Solving the photon diffusion equation by FEM with Lanczos Algorithm

Abstract

In this thesis, finite element method (FEM) with Lanczos algorithm under uniform-grid and multi-grid meshes, denoted as FEML-U and FEML-M algorithms respectively, are introduced to solve the photon diffusion equation. In FEML-U and FEML-M algorithms, an n-dimensional state-space system is established by FEM, and the output of this system is approximated by that of an m-dimensional reduced system, which is generated by Lanczos algorithm. The implementations of FEML-U and FEML-M algorithms simulate the output at detectors of two given organ-size phantoms, and the corresponding simulators are validated. It is shown that the space and time complexities of FEML-U and FEML-M simulators are about O(n) and O(m*n) respectively. Compared to the uniform-grid alternating direction implicit algorithm (ADI-U), it is demonstrated that FEML-U simulations to the given organ-size phantoms are more than 50 times faster. Furthermore, without essentially losing accuracy, FEML-M simulations use at most 30% of space and 75% of time of FEML-U simulations.