  This work suggests an interpolation-based stochastic collocation method for
the non-intrusive and adaptive construction of sparse polynomial chaos
expansions (PCEs). Unlike pseudo-spectral projection and regression-based
stochastic collocation methods, the proposed approach results in PCEs featuring
one polynomial term per collocation point. Moreover, the resulting PCEs are
interpolating, i.e., they are exact on the interpolation nodes/collocation
points. Once available, an interpolating PCE can be used as an inexpensive
surrogate model, or be post-processed for the purposes of uncertainty
quantification and sensitivity analysis. The main idea is conceptually simple
and relies on the use of Leja sequence points as interpolation nodes. Using
Newton-like, hierarchical basis polynomials defined upon Leja sequences, a
sparse-grid interpolation can be derived, the basis polynomials of which are
unique in terms of their multivariate degrees. A dimension-adaptive scheme can
be employed for the construction of an anisotropic interpolation. Due to the
degree uniqueness, a one-to-one transform to orthogonal polynomials of the
exact same degrees is possible and shall result in an interpolating PCE.
However, since each Leja node defines a unique Newton basis polynomial, an
implicit one-to-one map between Leja nodes and orthogonal basis polynomials
exists as well. Therefore, the in-between steps of hierarchical interpolation
and basis transform can be discarded altogether, and the interpolating PCE can
be computed directly. For directly computed, adaptive, anisotropic
interpolating PCEs, the dimension-adaptive algorithm is modified accordingly. A
series of numerical experiments verify the suggested approach in both low and
moderately high-dimensional settings, as well as for various input
distributions.
