An Inverse Boundary Value Problem for the Porous Medium Equation: Numerical Methods

Abstract

Inverse boundary value problems aim to infer a material's properties from boundary measurements. This article addresses an inverse problem for the porous medium equation, focusing on determining two coefficients characterizing the medium's diffusivity and porosity. As this problem has mainly been discussed in theory, a main contribution is the development of efficient numerical approaches for solving this problem. Based on the Dirichlet-to-Neumann map's injectivity, the diffusion coefficient's inversion can be related to the Calder\'on problem, leveraging its favorable properties. A key advantage of the proposed method is the decoupled inversion approach: After inferring the diffusion coefficient, the original problem only has to be solved for the porosity coefficient which can be done using a Bayesian inversion approach. This article lays the foundation for developing numerical methods solving the considered problem, highlighting the benefits of the decoupled strategy. Numerical frameworks are established for both the classical and Bayesian inversion of the diffusion coefficient. In the classical setting, noise-free measurements lead to a well-posed inverse problem. For noisy data, the Bayesian inversion maintains the problem's well-posedness.