Dimension-wise Multivariate Orthogonal Polynomials in General Probability Spaces
For researchers in uncertainty quantification, this provides a theoretical framework to handle dependent variables in polynomial expansions, though no empirical results or comparisons are provided.
This paper introduces a generalized polynomial dimensional decomposition (GPDD) for dependent random variables, extending existing methods that require independence. It proves mean-square convergence and completeness, offering a solution for high-dimensional stochastic problems with dependent variables.
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.