Lizao Li

Douglas N. Arnold, Lizao Li

The scalar and vector Laplacians are basic operators in physics and engineering. In applications, they show up frequently perturbed by lower-order terms. The effect of such perturbations on mixed finite element methods in the scalar case is well-understood, but that in the vector case is not. In this paper, we first show that surprisingly for certain elements there is degradation of the convergence rates with certain lower-order terms even when both the solution and the data are smooth. We then give a systematic analysis of lower-order terms in mixed methods by extending the Finite Element Exterior Calculus (FEEC) framework, which contains the scalar, vector Laplacian, and many other elliptic operators as special cases. We prove that stable mixed discretization remains stable with lower-order terms for sufficiently fine discretization. Moreover, we derive sharp improved error estimates for each individual variable. In particular, this yields new results for the vector Laplacian problem which are useful in applications such as electromagnetism and acoustics modeling. Further our results imply many previous results for the scalar problem and thus unifies them all under the FEEC framework.

Lizao Li, Rob Carver, Ignacio Lopez-Gomez et al.

Uncertainty quantification is crucial to decision-making. A prominent example is probabilistic forecasting in numerical weather prediction. The dominant approach to representing uncertainty in weather forecasting is to generate an ensemble of forecasts. This is done by running many physics-based simulations under different conditions, which is a computationally costly process. We propose to amortize the computational cost by emulating these forecasts with deep generative diffusion models learned from historical data. The learned models are highly scalable with respect to high-performance computing accelerators and can sample hundreds to tens of thousands of realistic weather forecasts at low cost. When designed to emulate operational ensemble forecasts, the generated ones are similar to physics-based ensembles in important statistical properties and predictive skill. When designed to correct biases present in the operational forecasting system, the generated ensembles show improved probabilistic forecast metrics. They are more reliable and forecast probabilities of extreme weather events more accurately. While this work demonstrates the utility of the methodology by focusing on weather forecasting, the generative artificial intelligence methodology can be extended for uncertainty quantification in climate modeling, where we believe the generation of very large ensembles of climate projections will play an increasingly important role in climate risk assessment.

Jack S. Hale, Lizao Li, Chris N. Richardson et al.

Containers are an emerging technology that hold promise for improving productivity and code portability in scientific computing. We examine Linux container technology for the distribution of a non-trivial scientific computing software stack and its execution on a spectrum of platforms from laptop computers through to high performance computing (HPC) systems. We show on a workstation and a leadership-class HPC system that when deployed appropriately there are no performance penalties running scientific programs inside containers. For Python code run on large parallel computers, the run time is reduced inside a container due to faster library imports. The software distribution approach and data that we present will help developers and users decide on whether container technology is appropriate for them. We also provide guidance for the vendors of HPC systems that rely on proprietary libraries for performance on what they can do to make containers work seamlessly and without performance penalty.