High-Order Uncertainty Propagation Enabled by Computational Differentiation
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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 intensive 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) Map initial uncertainty models into instantaneous uncertainty models by building a series-based state transition tensor model; and (3) Compute an approximate probability distribution function by solving the Liouville equation using the state transition tensor model. The resulting nonlinear probability distribution function (PDF) represents a Liouville approximation for the stochastic Fokker-Planck equation. Several applications are presented that demonstrate the effectiveness of the proposed mathematical developments. 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.