  The paper presents a novel methodology to build surrogate models of
complicated functions by an active learning-based sequential decomposition of
the input random space and construction of localized polynomial chaos
expansions, referred to as domain adaptive localized polynomial chaos expansion
(DAL-PCE). The approach utilizes sequential decomposition of the input random
space into smaller sub-domains approximated by low-order polynomial expansions.
This allows approximation of functions with strong nonlinearties,
discontinuities, and/or singularities. Decomposition of the input random space
and local approximations alleviates the Gibbs phenomenon for these types of
problems and confines error to a very small vicinity near the non-linearity.
The global behavior of the surrogate model is therefore significantly better
than existing methods as shown in numerical examples. The whole process is
driven by an active learning routine that uses the recently proposed $\Theta$
criterion to assess local variance contributions. The proposed approach
balances both \emph{exploitation} of the surrogate model and \emph{exploration}
of the input random space and thus leads to efficient and accurate
approximation of the original mathematical model. The numerical results show
the superiority of the DAL-PCE in comparison to (i) a single global polynomial
chaos expansion and (ii) the recently proposed stochastic spectral embedding
(SSE) method developed as an accurate surrogate model and which is based on a
similar domain decomposition process. This method represents general framework
upon which further extensions and refinements can be based, and which can be
combined with any technique for non-intrusive polynomial chaos expansion
construction.
