Advances in Uncertainty Quantification for the Natural Sciences

Uptake of carbon by terrestrial ecosystems is expected to be a major role player in future climate projections, and therefore it is integral that our forecasts are informed and quantify as many sources of uncertainty as possible. To incorporate these sources of uncertainty, a typical approach is to use process-based ecosystem models as the latent process in a statistical state space model. The inference for these models is challenging: ecosystem process models are over-parameterized and have little data available to constrain parameters and estimate latent states. Furthermore, for multivariate dynamical systems with few observations, particle filter approximations to the marginal log-likelihood contain Monte Carlo variance that cannot be reduced by increasing the number of particles. As a result, classical techniques such as particle Markov Chain Monte Carlo and iterated filtering become infeasible. In this talk, we discuss frameworks for biophysically realistic parameterizations of state space models, parameter estimation, model selection, and uncertainty quantification in the presence of irreducible Monte Carlo error through the applied lens of carbon cycle forecasting. 

Gaussian process (GP) models are commonly used in the field of surrogate modeling due to their ability to provide a predictive distribution, interpolate in deterministic settings, and provide a foundation for uncertainty. However, in cases where the number of observations in the training set is large, the ability to model using a GP becomes computationally intractable. As such, fast approximations of such models have become prominent throughout GP literature. One of the leading approaches is to construct a GP using a local neighbourhood for each testing point, which has come to be known as a local approximate GP). While this approach performs well in cases where the number of inputs is small, it may falter when this is not the case. In this work, we propose to use local Gaussian processes with structured mean functions. The proposed approach is used to emulate a loss of coolant model for a CANDU nuclear reactor. Issues of internal extrapolation in sparse sampling case naturally arises.  

Computationally expensive posterior distributions arise in a myriad of modern scientific settings. One such setting is Bayesian inverse problems with computer simulators, where each evaluation of the (unnormalized) posterior distribution $p$ requires a forward run of the expensive simulator that can take hundreds of CPU hours. This evaluation cost poses a challenge for many existing posterior samplers, where each sample taken requires at least one evaluation of the posterior $p$. Given a computational budget, a "cost-efficient" sampler is desired for effective posterior exploration with limited posterior evaluations. We thus propose a new sampling method called cost-Efficient Stein Points (ESPs). ESPs extend the Stein points in Chen et al. (2018; ICML), which employs the sequential minimization of the kernel Stein discrepancy to generate a sequence of posterior samples $x_1, x_2, \cdots$. To reduce posterior evaluations for optimizing a sample $x_i$, we leverage a Gaussian process surrogate on the kernel discrepancy that guides the selection of evaluation points. This "Bayesian optimization" strategy is then used sequentially to optimize $x_1, x_2, \cdots$ in a cost-efficient manner, where all previous posterior evaluations can be re-used for optimizing the current sample. We prove that, under mild conditions, ESPs converge in distribution to the desired posterior distribution. Finally, we demonstrate the cost-efficiency of ESPs over the state-of-the-art in a suite of numerical experiments and an application for Bayesian calibration of a biological oscillator process. 

Forest fires generate burning pieces of vegetation, called firebrands, and launch them into the air through columns of gas produced by the fire. After lofting into the air, firebrands are taken away by the ambient wind. At the time of landing, these burning firebrands can start separate fires away from the original fire. The secondary fires are called spot fires and the entire process is referred to as spotting. Quantifying the probability of spot fires at a large distance remains a major challenge in forecasting and managing wildfires. In this talk, we first show that atmospheric waves can significantly increase the spotting distance. Then, using the concept of inertial manifolds and large deviation theory, we introduce a mathematical method that quantifies the probability distribution of such rare extreme spot fires.

This work was supported by the National Science Foundation, the Algorithms for Threat Detection (ATD) program, through the award DMS-2220548.