JuliaCon 2026

Electron Probe Microanalysis (EPMA) is an imaging technique for the quantitative analysis of solid material samples relying on measurements of characteristic X-ray emission induced by electron irradiation.
The determination of the material constitutes an inverse problem, hence an efficient reconstruction requires differentiability of the forward model.
The mathematical model employed in EPMA is governed by a linear transport equation that for heterogeneous materials is commonly approximated using Monte Carlo simulation, where statistical noise complicates the computation of gradients.
For reconstruction, there exist surrogate models that are well tested in practice, but are very restrictive in the parametrization of the material, allowing only homogeneous or depth-layered materials, which ultimately limits the spatial resolution of quantitative analysis in EPMA.

In this short talk, we present an implementation of a deterministic, heterogeneous, and differentiable model for EPMA in Julia.
Reconstruction can then be implemented as a gradient-based optimization using the model as a PDE constraint.
Compatibility with algorithmic differentiation allows us to tailor the material parametrization to a set of quantities of interest, depending on the requirements of a specific sample.
Reconstruction results using realistic as well as synthetic measurements demonstrate potential for further development.

Additionally, we briefly discuss a structural similarity of the forward model in EPMA to the structure in which adjoint methods can be effectively applied for gradient computation. It allows us to "apply adjoints twice" leading also to a more efficient computation of the forward problem, ultimately accelerating reconstruction approaches.