  Fabrication process variations are a major source of yield degradation in the
nano-scale design of integrated circuits (IC), microelectromechanical systems
(MEMS) and photonic circuits. Stochastic spectral methods are a promising
technique to quantify the uncertainties caused by process variations. Despite
their superior efficiency over Monte Carlo for many design cases, these
algorithms suffer from the curse of dimensionality; i.e., their computational
cost grows very fast as the number of random parameters increases. In order to
solve this challenging problem, this paper presents a high-dimensional
uncertainty quantification algorithm from a big-data perspective. Specifically,
we show that the huge number of (e.g., $1.5 \times 10^{27}$) simulation samples
in standard stochastic collocation can be reduced to a very small one (e.g.,
$500$) by exploiting some hidden structures of a high-dimensional data array.
This idea is formulated as a tensor recovery problem with sparse and low-rank
constraints; and it is solved with an alternating minimization approach.
Numerical results show that our approach can simulate efficiently some ICs, as
well as MEMS and photonic problems with over 50 independent random parameters,
whereas the traditional algorithm can only handle several random parameters.
