编码汉克尔多项式混沌：主导多项式混沌模式的谱识别
Coded Hankel Polynomial Chaos: Spectral Identification of Dominant Polynomial-Chaos Modes
- School of Mathematical Science, University of Electronic Science and Technology of China（电子科技大学数学科学学院）
- School of Mathematics, Southwest Jiaotong University（西南交通大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究提出编码汉克尔多项式混沌（CH-PC）方法，用于谱识别主导多项式混沌模式，通过汉克尔矩阵编码阶数与节点，结合相位投票等实现精确恢复，在基准和达西问题中验证了其有效性。
AI中文摘要：
主导多项式混沌（PCE）模式的识别通常被表述为对采样多元多项式字典的稀疏回归问题。我们提出编码汉克尔多项式混沌（CH-PC），一种用于主导模式识别的互补谱公式。有限生成变换将PCE系数转换为系数生成多项式，沿几何相位轨道的求值产生有限指数和。其模型阶数和谱节点由低秩汉克尔矩阵编码，而坐标相移附加单位根标签，从中可恢复完整的多项式多重指标。坐标移位探针被组合为公共节点快照，当单个谱编码条件不佳时，独立相位编码提供冗余表示。对于有限观测，总体、有限数据和观测探针需区分：采样或求积误差与观测误差作为独立汉克尔扰动输入，随后与谱稳定性、离散解码和相位投票关联。对于张量积候选集，生成核可分解为一维和，无需组装完整多元PCE设计矩阵即可求值。在稀疏勒让德基准和随机达西问题上的数值实验表明，该方法可实现精确恢复、噪声稳定、通过相位持久性识别未知阶数，以及对PDE生成的感兴趣量的主导模式恢复。
英文摘要：
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.
