Repository: Freie Universit├Ąt Berlin, Math Department

Lattice-based kernel approximation and serendipitous weights for parametric PDEs in very high dimensions

Kaarnioja, Vesa and Kuo, Frances Y. and Sloan, Ian H. (2024) Lattice-based kernel approximation and serendipitous weights for parametric PDEs in very high dimensions. In: Monte Carlo and Quasi-Monte Carlo Methods 2022. Springer Verlag. (Submitted)

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Abstract

We describe a fast method for solving elliptic partial differential equations (PDEs) with uncertain coefficients using kernel interpolation at a lattice point set. By representing the input random field of the system using the model proposed by Kaarnioja, Kuo, and Sloan (SIAM J.~Numer.~Anal.~2020), in which a countable number of independent random variables enter the random field as periodic functions, it was shown by Kaarnioja, Kazashi, Kuo, Nobile, and Sloan (Numer.~Math.~2022) that the lattice-based kernel interpolant can be constructed for the PDE solution as a function of the stochastic variables in a highly efficient manner using fast Fourier transform (FFT). In this work, we discuss the connection between our model and the popular ``affine and uniform model'' studied widely in the literature of uncertainty quantification for PDEs with uncertain coefficients. We also propose a new class of weights entering the construction of the kernel interpolant -- \emph{serendipitous weights} -- which dramatically improve the computational performance of the kernel interpolant for PDE problems with uncertain coefficients, and allow us to tackle function approximation problems up to very high dimensionalities. Numerical experiments are presented to showcase the performance of the serendipitous weights.

Item Type:Book Section
Subjects:Mathematical and Computer Sciences > Mathematics > Applied Mathematics
Divisions:Department of Mathematics and Computer Science > Institute of Mathematics > Deterministic and Stochastic PDEs Group
ID Code:3104
Deposited By: Ulrike Eickers
Deposited On:19 Feb 2024 15:33
Last Modified:19 Feb 2024 15:33

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