  Uncertainty quantification based on generalized polynomial chaos has been
used in many applications. It has also achieved great success in
variation-aware design automation. However, almost all existing techniques
assume that the parameters are mutually independent or Gaussian correlated,
which is rarely true in real applications. For instance, in chip manufacturing,
many process variations are actually correlated. Recently, some techniques have
been developed to handle non-Gaussian correlated random parameters, but they
are time-consuming for high-dimensional problems. We present a new framework to
solve uncertainty quantification problems with many non-Gaussian correlated
uncertainties. Firstly, we propose a set of smooth basis functions to well
capture the impact of non-Gaussian correlated process variations. We develop a
tensor approach to compute these basis functions in a high-dimension setting.
Secondly, we investigate the theoretical aspect and practical implementation of
a sparse solver to compute the coefficients of all basis functions. We provide
some theoretical analysis for the exact recovery condition and error bound of
this sparse solver in the context of uncertainty quantification. We present
three adaptive sampling approaches to improve the performance of the sparse
solver. Finally, we validate our methods by synthetic and practical
electronic/photonic ICs with 19 to 57 non-Gaussian correlated variation
parameters. Our approach outperforms Monte Carlo by thousands of times in terms
of efficiency. It can also accurately predict the output density functions with
multiple peaks caused by non-Gaussian correlations, which are hard to capture
by existing methods.
