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Fast and Reliable Methods for Determining the Evolution of Uncertain Parameters in Differential Equations

Donald Estep
CSU Mathematics


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