  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 when
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.
