  We here consider the subset simulation method which approaches a failure
event using a decreasing sequence of nested intermediate failure events. The
method resembles importance sampling, which actively explores a probability
space by conditioning the next evaluation on the previous evaluations using a
Markov chain Monte Carlo (MCMC) algorithm. A Markov chain typically requires
many steps to estimate the target distribution, which is impractical with
expensive numerical models. Therefore, we propose to approximate each step of a
Markov chain locally with Gaussian process (GP) regression. Benchmark examples
of reliability analysis show that local approximations significantly improve
overall efficiency of subset simulation. They reduce the number of expensive
limit-state evaluations by over $80\%$. However, GP regression becomes
computationally impractical with increasing dimension. Therefore, to make our
use of a GP feasible, we employ the partial least squares (PLS) regression, a
gradient-free reduction method, locally to explore and utilize a
low-dimensional subspace within a Markov chain. Numerical experiments
illustrate a significant computational gain with maintained sufficient
accuracy.
