High-Dimensional Uncertainty Quantification via Rank- and Sample-Adaptive Tensor Regression

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.