  Multi-fidelity modelling arises in many situations in computational science
and engineering world. It enables accurate inference even when only a small set
of accurate data is available. Those data often come from a high-fidelity
model, which is computationally expensive. By combining the realizations of the
high-fidelity model with one or more low-fidelity models, the multi-fidelity
method can make accurate predictions of quantities of interest. This paper
proposes a new dimension reduction framework based on rotated multi-fidelity
Gaussian process regression and a Bayesian active learning scheme when the
available precise observations are insufficient. By drawing samples from the
trained rotated multi-fidelity model, the so-called supervised dimension
reduction problems can be solved following the idea of the sliced average
variance estimation (SAVE) method combined with a Gaussian process regression
dimension reduction technique. This general framework we develop can
effectively solve high-dimensional problems while the data are insufficient for
applying traditional dimension reduction methods. Moreover, a more accurate
surrogate Gaussian process model of the original problem can be obtained based
on our trained model. The effectiveness of the proposed rotated multi-fidelity
Gaussian process(RMFGP) model is demonstrated in four numerical examples. The
results show that our method has better performance in all cases and
uncertainty propagation analysis is performed for last two cases involving
stochastic partial differential equations.
