Fast and reliable methods for determining the evolution of uncertain parameters in differential equations

A very common problem in science and engineering is the determination of the effects of uncertainty or variation in parameters and data on the output of a deterministic nonlinear operator. For example, such variations may describe the effect of experimental error or may arise as part of a sensitivity analysis of the model. The Monte-Carlo Method is a justifiably popular tool for analyzing such effects. It does, however, require many evaluations of the operator and it is difficult to extract precise information about the accuracy of any particular result. In this paper, we borrow techniques from a posteriori error analysis for finite element methods to compute information about the derivative of an operator with respect to its parameters. These techniques employ the generalized Green’s function to describe how variation propagates into the solution around localized points in the parameter space. We show how this derivative information can be used either to create a higher order method or produce an error estimate for information computed from a given representation. In the latter case, this provides the basis for adaptive sampling according to the variation in the output values. Both the higher order method and the adaptive sampling method are generally orders of magnitude faster than Monte-Carlo methods in the case that the parameter space is not too high dimensional.