Seminar in Numerical Analysis: Vesa Kaarnioja (FU Berlin)
We describe a fast method for solving elliptic PDEs with uncertain coefficients using kernel-based interpolation over a rank-1 lattice point set . By representing the input random field of the system using a model proposed by Kaarnioja, Kuo, and Sloan , in which a countable number of independent random variables enter the random field as periodic functions, it is shown that the kernel interpolant can be constructed for the PDE solution (or some quantity of interest thereof) as a function of the stochastic variables in a highly efficient manner using fast Fourier transform. The method works well even when the stochastic dimension of the problem is large, and we obtain rigorous error bounds which are independent of the stochastic dimension of the problem. We also outline some techniques that can be used to further improve the approximation error and computational complexity of the method .
 V. Kaarnioja, Y. Kazashi, F. Y. Kuo, F. Nobile, and I. H. Sloan. Fast approximation by periodic kernel-based lattice-point interpolation with application in uncertainty quantification. Numer. Math. 150:33-77, 2022.
 V. Kaarnioja, F. Y. Kuo, and I. H. Sloan. Uncertainty quantification using periodic random variables. SIAM J. Numer. Anal. 58(2):1068-1091, 2020.
 V. Kaarnioja, F. Y. Kuo, and I. H. Sloan. Lattice-based kernel approximation and serendipitous weights for parametric PDEs in very high dimensions. Preprint 2023, arXiv:2303.17755 [math.NA].
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