Extreme event probability estimation using PDE-constrained optimization and large deviation theory, with application to tsunamis

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