This paper is devoted to the numerical resolution of an inverse Cauchy problem governed by Stokes equation modeling the airflow in the lungs. It consists in determining the air velocity and pressure on the artificial boundaries of the bronchial tree. This data completion problem is one of the highly ill-posed problems in the Hadamard sense (Hadamard in Lectures on Cauchy’s problem in linear partial differential equations. Dover, New York, 1953). This gives great importance to its numerical resolution and in particular to carry out stable numerical approaches, mostly in the case of noisy data. The main idea of this work is to extend some regularizing, stable and fast iterative algorithms for solving this problem based on the domain decomposition approach (Chakib et al. in Inverse Prob 35(1):015008, 2018). We discuss the efficiency and the feasibility of the proposed approach through some numerical tests performed using different domain decomposition algorithms. Finally, we opt for the Robin–Robin algorithm, which showed its performance, for the numerical simulation of the airflow in the bronchial tree configuration.

本文致力于通过模拟肺中气流的斯托克斯方程控制柯西逆问题的数值分辨率。它包括确定支气管树人工边界上的空气速度和压力。这个数据完成问题是Hadamard意义上的病态严重的问题之一（Hadamard在线性偏微分方程的Cauchy问题讲座中，Dover，纽约，1953年）。这对于其数值分辨率非常重要，尤其是在大多数数据嘈杂的情况下，尤其是要执行稳定的数值方法。这项工作的主要思想是基于域分解方法扩展一些正则化，稳定和快速的迭代算法来解决此问题（Chakib等人，In Prob 35（1）：015008，2018）。通过使用不同的域分解算法进行的一些数值测试，我们讨论了该方法的效率和可行性。最后，我们选择了Robin-Robin算法，该算法显示了其性能，用于支气管树结构中气流的数值模拟。