Coded Hankel Polynomial Chaos: Spectral Identification of Dominant Polynomial-Chaos Modes

Identifying dominant modes in polynomial chaos expansion (PCE) is typically framed as a sparse-regression problem over a large multivariate polynomial dictionary. This paper introduces coded Hankel polynomial chaos (CH-PC), a complementary spectral formulation that transforms PCE coefficient identification into a structured Hankel low-rank matrix problem.

The core methodology begins with a finite generating transform that converts PCE coefficients into a multivariate coefficient-generating polynomial. Evaluating this transform along a geometric phase orbit yields a finite exponential sum, allowing the active model order to be determined from the rank of associated Hankel matrices and spectral nodes to be recovered via matrix-pencil techniques. Because a single spectral node only reveals multi-indices through scalar phase combinations, the authors introduce coordinate phase shifts that multiply amplitudes by root-of-unity factors, reducing multivariate index recovery to independent coordinate-wise discrete decoding.

A key aspect of CH-PC is its computational profile compared to standard dictionary-based sparse regression. For tensor-product candidate sets, the generating kernel factorizes into one-dimensional sums. This enables kernel evaluation without assembling the full multivariate polynomial design matrix, significantly reducing memory and scaling overhead during probe generation.

For finite data, the framework rigorously separates sampling or quadrature errors from observation noise. These perturbations are tracked at the joint-Hankel level before propagating through spectral recovery, discrete decoding, and phase voting. When a single phase encoding is poorly conditioned, repeating the procedure over independent phases provides redundant representations and stabilizes support identification.

Numerical experiments on sparse Legendre benchmarks and a stochastic Darcy problem demonstrate several practical advantages of the CH-PC framework. The approach achieves exact recovery in noise-free settings, exhibits strong noise stabilization through joint-snapshot representations, and handles unknown model orders via phase persistence. Furthermore, it successfully recovers dominant modes for physical quantities of interest generated by partial differential equations.