Estimation of Diffusion Coefficient in Gas Exchange Process with in Human Respiration Via an Inverse Problem

This paper is intended to provide a stochastic approach involving the combined use of the Feynman-Kac formula and Monte Carlo method as a solution algorithm for estimating the timedependent effective diffusivity in a one-dimensional parabolic inverse problem. The inverse problem is purposed to design a mathematical model for the gas-diffusion process within the alveolar region of the human's lung. The model depends on a representative physical property of the alveolar region termed the effective diffusivity. In the present study, the functional form of the effective diffusivity is unknown a priori. The unknown effective diffusivity is approximated by the polynomial form. To modify the coefficients of the polynomial form of the unknown effective diffusivity, we introduce a deterministic optimization method based on least squares minimization. A numerical test is performed in order to show the efficiency and accuracy of the present work.