Calculus of variations for estimation in ODE-PDE landslide-like models with discrete-time asynchronous measurements

Motivated by some landslide models, and related estimation challenges, this paper presents an optimal estimation method for state and parameter in a special class of so-called ODE-PDE system based on the adjoint method for discrete-time asynchronous measurements. This system is described by a pair of coupled Ordinary Differential Equation (ODE) and Partial Differential Equation (PDE), with a mixed boundary condition for the PDE. The coupling appears both in the ODE and in the Neuman boundary condition of the PDE. For this system, initial conditions or state variables and some empirical parameters are assumed to be unknown and need to be estimated. The Lagrangian multiplier method is used to connect the dynamics of the system and the cost function defined as the least square error between the simulation values and the available measurements. The adjoint state method is applied to the objective functional to get the adjoint system and the gradients with respect to parameters and initial state. The cost functional is optimised, employing the steepest descent method to estimate parameters and initial state. This general approach is illustrated by two application examples corresponding to two different landslide models that validate the presented optimal estimation approach. The first one is about state and parameter estimation in an extended sliding-consolidation landslide model, and the second one is in the viscoplastic sliding-consolidation landslide model.

Automated summary

This paper presents an optimal estimation method for state and parameter in a special class of ODE-PDE system based on Lagrangian multiplier and adjoint state method. The effectiveness of this method is validated through two application examples of different landslide models.