2021/02/04 by Jonas Kiessling, Emanuel Ström, Raúl Tempone
Mathematics · Computer Science · #math.NA #cs.NA #stat.AP #stat.ML #msc:65D15 #msc:65C05 #msc:62P12
paper · pdf · doi:10.1098/rspa.2021.0236
arxiv created 2021/02/04 · arxiv updated 2022/01/19
We investigate the use of spatial interpolation methods for reconstructing the horizontal near-surface wind field given a sparse set of measurements. In particular, random Fourier features is compared to a set of benchmark methods including Kriging and Inverse distance weighting. Random Fourier features is a linear model β(\pmb x) = ∑k=1K βk eiωk \pmb x approximating the velocity field, with frequencies ωk randomly sampled and amplitudes βk trained to minimize a loss function. We include a physically motivated divergence penalty term |∇ ⋅ β(\pmb x)|2, as well as a penalty on the Sobolev norm. We derive a bound on the generalization error and derive a sampling density that minimizes the bound. Following (arXiv:2007.10683 [math.NA]), we devise an adaptive Metropolis-Hastings algorithm for sampling the frequencies of the optimal distribution. In our experiments, our random Fourier features model outperforms the benchmark models.