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jck: Did you play up the global warming part to increase your odds of funding?

Our grant application did not mention climate change, global warming, or any related concept. Our entire emphasis is on tracking the movement of the fire line, not on any broader scientific issues.

The subject of firestorms and global warming came up in the context of what we cannot do with current woodland fire models. I was trying to give the reporter an idea of the limits of our research program. Despite the interest of the reporter, firestorms and the influence of global warming are completely beyond our scope, and we did not propose to address these subjects. They were never even mentioned, and properly so.

It may help to remember that this is not a grant in atmospheric physics, nor is it climate science. We are merely mathematicians, seeking funding from the math division of NSF. Perhaps it may help to present our project summary here. The core idea of the proposal is in the second paragraph below.

Intellectual merit. This project will develop new methods for data-driven scientific computing. These methods will build on the principles of statistical data assimilation, which adjusts the state of a model in response to new data, while the model is running. Related methods are used in artificial intelligence, such as robotic vision, and many other areas. New methods for the estimation of the model state covariance, accelerated by the use of wavelets, will allow the treatment of large problems with many millions of variables on a laptop instead of a supercomputer. The new methodologies exploit the synergy of theory of random fields in spatial statistics with the power of Bayesian modeling and spectral analysis. The basic approach is that of ensembles, which are collections of independent simulations, which approximate the probability distribution of the model state. Since the targeted application is numerical solution of partial differential equations, the ensemble members are very large — easily millions of degrees of freedoms — which makes large ensembles impractical. Unlike sequential Monte Carlo methods, which require very large ensembles with thousands of members, and ensemble Kalman filters, which require tens or hundreds of members, the proposed wavelet-based methods could reduce the required ensemble size to just a handful of members.

A common feature of ensemble methods is that their accuracy increases with the ensemble size, and the investigators have already proved an asymptotic convergence with the ensemble size to the correct probability distribution in the finite dimensional case. Theoretical analysis of the ensemble methods in this project centers on the issue of stochastic convergence in the case of both high dimension of the system and large number of ensemble members. The general approach will be to prove the convergence in the large ensemble limit in infinitely dimensional spaces first, then consider the high-dimensional case as an approximation of the infinitely dimensional case, much as in standard error analysis of the numerical solution of partial differential equations, except that the convergence is stochastic spaces. Connections between probability measures on Sobolev spaces, random fields in spatial statistics, stochastic expansions will be exploited to effectively reduce the dimensionality of the system.

The application area of the project is data assimilation for coupled atmosphere-wildland fire models. While Bayesian data assimilation for the fire spread itself is amenable to variants of classical Monte-Carlo approaches, the high dimension of a coupled weather-fire model makes such methods infeasible. The investigators have already developed methods for position correction, which transform the state probability distribution closer to Gaussian in problems with coherent features, such as firelines, thus making Kalman filer type methods feasible. An additional challenge is that the state of the atmosphere must be consistent with the state and the intensity of the fire, and the influence of the fire builds up over time. Rather than modifying the state directly, this project will seek to modify the state of the fire model and then catch up with the evolution of the atmosphere, thus avoiding incompatibilities of the coupled model state.

Broader impact. This project will have impact on the state of the art of modeling wildland fires. Wildfires present a serious danger and cause significant damage every year, and managing wildland fires is a challenge of great importance to the society. The project will involve undergraduate and graduate students in seminars and classes, and K-12 students at STEM outreach events. The project will support a PhD student and a postdoc. The investigators will reach out to existing programs for successful participation of underrepresented groups in STEM-related disciplines, and their alumni will be strongly considered for involvement in this project.

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