Abstract
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 drastically decrease dimensionality with little loss in information. The Gaussian process emulation technique is combined with this dimension reduction approach. Theoretical properties of the approximation are explored. Our proposed approach provides an answer to the dimension reduction issue in emulation for a wide range of simulation problems that cannot be tackled using existing methods. The efficiency and accuracy of the proposed framework is demonstrated theoretically and compared with other methods on an elliptic partial differential equation (PDE) problem. We finally present a realistic application to tsunami modeling. The uncertainties in the bathymetry (seafloor elevation) are modeled as high-dimensional realizations of a spatial process using a geostatistical 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 include the effect of uncertainties in the bathymetry in tsunami early warnings and risk assessments.
Highlights
Simulators are widely employed, to reproduce physical processes and explore their behavior, in fields such as fluid dynamics or climate modeling
We proposed a joint framework for emulation of high-dimensional simulators with dimension reduction
Our method can be applied to uncertainty quantification for many purposes, such as risk assessment, sensitivity analysis, and calibration, with great perspectives in real-world applications
Summary
Simulators are widely employed, to reproduce physical processes and explore their behavior, in fields such as fluid dynamics or climate modeling. This implies that more evaluations of the simulator are required to train an accurate emulator when the number of input parameters d increases and the associated computational cost of constructing an emulator could increase dramatically as a result. In the following numerical studies, we select d as well as other parameters for the gKDR approach using simple trial-and-error or a more formal cross validation approach based on the predictive accuracy of the respective emulators. Throughout the simulations, the Gaussian processes for machine learning code using maximum likelihood method implemented by [30] is employed for the emulation, assuming a linear form mean function with intercept and a squared exponential correlation function
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