Joint state-parameter estimation and inverse problems governed by reaction–advection–diffusion type PDEs with application to biological Keller–Segel equations and pattern formation

Abstract

Inverse problems aim to find the causes of outcoming features knowing the consequences of a model by calibrating the model’s parameters to fit data. In this paper, we present a method that solves simultaneously the inverse problem and the state estimation problem associated with nondegenerate anisotropic reaction–advection–diffusion systems, combined with a smooth observation operator, and showcase it on two examples: a Keller–Segel system used for the chemotaxis, and a Turing system producing stable spatial patterns. The method is defined as an optimization problem that minimizes the misfit formulated with three different types of error: on the modelling choices, on the initial state assumption, and on the difference between data and the forward predictive model output. The resolution of the corresponding inverse problem relies on the rewriting of the variational system and involves solving the forward system while nullifying a vector-valued function that represents the optimality of the coefficients. From a numerical perspective, we approach the inverse problem by adjusting both the state and parameter vectors using sparse temporal data. Instead of employing a classical Newton algorithm, we exploit strategic numerical schemes to effectively handle the resulting coupled system. Numerical experiments in one- and two-dimensional physical domains have been performed with synthetic data to evaluate the efficiency of the proposed method, but also to describe the influence of hyperparameters on the inverse problem.