Speaker
Description
We consider general linear PDE boundary value problems in the strong form on arbitrary bounded Lipschitz domains. For such problems, we recently presented a scale of meshless greedy kernel-based collocation techniques [1]. The approximation spaces are incrementally constructed by carefully collecting Riesz-representers of (derivative operator) point-evaluation functionals. The approximants are obtained by generalized interpolation [2, Chap. 16]. The scale of methods naturally generalizes existing approaches of PDE approximation [3] as well as function approximation techniques [4,5]. Assuming well-posedness and a stability estimate of the given PDE problem, we can rigorously prove the convergence rates of the resulting approximation schemes [1]. Interestingly, those rates show that it is possible to break the curse of dimensionality and potentially reach high input dimensions. For cases with domains in small input space dimensions the schemes allow experimental comparison with, e.g., standard finite element methods. The strength of the procedure, however, is the ease of treating high-dimensional input space dimensions due to the mesh independence and omitting spatial integrals. When considering additional parametric inputs, the overall procedure can be interpreted as an apriori surrogate modelling approach. Herewith, we can especially address (non-affine) parametric geometries, moving source terms, or high-dimensional domains, which pose obstacles to standard model reduction techniques. We present numerical experiments demonstrating these aspects, especially exponential convergence rates for problems with smooth solutions and smooth kernels.
References:
[1] S. Dutta, M. W. Farthing, E. Perracchione, G. Savant, and M. Putti. A greedy non-intrusive reduced order model for shallow water equations. Journal of Computational Physics, 439:110378, 2021.
[2] R. Schaback. A greedy method for solving classes of PDE problems, 2019. arXiv 1903.11536.
[3] H. Wendland. Scattered Data Approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, 2005.
[4] T. Wenzel, G. Santin, and B. Haasdonk. Analysis of target data-dependent greedy kernel algorithms: Convergence rates for f-, f/P- and f.P-greedy. Constructive Approximation, 57(1):45–74,Feb 2023.
[5] Wenzel, T., Winkle, D., Santin, G. and Haasdonk, B.: Adaptive meshfree approximation for linear elliptic partial differential equations with PDE-greedy kernel methods. preprint arXiv:2207.13971v2, 2024.
