  Motivated by risk assessment of coastal flooding, we consider time-consuming
simulators with a spatial output. The aim is to perform sensitivity analysis
(SA), quantifying the influence of input parameters on the output. There are
three main issues. First, due to computational time, standard SA techniques
cannot be directly applied on the simulator. Second, the output is infinite
dimensional, or at least high dimensional if the output is discretized. Third,
the spatial output is non-stationary and exhibits strong local variations. We
show that all these issues can be addressed all together by using functional
PCA (FPCA). We first specify a functional basis, such as wavelets or B-splines,
designed to handle local variations. Secondly, we select the most influential
basis terms, either with an energy criterion after basis orthonormalization, or
directly on the original basis with a penalized regression approach. Then FPCA
further reduces dimension by doing PCA on the most influential basis
coefficients, with an ad-hoc metric. Finally, fast-to-evaluate metamodels are
built on the few selected principal components. They provide a proxy on which
SA can be done. As a by-product, we obtain analytical formulas for
variance-based sensitivity indices, generalizing known formula assuming
orthonormality of basis functions.
