  Stochastic collocation methods for approximating the solution of partial
differential equations with random input data (e.g., coefficients and forcing
terms) suffer from the curse of dimensionality whereby increases in the
stochastic dimension cause an explosion of the computational effort. We propose
and analyze a multilevel version of the stochastic collocation method that, as
is the case for multilevel Monte Carlo (MLMC) methods, uses hierarchies of
spatial approximations to reduce the overall computational complexity. In
addition, our proposed approach utilizes, for approximation in stochastic
space, a sequence of multi-dimensional interpolants of increasing fidelity
which can then be used for approximating statistics of the solution as well as
for building high-order surrogates featuring faster convergence rates. A
rigorous convergence and computational cost analysis of the new multilevel
stochastic collocation method is provided, demonstrating its advantages
compared to standard single-level stochastic collocation approximations as well
as MLMC methods. Numerical results are provided that illustrate the theory and
the effectiveness of the new multilevel method.
