629f Propagation of Uncertainties in Nonlinear Dynamic Models

Scott Ferson¹, Youdong Lin², George F. Corliss³, and Mark A. Stadtherr². (1) Applied Biomathematics, 100 North Country Road, Setauket, NY 11733, (2) Chemical and Biomolecular Engineering, University of Notre Dame, 181 Fitzpatrick Hall, Notre Dame, IN 46556, (3) Electrical and Computer Engineering, Marquette University, 1515 W. Wisconsin Avenue, Milwaukee, WI 53233

Engineering analysis and design problems, either static or dynamic, frequently involve uncertain parameters and inputs. Propagating these uncertainties through a complex model to determine their effect on system states and outputs can be a challenging problem, especially for nonlinear ODE models.

Probability distributions describing the uncertainties may not be known precisely, if known at all. If there are only upper and lower bounds on the uncertainties but no known probability distribution, then this can be represented by an interval. If there is some knowledge of the probability distribution, but it is uncertain, then this can be represented by a probability box (p-box). A p-box [1] is a pair (F, G) of bounding cumulative distribution functions identified with the class of all distributions {H: F(x) ≥ H(x) ≥ G(x), for all real x} whose graphs lie within the bounds. A p-box can represent epistemic uncertainty about a distribution for which there is insufficient detail available to specify its parameters or shape precisely. Operations that generalize the convolution of distributions can be defined for p-boxes such that a guarantee that p-boxes A and B enclose respective unknown distributions can be translated into a guarantee that the sum (or difference, product, etc.) of random variates drawn from these two distributions will have a distribution that is enclosed by a p-box that is easily computed from A and B. Arithmetic with p-boxes is thus possible and convenient, and has led to their wide use in probabilistic risk assessments where both stochastic variability and epistemic uncertainty (partial ignorance) are pervasive features.

Recently, Lin and Stadtherr [2] have described and implemented a new validating solver for parametric ODEs (VSPODE). Using VSPODE, it is possible to obtain a Taylor model representation [3] of the state variables and outputs in terms of the uncertain quantities. A Taylor model consists of a Taylor polynomial function and an interval remainder bound. In this presentation, we explore the use of Taylor models for propagating uncertainties in nonlinear ODE models. The case in which uncertainties are interval-valued has been discussed by Lin and Stadtherr [4]. We concentrate here on uncertainties represented by p-boxes. The approach is to decompose the p-box problem into a Cartesian product, each of whose elements is an interval problem. This allows us to obtain p-box representations of the uncertainties in the state variables and outputs. Examples are used to demonstrate the potential of this approach for studying the effect of uncertainties with imprecise probability distributions.

References

[1] S. Ferson, RAMAS Risk Calc 4.0 Software: Risk Assessment with Uncertain Numbers. Lewis Publishers, Boca Raton, FL.

[2] Y. Lin and M. A. Stadtherr, Validated Solution of ODEs with Parametric Uncertainties, 9th International Symposium on Process Systems Engineering/16th European Symposium on Computer Aided Process Engineering (PSE'06/ESCAPE-16), Garmisch-Partenkirchen, Germany, July 2006.

[3] A. Neumaier, Taylor Forms -- Use and Limits, Reliable Computing, 9, 43-79 (2002).

[4] Y. Lin and M. A. Stadtherr, Validated Solution of Initial Value Problems for ODEs with Interval Parameters, 2nd NSF Workshop on Reliable Engineering Computing, Savannah, Georgia, February 2006.