On Symmetric Kernel Collocation for Nonlinear PDEs

Abstract

This paper considers kernel-based approximation methods for nonlinear partial differential equations. To this end, the problem is formulated as an optimal-recovery generalized interpolation problem, that is, as an optimization problem in an RKHS with nonlinear functional constraints. This formulation provides the basis for a convergence analysis carried out directly in the RKHS and extends existing results by relaxing the uniqueness assumption on the PDE solution. In the nonunique case, the limiting object is characterized as a minimum-norm solution. Furthermore, a residual-greedy strategy for adaptive collocation point selection is proposed, and convergence of the resulting sequence of generalized interpolants is established. Numerical experiments for a stationary nonlinear heat equation illustrate the method and indicate that residual-greedy point selection can lead to markedly smaller PDE residuals than point sets selected according to fill-distance criteria.

Disclosure

“on) under Project No. 539436611 and Project No. 540080351. The authors used ChatGPT by OpenAI during the preparation of this paper for brainstorming, language revision, improving exposition, and support with numerical implemen- tation. All AI-assisted output was critically reviewed and revised by the authors. The mathematical arguments, proofs, numerical results, and scientific conclusions presented in this paper remain the authors’ responsibility. References [1] N. Aronszajn. “The”

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2026-08-24T00:34:12.480Z

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