2020/07/31 by Vesa Kaarnioja, Yoshihito Kazashi, Frances Y. Kuo +2 · 3 citations
Computer Science · Decision Sciences · Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Applied mathematics #Computer science #Differential equation #Interpolation (computer graphics) #Lattice (music) #Mathematical Approximation and Integration #Mathematical analysis #Mathematical optimization #Mathematics #Ordinary differential equation #Physics #Probabilistic and Robust Engineering Design #Rate of convergence #Solver #cs.NA #math.NA
paper · pdf · doi:10.1007/s00211-021-01242-3
published in Numerische Mathematik 150(1), 33-77 (Springer Science+Business Media) · 37 pages, 5 figures
arxiv created 2021/10/13 · openalex publication_date 2021/11/30 · openalex created_date 2021/12/06 · arxiv updated 2022/01/25 · openalex updated_date 2026/08/05
Abstract This paper deals with the kernel-based approximation of a multivariate periodic function by interpolation at the points of an integration lattice—a setting that, as pointed out by Zeng et al. (Monte Carlo and Quasi-Monte Carlo Methods 2004, Springer, New York, 2006) and Zeng et al. (Constr. Approx. 30: 529–555, 2009), allows fast evaluation by fast Fourier transform, so avoiding the need for a linear solver. The main contribution of the paper is the application to the approximation problem for uncertainty quantification of elliptic partial differential equations, with the diffusion coefficient given by a random field that is periodic in the stochastic variables, in the model proposed recently by Kaarnioja et al. (SIAM J Numer Anal 58(2): 1068–1091, 2020). The paper gives a full error analysis, and full details of the construction of lattices needed to ensure a good (but inevitably not optimal) rate of convergence and an error bound independent of dimension. Numerical experiments support the theory.