Greedy Kernel-based Surrogates for Approximating Parametric PDEs

Résumé

Multi-query simulation settings require efficient surrogates for the underlying processes. We consider a scale of kernel-based greedy schemes for approximating the solutions of PDEs, exemplified by elliptic boundary value problems [1]. The procedure is based on an optimal recovery/generalized interpolation framework in reproducing kernel Hilbert spaces and was coined PDE-β-greedy procedure, where the parameter β ≥ 0 is used in a greedy selection criterion and steers the degree of function adaptivity. Algebraic convergence rates have been obtained for Sobolev-space kernels and solutions of finite smoothness [1] and have recently been refined [4]. Exponential convergence rates can be proven for the case of an infinitely smooth kernel and solutions [2].

The scheme can be extended to parametric PDEs by the use of position-parameter product kernels [3]. In the surrogate modelling context, the resulting approach can be interpreted as an a priori model reduction approach, as no solution snapshots need to be precomputed. Numerical results show the efficiency of the approximation procedure for problems which occur as challenges for other parametric MOR procedures: non-affine geometry parametrizations, moving sources or high-dimensional domains.

References

[1] Wenzel, T., Winkle, D., Santin, G. and Haasdonk, B.: Adaptive meshfree approximation for linear elliptic partial differential equations with PDE-greedy kernel methods. BIT Numerical Mathematics, 65:1, 2025. https://doi.org/10.1007/s10543-025-01053-0.

[2] Vogel, M.-P.: Target dependent greedy sampling for Gaussian kernel PDE collocation, B.Sc. Thesis, University of Stuttgart, 2024. https://doi.org/10.18419/opus-15665

[3] Haasdonk, B., Wenzel, T., Santin, G.: Kernel-based Greedy Approximation of Parametric Elliptic Boundary Value Problems. ACOM, 2026. Accepted. Preprint https://arxiv.org/abs/2507.06731, 2025.

[4] Haasdonk, B., Santin, G., Wenzel, T., Winkle, D.: Refined rates of convergence for target-data dependent greedy generalized interpolation with Sobolev kernels, Applied Mathematics Letters, 2026. Accepted. Preprint arXiv:2601.20407

Bernard Haasdonk

Bernard Haasdonk is professor for "Numerical Mathematics" at the Institute of Applied Analysis and Numerical Simulation of the University of Stuttgart. 

Prior to this assignment he has studied Physics, Mathematics and Computer Science at the University of Freiburg and obtained his PhD in Machine Learning in 2005. He joined the Applied Mathematics Institute at the University of Freiburg as a Postdoc, spent some months at the Massachusetts Institute of Technology and moved to the University of Muenster in 2007. In 2009, he has joined the “Cluster in Simulation Technology SimTech" at the University of Stuttgart as Junior professor until taking up his current professorship in 2014. From 2014-2018 B. Haasdonk served as a german representative in the Management Committee of the "European Model Reduction Network" funded by the European Union. He is a member of the Cluster of Excellence "Data-integrated Simulation Sciences" and Fellow of the SC SimTech.

The research of his group is mainly devoted to surrogate modelling by model reduction and machine-learning methods. Especially reduced basis methods for parametrized problems and kernel methods for data-based modelling are methodological focus areas. Among others, some fields of application comprise optimization, control, inverse problems, transport problems, multiphysics systems, etc.