Least-squares dynamic approximation method for evolution of uncertainty in initial conditions of dynamical systems

We describe an approximation method to solve the probability density function transport equation, i.e., the Liouville equation, which is encountered in the evolution of uncertainty of the initial values of dynamical systems. A state-space based method is formulated using a least-squares technique that preserves the parabolic nature of the Liouville equation and is flexible in terms of accuracy of representation. This method is based on a global approximation in terms of analytical elementary functions with unknown parameters, whose evolution equations are determined by a global least-squares approximation. The realizability conditions of the probability density, i.e., the non-negativity and normalization conditions are enforced at all times. The method is successfully evaluated in a number of scenarios including the uncertainty evolution in a system governed by a Riccati equation and a particle moving in a fluid under the influence of Stokes drag force. The results obtained in our examples exhibit a reasonable good agreement when compared with the solution of the probability transport equation using the method of characteristics. The cost of the method is proportional to the cost of solving the deterministic system and the number of parameters used to approximate the probability density function, a feature that can make the present method very advantageous in comparison with other methods in problems involving a large number of dimensions.