Dimension reduction for emulation: application to the influence of bathymetry on tsunami heights

High accuracy complex computer models, also called simulators, require large resources in time and memory to produce realistic results. Statistical emulators are computationally cheap approximations of such simulators. They can be built to replace simulators for various purposes, such as the propagation of uncertainties from inputs to outputs or the calibration of some internal parameters against observations. However, when the input space is of high dimension, the construction of an emulator can become prohibitively expensive. In this paper, we introduce a joint framework merging emulation with dimension reduction in order to overcome this hurdle. The gradient-based kernel dimension reduction technique is chosen due to its ability to extract drastically lower dimensions with little loss in information. The Gaussian Process emulation technique is combined with this dimension reduction approach. Theoretical properties of the approximation are explored. We demonstrate both efficiency and accuracy theoretically and on an elliptic PDE. We finally present a realistic application to tsunami modeling. The uncertainties in the sea-floor elevation (bathymetry) are modeled as high-dimensional realizations of a spatial process using the INLA-SPDE approach. Our dimension-reduced emulation enables us to compute the impact of these uncertainties on resulting possible tsunami wave heights near-shore and on-shore. Considering an uncertain earthquake source, we observe a significant increase in the spread of uncertainties in the tsunami heights due to the contribution of the bathymetry uncertainties to the overall uncertainty budget. These results highlight the need to reduce uncertainties in the bathymetry in early warnings and hazard assessments.