Inverse modeling of diffusion-reaction processes with image-type measurement data

Abstract

The inverse source problem for 1D diffusion-reaction model is considered. The measurement data is given as the images of the concentration fields dynamics for the subset of the interacting species. These inverse problems arise in the study of the growing tissues (morphogenes theory), in the development of the tissue engineering technologies and in the other fields of modern mathematical biology. The sensitivity operator, composed of the ensemble of the independent adjoint problem solutions allow to transform the inverse problem to the family of nonlinear ill-posed integral equations. Each member of the family correspond to the image to structure operator that extracts certain features of the image. An equation from the family is solved with the Newton-Kantorovich-type algorithm combining truncated SVD and iterative regularization. Due to the design with the adjoint problems ensemble, the algorithm can be efficiently parallelized. The algorithm’s convergence and stability are illustrated numerically in Brusselator model case.