A Spline Dimensional Decomposition for Uncertainty Quantification in High Dimensions

This study debuts a new spline dimensional decomposition (SDD) for uncertainty quantification analysis of high-dimensional functions, including those endowed with high nonlinearity and nonsmoothness, if they exist, in proficient manner. The creates hierarchical expansion an output random variable interest respect to measure-consistent orthonormalized basis splines (B-splines) independent input variables. A dimensionwise space into orthogonal subspaces, each spanned by reduced set such orthonormal splines, results SDD. Exploiting the modulus smoothness, SDD approximation is shown converge mean-square correct limit. computational complexity method polynomial, as opposed exponential, thus alleviating curse dimensionality extent possible. Analytical formulae are proposed calculate second-moment properties truncated general terms coefficients involved. Numerical indicate that low-order nonsmooth functions calculates probabilistic characteristics accuracy matching or surpassing obtained high-order approximations from several existing methods. Finally, 34-dimensional eigenvalue demonstrates utility solving practical problems.