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

• URL: http://arxiv.org/abs/2608.16126v1
• Date: Mon, 17 Aug 2026 05:32:20 GMT
• Title: Coded Hankel Polynomial Chaos: Spectral Identification of Dominant Polynomial-Chaos Modes
• Abstract summary: Identification of dominant-chaos modes is usually formulated as a sparse-re problem on a sampled dictionary.<n>We develop coded Hankel chaos (CH-PC-), a complementary spectral formulation for dominant-mode identification.
• Score: 1.6292034288562798
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: 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.

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