Interpolation-based optimal knot selection in spline dimensional decomposition for uncertainty quantification in dynamical systems

Abstract

Forward uncertainty quantification in dynamical systems is challenging due to non-smooth or locally oscillating nonlinear behaviors. Spline dimensional decomposition addresses such nonlinearity by partitioning input coordinates via knot placement, yet its accuracy is highly sensitive to internal knot locations, and optimization-based strategies for determining these locations are computationally demanding. We propose a computationally efficient, interpolation-based method for optimal knot selection in spline dimensional decomposition. The proposed method involves three steps: (1) interpolating input-output profiles, (2) defining subinterval-based reference regions, and (3) selecting optimal knot locations at maximum gradient points within each region. This approach avoids costly iterative optimization while preserving high accuracy in approximating non-smooth and oscillating responses. In a modal analysis of a lower control arm, spline dimensional decomposition with the proposed knot selection achieves higher accuracy than versions with uniformly or randomly spaced knots, and a Gaussian process surrogate, showing the lowest relative variance error (2.89 %) in the first natural frequency distribution. Applications to mathematical functions and a ten-dimensional structural model demonstrates the proposed method's scalability and efficiency, yielding accurate second-moment statistics and reliability estimates with only a few hundred simulations.