Efficient Computational Methods to Treat Multiple Scattering in Electron Diffraction by Nanostructures

Our purpose is to extend the capabilities of surface structure determination methods, such as Low Energy Electron Diffraction, so they can be used for nanostructures. To treat non-periodic systems, a cluster approach is used. The main computational challenge consists in solving a Ax=b matrix-vector equation of large dimension. Since matrix inversion is both memory and compute-time demanding, we have developed and tested two fast iterative methods to solve the above equation: the Sparse-Matrix Canonical Grid (SMCG) method shifts the atoms to a regular space grid and makes use of FFT transformations while the Multi-Level Singular-Value Decomposition (MLSVD) performs fast rank determination and SV decomposition of A. For both these methods, the compute time scales as N x log$_{2}$N per iteration, where N is the number of atoms. These two methods complement each other in terms of the types of nanostructures that they handle best.