Greedy Kernel-Based Surrogates for Approximating Parametric PDEs

Abstract

Multi-query simulation settings require efficient surrogates for the underlying processes. We consider a family 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 named the PDE-β-greedy procedure, where the parameter β ≥ 0 is used in a greedy selection criterion and controls 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 through the use of position-parameter product kernels [3]. In the context of surrogate modeling, the resulting approach can be interpreted as an a priori model reduction method, as no solution snapshots need to be precomputed. Numerical results demonstrate the efficiency of the approximation procedure for problems that pose challenges for other parametric MOR methods: non-affine geometric 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 a professor of “Numerical Mathematics” at the Institute of Applied Analysis and Numerical Simulation at the University of Stuttgart.

Prior to this appointment, he studied physics, mathematics, and computer science at the University of Freiburg and earned his PhD in machine learning in 2005. He joined the Institute of Applied Mathematics at the University of Freiburg as a postdoctoral researcher, spent several months at the Massachusetts Institute of Technology, and moved to the University of Münster in 2007. In 2009, he joined the “Cluster in Simulation Technology SimTech” at the University of Stuttgart as an assistant professor until assuming his current professorship in 2014. From 2014 to 2018, B. Haasdonk served as the German representative on 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 a Fellow of the SC SimTech.

His research group focuses primarily on surrogate modeling using model reduction and machine learning methods. In particular, reduced basis methods for parameterized problems and kernel methods for data-driven modeling are key areas of focus. Among other fields, applications include optimization, control, inverse problems, transport problems, and multiphysics systems.