Mathematical modeling of parameter identification process of convection-diffusion transport models using the SVD-based Kalman filter

The paper addresses a problem of mathematical modeling of the process of identifying the coefficients of a partial differential equation in convection-diffusion transport models based on the results of noisy measurements of the function values. Identification process is performed using a new method belonging to the class of recurrent parameter identification methods based on optimal discrete Kalman-type filtering algorithms. One-dimensional models with constant coefficients, boundary conditions of first kind, or mixed boundary conditions of first and third kind are considered.
The proposed method is based on the transition from the initial continuous model with a partial differential equation to the model described by the state-space linear discrete-time dynamic system and the application of the maximum likelihood method to it with construction of an identification criterion (likelihood function) based on the values calculated by the SVD algorithm of the Kalman filtering. This filter is based on the singular value decomposition of error covariance matrix and works stably even in cases when it is close to singular. The SVD filter has proven itself well in solving various problems of discrete filtering and parameter identification. It has several advantages over the traditionally used conventional Kalman filter. The main of which is robustness against machine roundoff errors.
Computer modeling of parameter identification has been processed with the MATLAB system using a specialized software package. The results of numerical experiments confirm the efficiency of the proposed method and its advantages compared to the similar one based on the conventional Kalman filter.