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. \vskip .1in \noindent 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.