We present a model for dose calculation in photon radiotherapy based on deterministic transport equations. The model consists of two coupled equations, one for photon and one for electron transport and an equation for the absorbed dose. No assumptions are made with respect to the geometry or the homogeneity of the irradiated medium, so irradiation of any heterogenous medium can be simulated. To get a mathematically simpler model, approximations to the exact equations are presented which keep the essential physical contents of the exact equations, but which can be solved with less numerical effort. The results of dose calculations for simple examples are presented and discussed.

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We propose a new data analysis approach for the efficient post-processing of bundles of finite element data from numerical simulations. The approach is based on the mathematical principles of symmetry. We consider the case where simulations of an industrial product are contained in the space of surface meshes embedded in R 3 . Furthermore, we assume that distance preserving transformations exist, albeit unknown, which map simulation to simulation. In this setting, a discrete Laplace-Beltrami operator can be constructed on the mesh, which is invariant to isometric transformations and therefore valid for all simulations. The eigenfunctions of such an operator are used as a common basis for all (isometric) simulations. One can use the projection coefficients instead of the full simulations for further analysis. To extend the idea of invariance, we employ a discrete Fokker-Planck operator, that in the continuous limit converges to an operator invariant to a nonlinear transformation, and use its eigendecomposition accordingly.The data analysis approach is applied to time-dependent datasets from numerical car crash simulations. One observes that only a few spectral coefficients are necessary to describe the data variability, and low dimensional structures are obtained. The eigenvectors are seen to recover different independent variation modes such as translation, rotation, or global and local deformations. An effective analysis of the data from bundles of numerical simulations is made possible, in particular an analysis for many simulations in time. *

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