Dimension-wise Multivariate Orthogonal Polynomials in General Probability Spaces

TL;DR

This paper introduces a generalized polynomial dimensional decomposition (GPDD) that uses measure-consistent multivariate orthogonal polynomials for dependent variables, extending the applicability of PDD to more complex stochastic problems.

Contribution

The paper develops a new GPDD framework that does not require independence among variables, broadening the scope of polynomial decompositions in stochastic analysis.

Findings

Mathematical properties of GPDD are rigorously established.

GPDD demonstrates mean-square convergence including for infinitely many variables.

The approach is effective for high-dimensional problems with dependent variables.

Abstract

This paper puts forward a new generalized polynomial dimensional decomposition (PDD), referred to as GPDD, comprising hierarchically ordered measure-consistent multivariate orthogonal polynomials in dependent random variables. Unlike the existing PDD, which is valid strictly for independent random variables, no tensor-product structure is assumed or required. Important mathematical properties of GPDD are studied by constructing dimension-wise decomposition of polynomial spaces, deriving statistical properties of random orthogonal polynomials, demonstrating completeness of orthogonal polynomials for prescribed assumptions, and proving mean-square convergence to the correct limit, including when there are infinitely many random variables. The GPDD approximation proposed should be effective in solving high-dimensional stochastic problems subject to dependent variables.