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 parameters 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 non-parametric and non-intrusive 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 pre-computed 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 PDEs. In the Appendix, an nonlinear ODE is considered and the strategy is compared with two existing ones.

Modelling variability in cardiac electrophysiology: a moment-matching approach

“…This has not only the advantage of capturing all of the available information but also the disadvantage of increasing the computational cost of the inverse procedure because the number of time steps may be large. To tackle this issue, a time-step selection algorithm was developed and is described in [ 20 ]. It uses the pre-computed simulation database to approximate the sensitivities with respect to each parameter and for each time step.…”

“…This approximation is, however, not meant to reach the precision of finer methods such as MCMC. The interested reader is referred to [ 20 ] where more details are provided.…”

“…It is also well suited to our optimization problem for practical mathematical reasons. In Section 2.4.2. of [ 20 ], we propose an analysis of the error on the PDF estimation made by adopting the maximum entropy regularization. In Prop.…”

“…The estimated PDF is interpolated on the finer grid using kernel smoothing. This step is discussed in detail in [ 20 ]. In addition to the marginal densities, the estimated parameter moments μ m ( θ p ) may also be computed directly from the PDF: Then, one can compare μ 1 ( θ p ), the mean of parameter p and its standard deviation to their true values when known.…”

“…More precisely, the estimated PDF is the solution of a constrained optimization problem which is an adaptation of the maximum entropy principle [ 19 ]. The method, later referred to as observable moment matching (OMM), is detailed in [ 20 ]. Contrary to other approaches such as Monte–Carlo Markov Chains (MCMC) [ 21 ] or Approximate Bayesian computation [ 22 ], the present method does not guarantee to converge to the true parameters distribution.…”