PhD Defence Riccardo Bardin | On adaptive and flexible numerical methods for the approximation of radiative transfer problems

On adaptive and flexible numerical methods for the approximation of radiative transfer problems

Riccardo Bardin is a PhD student in the Department Mathematics of Computational Science. (Co)Promotors are prof.dr.ir. J.J.W. van der Vegt and dr. M. Schlottbom from the Faculty of Electrical Engineering, Mathematics and Computer Science.

This thesis presents significant advancements in the numerical approximation of radiative transfer problems, with a particular emphasis on adaptive, low-rank, and accelerated iterative techniques.

Firstly, an adaptive discontinuous Galerkin method is developed for the radiative transfer equation in slab geometry with isotropic scattering. This approach departs from traditional tensor-product discretizations by employing a non-tensor product discretization strategy, allowing for local mesh refinement. The adaptive scheme is guided by error estimators, effectively capturing phase-space singularities and ensuring optimal convergence rates in case of point singularities, and sub-optimal, but consistent with the behaviour of the true error, convergence rates for line discontinuities. This method shows, at least numerically, substantial improvements in computational efficiency while maintaining high accuracy.

The next achievement involves the development of a low-rank tensor product framework for the radiative transfer equation in slab geometry and with isotropic scattering, in order to address the computational complexity inherent to tensor-product discretizations. By leveraging low-rank structures, the approach reduces the dimensionality and memory requirements, combining a preconditioned Richardson iteration with a rank-compression technique that allows to control the ranks of the iterates. This framework proves to be reliable and flexible, since it admits various kinds of discretization and allows for error control in the energy norm using the Euclidean norm of computable quantities.

Finally, the thesis introduces an acceleration technique for iterative solvers used for general multidimensional anisotropic radiative transfer problems. By employing residual minimization over suitable subspaces, the method substantially accelerates the standard source iteration scheme. This enhancement is particularly relevant for scenarios involving highly forward-peaked scattering, where conventional solvers face convergence issues. The proposed method shows robust performance, offering a flexible and effective solution that significantly reduces the number of iterations required for convergence.

Together, these contributions represent a comprehensive and impactful advancement in the numerical treatment of radiative transfer problems, by addressing key challenges in adaptivity, computational efficiency, and solver acceleration.