Calculus of variations for estimation in ODE–PDE landslide-like models with discrete-time asynchronous measurements
Résumé
Motivated by some landslide models, and related estimation challenges, this paper presents an optimalestimation method for state and parameter in a special class of so-called ODE–PDE system based on theadjoint method for discrete-time asynchronous measurements. This system is described by a pair of cou-pled Ordinary Differential Equation (ODE) and Partial Differential Equation (PDE), with a mixed boundarycondition for the PDE. The coupling appears both in the ODE and in the Neuman boundary condition ofthe PDE. For this system, initial conditions or state variables and some empirical parameters are assumed tobe unknown and need to be estimated. The Lagrangian multiplier method is used to connect the dynam-ics of the system and the cost function defined as the least square error between the simulation valuesand the available measurements. The adjoint state method is applied to the objective functional to getthe adjoint system and the gradients with respect to parameters and initial state. The cost functional isoptimised, employing the steepest descent method to estimate parameters and initial state. This gen-eral approach is illustrated by two application examples corresponding to two different landslide modelsthat validate the presented optimal estimation approach. The first one is about state and parameter esti-mation in an extended sliding-consolidation landslide model, and the second one is in the viscoplastic sliding-consolidation landslide model.