High-frequency electromagnetic radiation
NWO - OCENW.KLEIN.183
The Radiative Transfer Equation (RTE) has been established as an important mathematical model in relevant applications, such as tumor treatment using radiation therapy, photoacoustic imaging, generation of white light, optics and astrophysics. For problems of practical relevance numerical approximations are required, but traditional methods suffer from the presence of non-smooth solutions and the curse of dimensionality, which describes the phenomenon that the dimensionality of the RTE enters the computational complexity exponentially, and prevents the use of the aforementioned methods in daily life. The main goal of the project is to break the curse of dimensionality by developing novel numerical methods for the RTE which adhere to the paradigm of low-rank tensor product calculus. In particular, the following approaches are addressed.
- Exploration of a new variational formulation of the RTE which allows to exploit the natural tensor product structure of the underlying phase-space.
- Development of low-rank tensor product approximations framework.
- Development of Uzawa iterations compatible with the tensor format and modified suitably in order to improve efficiency (through rank-control and preconditioning techniques).