By Shanyin Tong
Though extreme events like hurricanes, energy grid blackouts, dam breaks, earthquakes, and pandemics are infrequent, they have severe consequences [4]. Because estimating the probability of such events can inform strategies that mitigate their effects, scientists must develop methods to study the distribution tail of these occurrences.
Figure 1. Illustration of two-dimensional random parameter space with level sets (dotted blue lines) of rate functions \(I(\theta)\). Monte Carlo samples (blue dots) typically fall outside of the set of parameters that lead to extreme events (in red). Large deviation theory (LDT) indicates that the extreme event probability is dominated by the point \(\theta^\star(z)\): the least unlikely point that leads to an event of size \(z\) or larger. Finding this minimizer \(\theta^\star(z)\) amounts to solving a constrained optimization problem.
Explicitly calculating small probabilities is generally infeasible, particularly when the probability density depends on complex dynamics and the random variables that enter these dynamical systems are high dimensional. Standard Monte Carlo methods are inefficient for the exploration of probability tails because most samples are far from the extreme event set (see Figure 1). Sampling that is guided by tailored proposal distributions can improve this analysis [1]. However, we are interested in estimation methods that do not rely on sampling and can thus be applied to scenarios wherein the tail probability estimation is merely the inner part of an optimization, control, or design problem [5].
We use tsunami hazard assessment [7] to illustrate our proposed methods. For earthquake-induced tsunamis, the instantaneous seafloor deformation displaces the water columns over large regions and generates travelling tsunami waves. Since the up-and-down displacement patterns are irregular and unpredictable, we model them as a random field that is denoted by \(\theta\). The displacement of water caused by \(\theta\) results in tsunami waves, which interact with the ocean floor and land masses as they travel. We observe an increased water level when these waves reach the shore; the parameter-to-event map \(F(\theta)\) summarizes this entire dynamic process. Tsunami hazard assessment aims to estimate the probability that the maximal tsunami water height on shore is at or above a given threshold of \(z\) meters, i.e., the probability \(P(F(\theta)\ge z)\).
Our proposed method uses ideas from large deviation theory (LDT) to connect probability estimation with optimization [2, 3, 6]. The approach first computes the most important (in this case, the least unlikely) point in the extreme event set \(\{\theta:F(\theta)\ge z\}\); i.e., the point that minimizes the rate function \(I(\theta)\). This rate function is a convex function that is fully determined by the probability distribution of \(\theta\). Under reasonable assumptions, LDT implies that the minimizer \(\theta^\star(z)\) (see Figure 1) holds crucial information to estimate the probability \(P(F(\theta)\ge z)\). Because the parameter-to-event map \(F(\theta)\) in our application involves the solution of the shallow water equations, computing this minimizer is an optimization problem that is governed by a partial differential equation (PDE) constraint.
After solving this optimization problem, we utilize local curvature information of the extreme event set boundary \(F(\theta)=z\) at \(\theta^\star\) to obtain an accurate estimate of the probability \(P(F(\theta)\ge z)\). This is a sampling-free method that one can show to be asymptotically exact for \(z\to\infty\). Figure 2 depicts our rare event estimation results, which use a one-dimensional shallow water equation model for the sake of simplicity. The LDT-based approach allows for accurate estimation of tail probabilities by solving a sequence of optimization problems with different thresholds \(z\).
Figure 2. Probability estimation for the tsunami problem. The event threshold \(z\) is plotted on the x-axis and the probability of events of size at least \(z\) is on the y-axis. The blue line depicts results that are obtained via standard Monte Carlo estimation with \(10^5\) samples, which does not allow accurate probability estimation below \(10^{-4}\), as seen from the 95 percent confidence intervals (blue dotted line). The red line depicts the estimation that results from large deviation theory (LDT) optimization combined with curvature information of \(F(\theta)=z\).
This work also motivates the formulation and analysis of new classes of PDE-constrained optimization problems. The shallow water equations—a nonlinear hyperbolic conservation law—govern the tsunami LDT optimization problem. Researchers do not often study similar problems in PDE optimization, which are challenging for multiple reasons — like the possible occurrence of shocks in the solution and difficulties that arise in the computation of adjoint-based gradients. Additionally, the parameter-to-event map \(F\) measures the maximal water height over time, making this situation a time-optimal control problem [6].
Shanyin Tong presented this research during a minisymposium presentation at the 2021 SIAM Conference on Computational Science and Engineering, which took place virtually in March.
References
[1] Bucklew, J. (2004). Introduction to Rare Event Simulation. New York, NY: Springer-Verlag.
[2] Dematteis, G., Grafke, T., & Vanden-Eijnden, E. (2018). Rogue waves and large deviations in deep sea. PNAS, 115(5), 855-860.
[3] Dematteis, G., Grafke, T., & Vanden-Eijnden, E. (2019). Extreme event quantification in dynamical systems with random components. SIAM/ASA J. Uncer. Quant., 7(3), 1029-1059.
[4] Farazmand, M., & Sapsis, T.P. (2018, January 29). Physics-based probing and prediction of extreme events. SIAM News, 51(1), p. 1.
[5] Tong, S., Subramanyam, A., & Rao, V. (2020). Optimization under rare chance constraints. Preprint, arXiv:2011.06052.
[6] Tong, S., Vanden-Eijnden, E., & Stadler, G. (2020). Extreme event probability estimation using PDE-constrained optimization and large deviation theory, with application to tsunamis. Preprint, arXiv:2007.13930.
[7] Williamson, A., Rim, D., Adams, L.M., LeVeque, R.J., Melgar, D., & González, F.I. (2020). A source clustering approach for efficient inundation modeling and regional scale probabilistic tsunamic hazard assessment. Front. Earth Sci.
Shanyin Tong is a Ph.D. candidate at New York University's Courant Institute of Mathematical Sciences, where she is advised by Georg Stadler and Eric Vanden-Eijnden. Her research focuses on applied and computational mathematics — particularly rare events, uncertainty quantification, PDE-constrained optimization, stochastic optimization, and inverse problems.