Sparse Approximation Method for Accurate Uncertainty Propagation through a Nonlinear System

A computationally efficient approach is presented to propagate initial condition uncertainty, represented by a probability density function (PDF), through a nonlinear system. The log-PDF is assumed to be a linear combination of basis functions, and the amplitudes of these functions are computed by requiring the approximation to satisfy the Fokker-Planck-Kolmogorov equation (FPKE). Sparse approximation tools are utilized to trade off between FPKE error and the sparsity of the log-PDF model. The approach derives insights from known analytical stationary solutions of FPKE to construct multidimensional basis functions that accurately represent the state PDF. This judicious construction of basis functions helps alleviate the curse of dimensionality associated with the growth of basis functions in multidimensional space. Two nonlinear oscillators and a two-body problem are considered to demonstrate the efficacy of the proposed approach. Simulation results show that this approach effectively propagates uncertainty through both non-conservative and conservative systems.