Multi-fidelity sparse-grid-based surrogate models

Abstract

In this thesis, we look for multi-fidelity surrogate method to provide an approximation of the output of an input-output relationship using as few model evaluations as possible for uncertainty quantification purpose. We explore the use of multi-fidelity modelling within uncertainty quantification, for which we combine information from simulation models within a hierarchy of fidelity, in seeking accurate and efficient approximation of multi-fidelity model at reduced computational cost. 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. At first, our approach provides a framework to sample the input parameter space on a sparse grid, and then use sparse grid interpolation as our surrogate. For accurate but expensive high-fidelity models, however, even performing the number of simulations needed for fitting a surrogate may be too expensive. Thus, multi-fidelity is an useful approach to reduce the number of high-fidelity simulations. 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 via the combination technique in order to achieve accuracy at a reasonable cost. We focus attention on multi-fidelity sparse grids in which fidelities are combined inside the surrogate model. We present the multi-fidelity approach via a careful study and generalisation of the sparse grid combination technique and give a general formula for the multi-fidelity approach. The combination technique as a method to achieve a function representation on a sparse grid without having to work with a hierarchical basis. Study of the combination technique often assumes that approximations satisfy an error splitting model. The combination technique is competitive to the classical sparse grid approach with respect to quality and run time and give proof that the combined interpolant is equivalent to the hierarchical sparse grid interpolant.\\ Instead of relying on error and cost rates, an optimisation problem with an analytical result optimal balance the model evaluations across the high-fidelity model and an arbitrary number of low-fidelity models with respect to error and costs. The error analysis of convergence on arbitrary dimensions is applicable to general use cases of the multi-fidelity approach.\\ Our results consists of both theoretical and numerical parts. In the numerical part, we look at the application of the multi-fidelity sparse grid model to the solution of several test functions and partial differential equation (PDE) models particularly focusing on numerical aspects like stability, order of approximation and error convergence. Comparing the mathematical results and the numerical results, some of numerical test cases can obtain the same results as the theoretical analysis. Lastly, a multi-fidelity sparse grid surrogate model was constructed for the Hokkaido-Nansei-Oki tsunami modelling for uncertainty quantification. We demonstrate the experimental results in Okushiri wave flume, which reproduce the maximum value of the time-dependent average tsunami height on top of area of interest. We illustrate the multi-fidelity approach with the number of uncertain input parameters to quantify the uncertainty in the output of tsunami wave shape and properly report the achieved savings. The numerical results can be up to given accuracy with a reduced cost. Show more