Year: 2024
Type: article
Abstract: . This work is concerned with the kernel-based approximation of a complex-valued function from data, where the response function of a partial differential equation in the frequency domain is of particular interest. In this setting, kernel methods are employed more and more frequently; however, standard kernels do not perform well. Moreover, the role and mathematical implications of the underlying pair of kernels, which arise naturally in the complex-valued case, remain to be addressed. We introduce new reproducing kernel Hilbert spaces of complex-valued functions and formulate the problem of complex-valued interpolation with a kernel pair as minimum-norm interpolation in these spaces. Moreover, we combine the interpolant with a low-order rational function, where the order is adaptively selected based on a new model selection criterion. Numerical results on examples from different fields, including electromagnetics and acoustics examples, illustrate the performance of the method in comparison to available rational approximation methods. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://zenodo.org/records/12601365 and in the supplementary materials ( M158890_SuppMat.pdf [401KB]). [Formula: see text]
DOI: 10.1137/23m1588901
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G4403445835, G4547377733
OpenAlex ID: W4405005972