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Directors:
Randolph E. Bank
Philip E. Gill
Michael Holst

Administrative Contact:
Terry Le

Office: AP&M 7431
Phone: (858)534-9813
Fax: (858)534-5273
E-mail: tele@ucsd.edu
Fast and Reliable Methods for Determining the Evolution of Uncertain Parameters in Differential Equations

Donald Estep
CSU Mathematics

Abstract:

An important problem in science and engineering is the determinationof the effects of uncertainty or variation in parameters and data onthe output of a deterministic nonlinear operator. The Monte-Carlomethod is a widely used tool for determining such effects. It employsrandom sampling of the input space in order to produce a pointwiserepresentation of the output. It is a robust and easily implementedtool. Unfortunately, it generally requires sampling the operator verymany times. Moreover, standard analysis provides only asymptotic ordistributional information about the error computed from a particularrealization.

We present an alternative approach for this problem that is based ontechniques borrowed from a posteriori error analysis for finiteelement methods. Our approach allows the efficient computation of thegradient of a quantity of interest with respect to parameters atsample points. This derivative information is used in turn to producean error estimate for the information, thus providing a basis for bothdeterministic and probabilistic adaptive sampling algorithms. Thedeterministic adaptive sampling method can be orders of magnitudefaster than Monte-Carlo sampling in case of a moderate number ofparameters. The gradient can also be used to compute usefulinformation that cannot be obtained easily from a Monte-Carlo sample.For example, the adaptive algorithm yields a natural dimensionalreduction in the parameter space where applicable.

Thursday, December 1, 2005
4:00PM AP&M 7421