XRF map identification problems based on a PDE electrodeposition model

In this paper we focus on the following map identification problem (MIP): given a morphochemical reaction–diffusion (RD) PDE system modeling an electrodepostion process, we look for a time t*, belonging to the transient dynamics and a set of parameters p, such that the PDE solution, for the morphology h(x, y, t^*; p) and for the chemistry \theta(x, y, t^*; p) approximates a given experimental map M*. Towards this aim, we introduce a numerical algorithm using singular value decomposition (SVD) and Frobenius norm to give a measure of error distance between experimental maps for h and θ and simulated solutions of the RD-PDE system on a fixed time integration interval. The technique proposed allows quantitative use of microspectroscopy images, such as XRF maps. Specifically, in this work we have modelled the morphology and manganese distributions of nanostructured components of innovative batteries and we have followed their changes resulting from ageing under operating conditions. The availability of quantitative information on space-time evolution of active materials in terms of model parameters will allow dramatic improvements in knowledge-based optimization of battery fabrication and operation.