High-Dimensional Uncertainty Quantification via Tensor Regression with Rank Determination and Adaptive Sampling

TL;DR

This paper introduces a tensor regression approach with automatic rank determination and adaptive sampling to efficiently quantify uncertainty in high-dimensional electronic and photonic circuit simulations, reducing computational costs.

Contribution

It presents a novel tensor regression method that automatically determines tensor rank and adaptively selects simulation samples, addressing key challenges in high-dimensional uncertainty quantification.

Findings

Accurately captures uncertainty with 19-100 variables using only 100-600 samples.

Efficiently solves the optimization problem via an alternating minimization solver.

Validated on synthetic and circuit benchmarks demonstrating effectiveness.

Abstract

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.