  Quantifying uncertainties in physical or engineering systems often requires a
large number of simulations of the underlying computer models that are
computationally intensive. Emulators or surrogate models are often used to
accelerate the computation in such problems, and in this regard the Gaussian
Process (GP) emulator is a popular choice for its ability to quantify the
approximation error in the emulator itself. However, a major limitation of the
GP emulator is that it can not handle problems of very high dimensions, which
is often addressed with dimension reduction techniques. In this work we hope to
address an issue that the models of interest are so complex that they admit
different low dimensional structures in different parameter regimes. Building
upon the active subspace method for dimension reduction, we propose a clustered
active subspace method which identifies the local low-dimensional structures as
well as the parameter regimes they are in (represented as clusters), and then
construct low dimensional and local GP emulators within the clusters.
Specifically we design a clustering method based on the gradient information to
identify these clusters, and a local GP construction procedure to construct the
GP emulator within a local cluster. With numerical examples, we demonstrate
that the proposed method is effective when the underlying models are of complex
low-dimensional structures.
