Large-eddy simulation (LES) is used to model turbulent winds in a nominally neutral atmospheric boundary layer at varying mesh resolutions. The boundary layer is driven by wind shear with zero surface heat flux and is capped by a stable inversion. Because of entrainment the boundary layer is in a weakly stably stratified regime. The simulations use meshes varying from 1282× 64 to 10242× 512 grid points in a fixed computational domain of size (2560, 2560, 896) m. The subgrid-scale (SGS) parameterizations used in the LES vary with the mesh spacing. Low-order statistics, spectra, and structure functions are compared on the different meshes and are used to assess grid convergence in the simulations. As expected, grid convergence is primarily achieved in the middle of the boundary layer where there is scale separation between the energy-containing and dissipative eddies. Near the surface second-order statistics do not converge on the meshes studied. The analysis also highlights differences between one-dimensional and two-dimensional velocity spectra; differences are attributed to sampling errors associated with aligning the horizontal coordinates with the vertically veering mean wind direction. Higher-order structure functions reveal non-Gaussian statistics on all scales, but are highly dependent on the mesh resolution. A generalized logarithmic law and a k−1spectral scaling regime are identified with mesh-dependent parameters in agreement with previously published results.
We study the scaling limits of stochastic gradient descent (SGD) with constant step‐size in the high‐dimensional regime. We prove limit theorems for the trajectories of summary statistics (i.e., finite‐dimensional functions) of SGD as the dimension goes to infinity. Our approach allows one to choose the summary statistics that are tracked, the initialization, and the step‐size. It yields both ballistic (ODE) and diffusive (SDE) limits, with the limit depending dramatically on the former choices. We show a critical scaling regime for the step‐size, below which the effective ballistic dynamics matches gradient flow for the population loss, but at which, a new correction term appears which changes the phase diagram. About the fixed points of this effective dynamics, the corresponding diffusive limits can be quite complex and even degenerate. We demonstrate our approach on popular examples including estimation for spiked matrix and tensor models and classification via two‐layer networks for binary and XOR‐type Gaussian mixture models. These examples exhibit surprising phenomena including multimodal timescales to convergence as well as convergence to sub‐optimal solutions with probability bounded away from zero from random (e.g., Gaussian) initializations. At the same time, we demonstrate the benefit of overparametrization by showing that the latter probability goes to zero as the second layer width grows.
more » « less- NSF-PAR ID:
- 10486935
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Communications on Pure and Applied Mathematics
- Volume:
- 77
- Issue:
- 3
- ISSN:
- 0010-3640
- Format(s):
- Medium: X Size: p. 2030-2080
- Size(s):
- ["p. 2030-2080"]
- Sponsoring Org:
- National Science Foundation
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