  Fabrication process variations can significantly influence the performance
and yield of nano-scale electronic and photonic circuits. Stochastic spectral
methods have achieved great success in quantifying the impact of process
variations, but they suffer from the curse of dimensionality. Recently,
low-rank tensor methods have been developed to mitigate this issue, but two
fundamental challenges remain open: how to automatically determine the tensor
rank and how to adaptively pick the informative simulation samples. This paper
proposes a novel tensor regression method to address these two challenges. We
use a $\ell_{q}/ \ell_{2}$ group-sparsity regularization to determine the
tensor rank. The resulting optimization problem can be efficiently solved via
an alternating minimization solver. We also propose a two-stage adaptive
sampling method to reduce the simulation cost. Our method considers both
exploration and exploitation via the estimated Voronoi cell volume and
nonlinearity measurement respectively. The proposed model is verified with
synthetic and some realistic circuit benchmarks, on which our method can well
capture the uncertainty caused by 19 to 100 random variables with only 100 to
600 simulation samples.
