Triple Products of Correlated Multivariate Hermite Polynomials
Abstract
Stochastic Galerkin schemes (SGS) are used to approximate solutions to differential equations with stochastic parameters. In particular, they express the solution with an infinite orthogonal functional basis of the parameter space and derive a deterministic system of PDEs; the solution of this deterministic system can be used to produce a {surrogate} approximation of the original stochastic equations. When the parameters are correlated, the infinite series representation of the solution does not provably converge for all joint distributions; further, classic decorrelating transformations (e.g., Roseblatt) often introduce known numerical instabilities. Rahman (2017) established convergence in probability and distribution for correlated {Gaussian} parameters when the basis functionals are {multivariate Hermite polynomials} (MHPs), where these polynomials are only {weakly orthogonal} with respect to the underlying joint density. Due to weak orthogonality, the Nth moments of MHPs introduced during Galerkin projection do not trivially decompose into products of one-dimensional expectations and have no closed-form expression for N > 2; consequently, these moments are typically estimated via brute-force sampling and quickly become prohibitively expensive to compute. In this paper, we introduce and prove several foundational results for MHPs. These results culminate in an equation that can be used to calculate the N = 3 moments (triple products) exactly, whose implementation dramatically reduces the computational bottleneck otherwise inherent to SGS with correlated Gaussian parameters.
Related
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BibTeX
@misc{lyman2026triple,
title = {{Triple Products of Correlated Multivariate Hermite Polynomials}},
author = {Laura Lyman},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30204},
eprint = {2610.30204},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30204},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Triple Products of Correlated Multivariate Hermite Polynomials
AU - Laura Lyman
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30204
UR - https://arxiv.org/abs/2610.30204
AB - Stochastic Galerkin schemes (SGS) are used to approximate solutions to differential equations with stochastic parameters. In particular, they express the solution with an infinite orthogonal functional basis of the parameter space and derive a deterministic system of PDEs; the solution of this deterministic system can be used to produce a {surrogate} approximation of the original stochastic equations. When the parameters are correlated, the infinite series representation of the solution does not provably converge for all joint distributions; further, classic decorrelating transformations (e.g., Roseblatt) often introduce known numerical instabilities. Rahman (2017) established convergence in probability and distribution for correlated {Gaussian} parameters when the basis functionals are {multivariate Hermite polynomials} (MHPs), where these polynomials are only {weakly orthogonal} with respect to the underlying joint density. Due to weak orthogonality, the Nth moments of MHPs introduced during Galerkin projection do not trivially decompose into products of one-dimensional expectations and have no closed-form expression for N > 2; consequently, these moments are typically estimated via brute-force sampling and quickly become prohibitively expensive to compute. In this paper, we introduce and prove several foundational results for MHPs. These results culminate in an equation that can be used to calculate the N = 3 moments (triple products) exactly, whose implementation dramatically reduces the computational bottleneck otherwise inherent to SGS with correlated Gaussian parameters.
ER -