The radiative transfer equation (RTE) has been established as an important mathematical model in relevant societal applications, such as tumor treatment, photoacoustic imaging, and generation of white light. For problems of practical relevance numerical approximations are required. Since the RTE is high-dimensional and features non-smooth solutions, the construction and analysis of efficient numerical methods faces serious mathematical challenges. The main objective of this proposal is the development of novel numerical methods for the RTE that break the curse of dimensionality. We will build upon our recently derived variational formulation of the RTE that, for the first time, decouples the phase-space such that the involved operators are short sums of tensor products. This formulation is the natural starting point to explore and develop powerful low-rank tensor product approximations for the RTE that have the potential to reduce the computational complexity tremendously. We propose to develop low-rank tensor product approximations using tensors of order 2, which can reduce the computational complexity essentially to that of three-dimensional problems. Also, for certain parameter configurations, tensors of higher order will be used that can reduce the computational complexity to that of one-dimensional problems. For the efficient solution of the resulting saddle-point problems, we propose to develop preconditioned Uzawa iterations that are compatible with the used tensor format. The explicit form of the Uzawa iterations will be used to develop rank control techniques, which iteratively reveal the optimal rank given a desired tolerance and which give a proper balance between accuracy of the intermediate approximations in the iterative procedure and computational complexity.

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Photonic crystals are optical nanostructures that contain a bandgap, a range of frequencies where light is perfectly reflected, which makes them well suited to act as a back reflector in solar cells. Photonic crystals are difficult to manufacture, and manufacturing inaccuracies can have a significant impact on their optical properties. This project is the first to use the actual geometry of manufactured photonic crystals, obtained from X-ray experiments, to compute the behavior of light in these nanostructures. These computations give the first detailed insight into the effects of manufacturing inaccuracies of the behavior of light in photonic crystals.