Estimation Problems In Advective-Diffusivereactive Phenomena Using Meshless Numerical Methods Combined With Bayesian Inference

Abstract

The mathematical modeling of advective-diffusive-reactive phenomena finds numerous applications in various scientific fields, such as the transport of pollutants and adsorption columns. Mesh reduction techniques have proven efficient and have been gaining prominence in the literature. However, despite all the progress observed, some things could be improved in dealing with complex partial differential equations. With these limitations, variations of these methods emerged and sought to deal with complex systems. The present thesis proposal involves the development of a numerical method that combines the Eulerian-Lagrangian Method (ELM) with the Method of Fundamental Solutions (MFS) to solve a series of examples modeled by the transport equation. In addition, Bayesian inference methodologies, such as particle filters, which allow the estimation of states and model parameters, will be considered, providing an inverse approach to the problem. The results contemplated the solution of benchmark cases, which have an analytical solution for evaluating the proposed method, showing accurate and stable results when tested against different Peclet numbers between 0.5-200. The method sensitivity to parameters, such as node number and positioning, was also investigated. Its performance was evaluated against metrics such as root mean squared error and absolute error. We also performed manipulations to original models to address the reaction term and extend the cases to high-dimensionalities and complex geometries using the proposed methodology.