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

High accuracy complex computer models, or 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. 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. We observe a significant increase in the spread of uncertainties in the tsunami heights due to the contribution of the bathymetry uncertainties. These results highlight the need to include the effect of uncertainties in the bathymetry in tsunami early warnings and risk assessments.Comment: 26 pages, 8 figures, 2 table

Sequential Design with Mutual Information for Computer Experiments (MICE): Emulation of a Tsunami Model

Computer simulators can be computationally intensive to run over a large number of input values, as required for optimization and various uncertainty quantification tasks. The standard paradigm for the design and analysis of computer experiments is to employ Gaussian random fields to model computer simulators. Gaussian process models are trained on input-output data obtained from simulation runs at various input values. Following this approach, we propose a sequential design algorithm, MICE (Mutual Information for Computer Experiments), that adaptively selects the input values at which to run the computer simulator, in order to maximize the expected information gain (mutual information) over the input space. The superior computational efficiency of the MICE algorithm compared to other algorithms is demonstrated by test functions, and a tsunami simulator with overall gains of up to 20% in that case

Efficient spatial modelling using the SPDE approach with bivariate splines

Gaussian fields (GFs) are frequently used in spatial statistics for their versatility. The associated computational cost can be a bottleneck, especially in realistic applications. It has been shown that computational efficiency can be gained by doing the computations using Gaussian Markov random fields (GMRFs) as the GFs can be seen as weak solutions to corresponding stochastic partial differential equations (SPDEs) using piecewise linear finite elements. We introduce a new class of representations of GFs with bivariate splines instead of finite elements. This allows an easier implementation of piecewise polynomial representations of various degrees. It leads to GMRFs that can be inferred efficiently and can be easily extended to non-stationary fields. The solutions approximated with higher order bivariate splines converge faster, hence the computational cost can be alleviated. Numerical simulations using both real and simulated data also demonstrate that our framework increases the flexibility and efficiency.Comment: 26 pages, 7 figures and 3 table

Linked Gaussian Process Emulation for Systems of Computer Models Using Matérn Kernels and Adaptive Design

The state-of-the-art linked Gaussian process offers a way to build analytical emulators for systems of computer models. We generalize the closed form expressions for the linked Gaussian process under the squared exponential kernel to a class of Mat\'ern kernels, that are essential in advanced applications. An iterative procedure to construct linked Gaussian processes as surrogate models for any feed-forward systems of computer models is presented and illustrated on a feed-back coupled satellite system. We also introduce an adaptive design algorithm that could increase the approximation accuracy of linked Gaussian process surrogates with reduced computational costs on running expensive computer systems, by allocating runs and refining emulators of individual sub-models based on their heterogeneous functional complexity

Deep Gaussian Process Emulation using Stochastic Imputation

We propose a novel deep Gaussian process (DGP) inference method for computer model emulation using stochastic imputation. By stochastically imputing the latent layers, the approach transforms the DGP into the linked GP, a state-of-the-art surrogate model formed by linking a system of feed-forward coupled GPs. This transformation renders a simple while efficient DGP training procedure that only involves optimizations of conventional stationary GPs. In addition, the analytically tractable mean and variance of the linked GP allows one to implement predictions from DGP emulators in a fast and accurate manner. We demonstrate the method in a series of synthetic examples and real-world applications, and show that it is a competitive candidate for efficient DGP surrogate modeling in comparison to the variational inference and the fully-Bayesian approach. A $\texttt{Python}$ package $\texttt{dgpsi}$ implementing the method is also produced and available at https://github.com/mingdeyu/DGP

Multi-level emulation of tsunami simulations over Cilacap, South Java, Indonesia

Carrying out a Probabilistic Tsunami Hazard Assessment (PTHA) requires a large number of simulations done at a high resolution. Statistical emulation builds a surrogate to replace the simulator and thus reduces computational costs when propagating uncertainties from the earthquake sources to the tsunami inundations. To reduce further these costs, we propose here to build emulators that exploit multiple levels of resolution and a sequential design of computer experiments. By running a few tsunami simulations at high resolution and many more simulations at lower resolutions we are able to provide realistic assessments whereas, for the same budget, using only the high resolution tsunami simulations do not provide a satisfactory outcome. As a result, PTHA can be considered with higher precision using the highest spatial resolutions, and for impacts over larger regions. We provide an illustration to the city of Cilacap in Indonesia that demonstrates the benefit of our approach

Robust uncertainty quantification of the volume of tsunami ionospheric holes for the 2011 Tohoku-Oki earthquake: towards low-cost satellite-based tsunami warning systems

We develop a new method to analyze the total electron content (TEC) depression in the ionosphere after a tsunami occurrence. We employ Gaussian process regression to accurately estimate the TEC disturbance every 30 s using satellite observations from the global navigation satellite system (GNSS) network, even over regions without measurements. We face multiple challenges. First, the impact of the acoustic wave generated by a tsunami onto TEC levels is nonlinear and anisotropic. Second, observation points are moving. Third, the measured data are not uniformly distributed in the targeting range. Nevertheless, our method always computes the electron density depression volumes, along with estimated uncertainties, when applied to the 2011 Tohoku-Oki earthquake, even with random selections of only 5 % of the 1000 GPS Earth Observation Network System receivers considered here over Japan. Also, the statistically estimated TEC depression area mostly overlaps the range of the initial tsunami, which indicates that our method can potentially be used to estimate the initial tsunami. The method can warn of a tsunami event within 15 min of the earthquake, at high levels of confidence, even with a sparse receiver network. Hence, it is potentially applicable worldwide using the existing GNSS network

Probabilistic Landslide Tsunami Estimation in the Makassar Strait, Indonesia, Using Statistical Emulation

This paper presents a significant advancement in the understanding of tsunamigenic landslide hazard across the length of the Makassar Strait in Indonesia. We use statistical emulation across the length of the continental slope to conduct a probabilistic assessment of tsunami hazard on a regional scale, across 14 virtual coastal gauges. Focusing on the potential maximum wave amplitudes (distance between the wave crest and the still‐water level) from possible tsunamigenic landslide events, we generate predictions from Gaussian Process emulators fitted to input‐outputs from 50 training scenarios. We show that the most probable maximum wave amplitudes in the majority of gauges are between 1 and 5 m, with the maximum predicted amplitudes reaching values of up to 10 m on the eastern coast, and up to 50 m on the western coast. We also explore the potential use of Gaussian multivariate copulas to sample emulator prediction input values to create a more realistic distribution of volumes along the continental slope. The novel use of statistical emulation across a whole slope enables the probabilistic assessment of tsunami hazard due to landslides on a regional scale. This area is of key interest to Indonesia since the new capital will be established in the East Kalimantan region on the western side of the Makassar Strait