  For the reliable fabrication of the current and next generation of
nanostructures it is essential to be able to determine their material
composition and dimensional parameters. Using the grazing incidence X-ray
fluoresence technique, which is taking advantage of the X-ray standing wave
field effect, nanostructures can be investigated with a high sensitivity with
respect to the structural and elemental composition. This is demonstrated using
lamellar gratings made of Si$_3$N$_4$. Rigorous field simulations obtained from
a Maxwell solver based on the finite element method allow to determine the
spatial distribution of elemental species and the geometrical shape with sub-nm
resolution. The increasing complexity of nanostructures and demanded
sensitivity for small changes quickly turn the curse of dimensionality for
numerical simulation into a problem which can no longer be solved rationally
even with massive parallelisation. New optimization schemes, e.g. machine
learning, are required to satisfy the metrological requirements. We present
reconstruction results obtained with a Bayesian optimization approach to reduce
the computational effort.
