A Rocky Opportunity

By Michael Pilosov

Kyle Mandli’s presentation at a minisymposium at the 2018 SIAM Annual Meeting began differently than most. Shaky footage from an antiquated cellphone filled the first slide and was captioned with an ominous harbinger for the rest of the talk: “Can we model this?”

On screen the audience is witness to a rocky river basin bordered by lush jungle. In the distance a faint mist is visible, and the footage slowly zooms in on it. Soon the pixels resolve and it becomes clear that something bad is about to occur.

Water is rushing towards the screen, violently cresting and crashing into the boulders in the shallow creek. There is an unimaginably complex interplay of geometries and forces at work here. Occurrences of this nature are becoming more common and pose a direct threat to coastal communities. Mandli focused on the role mathematics needs to play in addressing such immediate risks

“Can we model this?” The question sits heavy in the room as the footage concludes. Turbulent flow is notoriously difficult to model, a formidable challenge for any computational scientist. Coastline development creates turbulence. Predictions of a storm’s path come with every hurricane forecast that adorns evening news channels. However, storm surge is the factor that is most responsible for mortality.

Storm surge that overwhelms existing protection strategies (such as levees and dams), is a particularly devastating outcome. Many communities historically sprang up around the waters these structures were built to contain. The engineering marvels of the last century should not inspire confidence in our ability to prevent tragedies in a warmer, less predictable world.

Owing to rising average global temperatures, greater levels of uncertainty pervade the predictions that existing models are capable of producing. This is partially because they are trained on data from a cooler ocean (and planet). The implications of extrapolation for long-term climate predictions are unfortunately grim. Such models are necessary to test hypothetical strategies and predict the efficacy of our response to climate change.

“Will dangerous storms become more frequent? Will they become more powerful? Can we forecast events? How do we protect ourselves?” These are the questions that we as audience members are implored to consider. This talk makes the case that we cannot delay progress with inaction by presenting numerous examples. The consequences of failing to adequately apply our collective knowledge towards these questions are too dire.

This justifies the necessity of applying uncertainty quantification to geological problems. The second half of the talk concentrated on demonstrating some examples of present capabilities in development. Mandli discussed the challenges that arise from the computational scales involved and the interweaving dependencies among models. Limitations on resources and time can also often constrict our abilities to make good predictions. Scientists must be creative in their approaches.

One approach in the relevant literature is the use of surrogate models. In the context of models, surrogates serve as approximations of some complex underlying physics. They may be less globally accurate but still capable of producing acceptable estimates of quantities of interest for policymakers.

For example, it may be unnecessary to resolve uncertainties in bathymetry (depth) everywhere to predict surge where it matters. Protective measures are near the coast, so the surface friction coefficients in the shallow waters are more important factors for accurate predictions. Computational scientists can use approximations from surrogate models to get rough estimates of certain quantities so that effort can be expended elsewhere.

Surrogate models are advantageous because they are computationally cheaper, allowing researchers to run more “what-if” scenarios in the same amount of time in order to gain useful information. This knowledge can then be incorporated into high-fidelity simulations in order to improve their efficiency. Ideally, this would also improve their predictive capabilities, but in such early days it is hard to make general statements in that regard. Tradeoffs like these are actively being studied and more tools and frameworks for assimilating data from different sources and resolutions are being developed.

The difficulty of quantifying uncertainty in complex models must be met with an urgency that reflects the consequences of inaction. Fortunately, the pressing need for better approaches presents an opportunity for impactful research in a very novel field. Uncertainty quantification (UQ), Mandli argues, will be a quintessential tool for analyzing predictions concerned with important questions.

Any early-career mathematician with a computational interest should consider pursuing UQ. There is a necessity for massive improvements and creative approaches to problem solving. The use and awareness of tools developed in the UQ field are increasing each day.