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

Abstract

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. In this paper, we present an alternative approach for ascertaining the effects of variations and uncertainty in parameters in a reaction-diffusion equation on the solution. The approach is based on techniques borrowed from a posteriori error analysis for finite element methods. The generalized Green’s function solving the adjoint problem is used to efficiently compute the gradient of a quantity of interest with respect to parameters at sample points in the parameter space. This derivative information is used in turn to produce an error estimate for the information computed from the sample points. The error estimate provides the 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. After describing the new methodology in detail, we apply the new method to analyze the parameter sensitivity of a predator–prey model with a Holling II functional response that has six parameters.