Automatic Determination of Set of Multivariate Basis Polynomials Based on Recursion and Application of LSPCR to High-Dimensional Uncertainty Quantification of Multi-Conductor Transmission Lines

Abstract

Uncertainty quantification (UQ) in polynomial chaos expansion (PCE) suffers from the curse of dimensionality, which is fundamentally reflected in that the number of the PCE coefficients to be found grows rapidly as the number of random inputs increases, and thus the number of samples required increases dramatically. Regardless of the approach taken to alleviate this disaster, we are usually faced with the technical challenge of automatically and programmatically determining the basis functions during the PCE implementation, which are composed of a large number of multivariate polynomials. To address this problem, this paper proposes an algorithm based on the recursive idea for automatically determining the basis functions, and then combines this algorithm with the latest weight-based regression called least squares polynomial chaos regression (LSPCR), develops the six algorithms for the LSPCR in the medium- and high-dimensional cases, and evaluates their performance in the electromagnetic coupling problem of the transmission lines (TLs) in this case. Subsequently, the performance of six algorithms for LSPCR are compared in the medium- and high-dimensional, low-order cases, each algorithm consisting of a sampling strategy selected from asymptotic sampling (AS), standard sampling (SS), and coherence-optimal sampling (COS), paired with a norm problem chosen from either least squares optimization (LSO) or $\ell _{\mathbf {1}}$ -minimization ( $\ell _{\mathbf {1}}$ -M) problems. Numerical experiments demonstrate the excellence of the proposed algorithm. Furthermore, the results also show that only the algorithms that employ the SS (SSs) and the algorithms based on the COS (COSs) require no more than 1.4% of the total Monte Carlo (MC) computation time when producing similarly accurate results to the MC, regardless of the chosen norm problem.

Keywords

• Uncertainty quantification (UQ)
• polynomial chaos expansion (PCE)
• least squares polynomial chaos regression
• high dimensionality
• transmission lines (TLs)