HIGH-ORDER UNCERTAINITY PROPAGATION USING STATE TRANSITION TENSOR SERIES
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Modeling and simulation for complex applications in science and engineering develop behavior predictions based on mechanical loads. Imprecise knowledge of the model parameters or external force laws alters the system response from the assumed nominal model data. As a result, one seeks algorithms for generating insights into the range of variability that can be the expected due to model uncertainty. Two issues complicate approaches for handling model uncertainty. First, most systems are fundamentally nonlinear, which means that closed-form solutions are not available for predicting the response or designing control and/or estimation strategies. Second, series approximations are usually required, which demands that partial derivative models are available. Both of these issues have been significant barriers to previous researchers, who have been forced to invoke computationally intense Monte-Carlo methods to gain insight into a systems nonlinear behavior through a massive sampling process. These barriers are overcome by introducing three strategies: (1) Computational differentiation that automatically builds exact partial derivative models; (2) Automatic development of state transition tensor series-based solution for mapping initial uncertainty models into instantaneous uncertainty models; and (3) Development of nonlinear transformations for mapping an initial probability distribution function into a current probability distribution function for computing fully nonlinear statistical system properties. The resulting nonlinear probability distribution function (pdf) represents a Liouiville approximation for the stochastic Fokker Planck equation. Several applications, using nonlinear systems, are presented that demonstrate the effectiveness of the proposed mathematical developments, where the solution is validated by using a Mont-Carlo simulation method. A nonlinear covariance algorithm is presented that uses a closed-form solution for evaluating the expectation values of high-order tensor models for Gaussian variables. The general modeling methodology is expected to be broadly useful for science and engineering applications in general, as well as grand challenge problems that exist at the frontiers of computational science and mathematics.
