Shravan Hanasoge

Shravan Hanasoge

Differential equations may possess coefficients that vary on a spectrum of scales. Because coefficients are typically multiplicative in real space, they turn into convolution operators in spectral space, mixing all wavenumbers. However, in many applications, only the largest scales of the solution are of interest and so the question turns to whether it is possible to build effective coarse-scale models of the coefficients in such a manner that the large scales of the solution are left intact. Here we apply the method of numerical homogenization to deterministic linear equations to generate sub-grid-scale models of coefficients at desired frequency cutoffs. We use the Fourier basis to project, filter and compute correctors for the coefficients. The method is tested in 1D and 2D scenarios and found to reproduce the coarse scales of the solution to varying degrees of accuracy depending on the cutoff. We relate this method to mode-elimination Renormalization Group (RG) and discuss the connection between accuracy and the cutoff wavenumber. The tradeoff is governed by a form of the uncertainty principle for convolutions, which states that as the convolution operator is squeezed in the spectral domain, it broadens in real space. As a consequence, basis sparsity is a high virtue and the choice of the basis can be critical.

Rasha Alshehhi, Chris S. Hanson, Laurent Gizon et al.

The inversion of ring fit parameters to obtain subsurface flow maps in ring-diagram analysis for 8 years of SDO observations is computationally expensive, requiring ~3200 CPU hours. In this paper we apply machine learning techniques to the inversion in order to speed up calculations. Specifically, we train a predictor for subsurface flows using the mode fit parameters and the previous inversion results, to replace future inversion requirements. We utilize Artificial Neural Networks as a supervised learning method for predicting the flows in 15 degree ring tiles. To demonstrate that the machine learning results still contain the subtle signatures key to local helioseismic studies, we use the machine learning results to study the recently discovered solar equatorial Rossby waves. The Artificial Neural Network is computationally efficient, able to make future flow predictions of an entire Carrington rotation in a matter of seconds, which is much faster than the current ~31 CPU hours. Initial training of the networks requires ~3 CPU hours. The trained Artificial Neural Network can achieve a root mean-square error equal to approximately half that reported for the velocity inversions, demonstrating the accuracy of the machine learning (and perhaps the overestimation of the original errors from the ring-diagram pipeline). We find the signature of equatorial Rossby waves in the machine learning flows covering six years of data, demonstrating that small-amplitude signals are maintained. The recovery of Rossby waves in the machine learning flow maps can be achieved with only one Carrington rotation (27.275 days) of training data. We have shown that machine learning can be applied to, and perform more efficiently than the current ring-diagram inversion. The computation burden of the machine learning includes 3 CPU hours for initial training, then around 0.0001 CPU hours for future predictions.

Shravan Hanasoge, Umang Agarwal, Kunj Tandon et al.

Determining the pressure differential required to achieve a desired flow rate in a porous medium requires solving Darcy's law, a Laplace-like equation, with a spatially varying tensor permeability. In various scenarios, the permeability coefficient is sampled at high spatial resolution, which makes solving Darcy's equation numerically prohibitively expensive. As a consequence, much effort has gone into creating upscaled or low-resolution effective models of the coefficient while ensuring that the estimated flow rate is well reproduced, bringing to fore the classic tradeoff between computational cost and numerical accuracy. Here we perform a statistical study to characterize the relative success of upscaling methods on a large sample of permeability coefficients that are above the percolation threshold. We introduce a new technique based on Mode-Elimination Renormalization-Group theory (MG) to build coarse-scale permeability coefficients. Comparing the results with coefficients upscaled using other methods, we find that MG is consistently more accurate, particularly so due to its ability to address the tensorial nature of the coefficients. MG places a low computational demand, in the manner that we have implemented it, and accurate flow-rate estimates are obtained when using MG-upscaled permeabilities that approach or are beyond the percolation threshold.