An important 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. The Monte-Carlo method is a widely used tool for determining such effects. It employs random sampling of the input space in order to produce a pointwise representation of the output. It is a robust and easily implemented tool. Unfortunately, it generally requires sampling the operator very many times. Moreover, standard analysis provides only asymptotic or distributional information about the error computed from a particular realization.

We present an alternative approach for this problem that is based on techniques borrowed from a posteriori error analysis for finite element methods. Our approach allows the efficient computation of the gradient of a quantity of interest with respect to parameters at sample points. This derivative information is used in turn to produce an error estimate for the information, thus providing a basis for both deterministic and probabilistic adaptive sampling algorithms. The deterministic adaptive sampling method can be orders of magnitude faster than Monte-Carlo sampling in case of a moderate number of parameters. The gradient can also be used to compute useful information that cannot be obtained easily from a Monte-Carlo sample. For example, the adaptive algorithm yields a natural dimensional reduction in the parameter space where applicable.