

HighOrder Uncertainty Propagation Enabled by Computational Differentiation
incollection
  

Author(s)
Ahmad Bani Younes
, James Turner
, Manoranjan Majji
, John Junkins

Published in Recent Advances in Algorithmic Differentiation

Editor(s) Shaun Forth, Paul Hovland, Eric Phipps, Jean Utke, Andrea Walther 
Year 2012 
Publisher Springer 
Abstract 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 closedform 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 MonteCarlo methods to gain insight into a system’s 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 seriesbased 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 FokkerPlanck 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. 
CrossReferences Forth2012RAi 
AD Theory and Techniques Higher Order, Uncertainties 
BibTeX
@INCOLLECTION{
Younes2012HOU,
title = "HighOrder Uncertainty Propagation Enabled by Computational Differentiation",
doi = "10.1007/9783642300233_23",
author = "Ahmad Bani Younes and James Turner and Manoranjan Majji and John Junkins",
abstract = "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 closedform 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 MonteCarlo methods to gain insight into a system’s
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 seriesbased 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 FokkerPlanck 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.",
pages = "251260",
crossref = "Forth2012RAi",
booktitle = "Recent Advances in Algorithmic Differentiation",
series = "Lecture Notes in Computational Science and Engineering",
publisher = "Springer",
address = "Berlin",
volume = "87",
editor = "Shaun Forth and Paul Hovland and Eric Phipps and Jean Utke and Andrea Walther",
isbn = "9783540689355",
issn = "14397358",
year = "2012",
ad_theotech = "Higher Order, Uncertainties"
}
 
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