TY - JOUR

T1 - Extreme Event Probability Estimation Using Pde-Constrained Optimization And Large Deviation Theory, With Application To Tsunamis

AU - Tong, Shanyin

AU - Vanden-Eijnden, Eric

AU - Stadler, Georg

N1 - Funding Information:
Tong and Stadler were partially supported by the US National Science Foundation (NSF) through grants DMS #1723211 and EAR #1646337, and by the SciDAC program funded by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research, and Biological and Environmental Research Programs. Vanden-Eijnden was supported in part by the NSF Materials Research Science and Engineering Center Program grant DMR #1420073, by NSF grant DMS #152276, by the Simons Collaboration on Wave Turbulence, grant #617006, and by ONR grant #N4551-NV-ONR. MSC2020: 35Q93, 60F10, 60H35, 65K10, 76B15. Keywords: extreme events, probability estimation, PDE-constrained optimization, large deviation theory, tsunamis.
Publisher Copyright:
© 2021, Communications in Applied Mathematics and Computational Science. All Rights Reserved.

PY - 2021

Y1 - 2021

N2 - We propose and compare methods for the analysis of extreme events in complex systems governed by PDEs that involve random parameters, in situations where we are interested in quantifying the probability that a scalar function of the system’s solution is above a threshold. If the threshold is large, this probability is small and its accurate estimation is challenging. To tackle this difficulty, we blend theoretical results from large deviation theory (LDT) with numerical tools from PDE-constrained optimization. Our methods first compute parameters that minimize the LDT-rate function over the set of parameters leading to extreme events, using adjoint methods to compute the gradient of this rate function. The minimizers give information about the mechanism of the extreme events as well as estimates of their probability. We then propose a series of methods to refine these estimates, either via importance sampling or geometric approximation of the extreme event sets. Results are formulated for general parameter distributions and detailed expressions are provided for Gaussian distributions. We give theoretical and numerical arguments showing that the performance of our methods is insensitive to the extremeness of the events we are interested in. We illustrate the application of our approach to quantify the probability of extreme tsunami events on shore. Tsunamis are typically caused by a sudden, unpredictable change of the ocean floor elevation during an earthquake. We model this change as a random process, which takes into account the underlying physics. We use the one-dimensional shallow water equation to model tsunamis numerically. In the context of this example, we present a comparison of our methods for extreme event probability estimation, and find which type of ocean floor elevation change leads to the largest tsunamis on shore

AB - We propose and compare methods for the analysis of extreme events in complex systems governed by PDEs that involve random parameters, in situations where we are interested in quantifying the probability that a scalar function of the system’s solution is above a threshold. If the threshold is large, this probability is small and its accurate estimation is challenging. To tackle this difficulty, we blend theoretical results from large deviation theory (LDT) with numerical tools from PDE-constrained optimization. Our methods first compute parameters that minimize the LDT-rate function over the set of parameters leading to extreme events, using adjoint methods to compute the gradient of this rate function. The minimizers give information about the mechanism of the extreme events as well as estimates of their probability. We then propose a series of methods to refine these estimates, either via importance sampling or geometric approximation of the extreme event sets. Results are formulated for general parameter distributions and detailed expressions are provided for Gaussian distributions. We give theoretical and numerical arguments showing that the performance of our methods is insensitive to the extremeness of the events we are interested in. We illustrate the application of our approach to quantify the probability of extreme tsunami events on shore. Tsunamis are typically caused by a sudden, unpredictable change of the ocean floor elevation during an earthquake. We model this change as a random process, which takes into account the underlying physics. We use the one-dimensional shallow water equation to model tsunamis numerically. In the context of this example, we present a comparison of our methods for extreme event probability estimation, and find which type of ocean floor elevation change leads to the largest tsunamis on shore

KW - PDE-constrained optimization

KW - extreme events

KW - large deviation theory

KW - probability estimation

KW - tsunamis

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U2 - 10.2140/camcos.2021.16.181

DO - 10.2140/camcos.2021.16.181

M3 - Article

AN - SCOPUS:85119927665

SN - 1559-3940

VL - 16

SP - 181

EP - 225

JO - Communications in Applied Mathematics and Computational Science

JF - Communications in Applied Mathematics and Computational Science

IS - 2

ER -