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

TL;DR

This paper introduces a novel tensor regression approach for high-dimensional uncertainty quantification that automatically determines tensor rank and adaptively selects simulation samples, significantly reducing computational costs.

Contribution

It proposes a new tensor regression framework with rank determination and sample selection strategies, advancing high-dimensional uncertainty quantification methods.

Findings

Successfully applied to 19-dimensional and 57-dimensional problems.

Achieved accurate uncertainty capture with fewer samples.

Demonstrated effectiveness in complex high-dimensional systems.

Abstract

Uncertainty quantification based on stochastic spectral methods suffers from the curse of dimensionality. This issue was mitigated recently by low-rank tensor methods. However, there exist two fundamental challenges in low-rank tensor-based uncertainty quantification: how to automatically determine the tensor rank and how to pick the simulation samples. This paper proposes a novel tensor regression method to address these two challenges. Our method uses an $\ell_{q}/ \ell_{2}$-norm regularization to determine the tensor rank and an estimated Voronoi diagram to pick informative samples for simulation. The proposed framework is verified by a 19-dim phonics bandpass filter and a 57-dim CMOS ring oscillator, capturing the high-dimensional uncertainty well with only 90 and 290 samples respectively.