Coded Hankel Polynomial Chaos: Spectral Identification of Dominant Polynomial-Chaos Modes
Authors: Zhiliang Deng, Xiaomei Yang
Organizations: School of Mathematical Science, University of Electronic Science and Technology of China · School of Mathematics, Southwest Jiaotong University
Identification of dominant polynomial-chaos modes is usually formulated as a sparse-regression problem on a sampled multivariate polynomial dictionary. We develop coded Hankel polynomial chaos (CH-PC), a complementary spectral formulation for dominant-mode identification. A finite generating transform converts PCE coefficients into a coefficient-generating polynomial, and evaluation along a geometric phase orbit produces a finite exponential sum. Its model order and spectral nodes are encoded by low-rank Hankel matrices, while coordinate phase shifts attach root-of-unity labels from which the full polynomial multi-indices are recovered. Coordinate-shifted probes are combined as common-node snapshots, and independent phase encodings provide redundant representations when a single spectral encoding is poorly conditioned. For finite observations, population, finite-data, and observed probes are kept distinct: sampling or quadrature error and observation error enter as separate Hankel perturbations, which are then connected to spectral stability, discrete decoding, and phase voting. For tensor-product candidate sets, the generating kernel factorizes into one-dimensional sums and can be evaluated without assembling the full multivariate PCE design matrix. Numerical experiments on sparse Legendre benchmarks and a stochastic Darcy problem illustrate exact recovery, noise stabilization, unknown-order identification by phase persistence, and dominant-mode recovery for a PDE-generated quantity of interest.
Figures & tables
Figure 1: Singular values of the joint Hankel matrix in Example 1. The sharp gap after the fourth singular value agrees with the true model order s=4 .
Figure 2: Exact-support frequency versus relative RMS observation noise for Example 2. Joint snapshots improve the one-phase decoder, and multi-phase voting provides further stabilization.
Figure 3: Unknown-order recovery versus relative RMS observation noise in Example 3. The true order is not supplied to the algorithm.
Figure 4: Unknown-order recovery at 30% noise as the number of phase encodings increases. Error bars show variation across independent phase ensembles.
Figure 5: Exact recovery frequency of the four reference dominant modes for the stochastic Darcy problem. All three methods recover the same dominant support throughout the tested noise range.
Center for Advanced Intelligence Project (AIP), RIKEN, Osaka, Japan · 2Graduate School of Information Science and Technology, The University of Osaka, Osaka, Japan