A Moment-Matching Method to Study the Variability of Phenomena Described by Partial Differential Equations

Abstract

Many phenomena are modeled by deterministic differential equations, whereas the observation of these phenomena, in particular in life sciences, exhibits an important variability. This paper addresses the following question: how can the model be adapted to reflect the observed variability? Given an adequate model, it is possible to account for this variability by allowing some parameters to adopt a stochastic behavior. Finding the parameter probability density function that explains the observed variability is a difficult stochastic inverse problem, especially when the computational cost of the forward problem is high. In this paper, a nonparametric and nonintrusive procedure based on offline computations of the forward model is proposed. It infers the probability density function of the uncertain parameters from the matching of the statistical moments of observable degrees of freedom (DOFs) of the model. This inverse procedure is improved by incorporating an algorithm that selects a subset of the model DOFs that both reduces its computational cost and increases its robustness. This algorithm uses the precomputed model outputs to build an approximation of the local sensitivities. The DOFs are selected so that the maximum information on the sensitivities is conserved. The proposed approach is illustrated with elliptic and parabolic partial differential equations.