Uncertainty Quantification Problems in Tsunami Modeling and Reduced Order Models for Hyperbolic Partial Differential Equations
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In this thesis, we consider an uncertainty quantification (UQ) problem that arises from tsunami modeling, namely the probabilistic tsunami hazard assessment (PTHA) problem. The goal of PTHA is to compute the probability of inundation at coastal communities, and the uncertainty originates from the unknown slip distribution of potential tsunamigenic earthquakes. First, we show that the Karhunen-Lo`eve (K-L) expansion can be used to gener- ate a wide range of random earthquake scenarios that represent this uncertainty well. Then we propose a multi-resolution approach to estimate the inundation: since it is computation- ally expensive to accurately estimate the inundation resulting from each scenario by using only fine-grid runs, many cheap coarse-grid runs are used instead to bulid an approximation. For physical models that involve hyperbolic partial differential equations (PDEs), dimen- sionality reduction techniques such as the K-L expansion or multi-resolution approaches face limitations due to the fact that snapshot matrices built from solutions often exhibit slow de- cay in singular values, whereas fast decay is crucial for the success of many projection-based model reduction methods. To overcome this problem, we build on previous work in symme- try reduction [Rowley and Marsden, Physica D (2000), pp. 1-19] and propose an iterativealgorithm we call transport reversal that decomposes the snapshot matrix into multiple shift- ing profiles, each with a corresponding speed in 1D. Its applicability to typical hyperbolic problems is demonstrated through numerical examples, and other natural extensions that modify the shift operator are considered. Transport or wave phenomena are much more complicated in multiple spatial dimensions, and in our approach to extend the transport reversal algorithm to higher dimensions it becomes crucial to generalize the large time-step (LTS) operators [LeVeque, SIAM J. Numer. Anal. (1985), pp.1051-1073]. For this purpose, we introduce a dimensional splitting method using the Radon transform, that enables the transport reversal introduced above for 1D to be extended to higher spatial dimensions. This dimensional splitting is of interest in its own right, and its applications to the solution of acoustic equation, absorbing boundary condition and displacement interpolation are illustrated. This splitting method requires inverting the Radon transform, and a method for inversion using conjugate gradient algorithm will be discussed.
- Tsunami Modeling 
- Applied mathematics