stability of convergence rates: Kernel interpolation on non-Lipschitz domains
Tizian Wenzel, Gabriele Santin, Bernard Haasdonk
Error estimates for kernel interpolation in Reproducing Kernel Hilbert Spaces (RKHS) usually assume quite restrictive properties on the shape of the domain, especially in the case…
Analysis of target data-dependent greedy kernel algorithms: Convergence rates for -, - and -greedy
Tizian Wenzel, Gabriele Santin, Bernard Haasdonk
Data-dependent greedy algorithms in kernel spaces are known to provide fast converging interpolants, while being extremely easy to implement and efficient to run. Despite this expe…
Analysis of Structured Deep Kernel Networks
Tizian Wenzel, Gabriele Santin, Bernard Haasdonk
In this paper, we leverage a recent deep kernel representer theorem to connect kernel based learning and (deep) neural networks in order to understand their interplay. In particula…
A full order, reduced order and machine learning model pipeline for efficient prediction of reactive flows
Pavel Gavrilenko, Bernard Haasdonk, Oleg Iliev +5
We present an integrated approach for the use of simulated data from full order discretization as well as projection-based Reduced Basis reduced order models for the training of ma…
Structured Deep Kernel Networks for Data-Driven Closure Terms of Turbulent Flows
Tizian Wenzel, Marius Kurz, Andrea Beck +2
Standard kernel methods for machine learning usually struggle when dealing with large datasets. We review a recently introduced Structured Deep Kernel Network (SDKN) approach that…
Biomechanical surrogate modelling using stabilized vectorial greedy kernel methods
Bernard Haasdonk, Tizian Wenzel, Gabriele Santin +1
Greedy kernel approximation algorithms are successful techniques for sparse and accurate data-based modelling and function approximation. Based on a recent idea of stabilization of…
A novel class of stabilized greedy kernel approximation algorithms: Convergence, stability & uniform point distribution
Tizian Wenzel, Gabriele Santin, Bernard Haasdonk
Kernel based methods provide a way to reconstruct potentially high-dimensional functions from meshfree samples, i.e., sampling points and corresponding target values. A crucial ing…