Uncertainty quantification by optimal spline dimensional decomposition

Abstract

AbstractAn optimal version of spline dimensional decomposition (SDD) is unveiled for general high‐dimensional uncertainty quantification analysis of complex systems subject to independent but otherwise arbitrary probability measures of input random variables. The resulting method involves optimally derived knot vectors of basis splines (B‐splines) in some or all coordinate directions, whitening transformation producing measure‐consistent orthonormalized B‐splines equipped with optimal knots, and Fourier‐spline expansion of a general high‐dimensional output function of interest. In contrast to standard SDD, there is no need to select the knot vectors uniformly or intuitively. The generation of optimal knot vectors can be viewed as an inexpensive preprocessing step toward creating the optimal SDD. Analytical formulas are proposed to calculate the second‐moment properties by the optimal SDD method for a general output random variable in terms of the expansion coefficients involved. It has been shown that the computational complexity of the optimal SDD method is polynomial, as opposed to exponential, thus mitigating the curse of dimensionality by a discernible magnitude. Numerical results affirm that the optimal SDD method developed is more precise than polynomial chaos expansion, sparse‐grid quadrature, and the standard SDD method in calculating not only the second‐moment statistics, but also the cumulative distribution function of an output random variable. More importantly, the optimal SDD outperforms standard SDD by sustaining nearly identical computational cost.