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 $\beta(\pmb x) = \sum_{k=1}^K \beta_k e^{i\omega_k \pmb x}$ approximating the velocity field, with frequencies $\omega_k$ randomly sampled and amplitudes $\beta_k$ trained to minimize a loss function. We include a physically motivated divergence penalty term $|\nabla \cdot \beta(\pmb x)|^2$, 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.
翻译:我们调查了利用空间内插法重建水平近表面风场的方法,因为测量量很少。 特别是, 随机的Fourier特征与一套基准方法比较, 包括克里格和反距离加权。 随机的Fourier特征是一个线性模型$\beta( pmbx) =\ sum ⁇ k=1 ⁇ K\\k\ k\ ⁇ k e ⁇ i\ i\ omega_k\ pmb x}, 以随机抽样频率$\omega_k$k k$k 来重建速度场。 随机抽样的频率为$\beta_ k, 并训练振幅 $\ beta_ k_ kt 来尽量减少损失功能。 我们使用一个有物理动机的偏差罚款术语 $ ⁇ nabla\ cdot\ cdotat\ beta( pmbxx) $2$, 是对Sobolev 规范的罚款。 我们从一般误差和取样密度中得出一个最小的密度。 随后 (ariv: 2007. 10683 [math.NA], 我们设计一个适应的Metopier 模型- hasting- hasting slogting sloging sloging exmlations exmlations expealbislations slations slations slations ex