Fast methods for determining the evolution of uncertain parameters in reaction-diffusion equations Original Research Article

An important problem in mathematical modeling 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 in measured parameter values or may arise as part of a sensitivity analysis of the model. 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 with relatively low dependence on the number of parameters. Unfortunately, it generally requires sampling the operator very many times at a significant cost, especially when the model is expensive to evaluate. Moreover, standard analysis provides only asymptotic or distributional information about the error computed from a particular realization.