Graduate student seminar: Multi-fidelity sparse grids for uncertainty quantification

In this work, we look for a surrogate method to provide an approximation of the output of an input-output relationships using as few model evaluations as possible. Simulations are often aimed at exploring the complex relationships between input and output variables, however, when models of real-world engineering systems are complex and the need for multiple realizations, as in uncertainty quantification or optimization, simulations run may be computationally expensive. We aim to make an accurate and efficient surrogate approximation to reduce the cost of simulation model. Multi-fidelity methods have been developed on multi-fidelity Monte Carlo, multi-fidelity Kriging and multi-fidelity sparse grids to accelerate the estimation of outputs of high-fidelity models. Our multi-fidelity approach provides a framework to sample the input parameter space on a sparse grid, and then use sparse grid interpolation as our surrogate. To further increase the efficiency, a multi-fidelity approach combine results from accurate and expensive high-fidelity simulations with inexpensive but less accurate low-fidelity simulations in order to achieve accuracy at a reasonable cost. We look at the application of the multi-fidelity sparse grids to the solution of several test functions particularly focusing on numerical aspects like stability, order of approximation and error convergence. Also, a multi-fidelity sparse grid surrogate model was constructed for the Hokkaido-Nansei-Oki tsunami. We demonstrate the experimental results in Okushiri wave flume, which reproduce the maximum value of the time-dependent average tsunami height on top of the Monai zone in Okushiri Island in 1993. We illustrate our approach of the number of uncertain input parameters to quantify the uncertainty in the output of tsunami wave shape and properly report the achieved savings. Our results consists of both theoretical and numerical parts. In the theoretical part, we discuss the error analysis of convergence on arbitrary dimensions. The numerical results can be up to given accuracy with a reduced cost.