This chapter presents a linear scaling ab initio method based on multiple scattering theories that shows clear advantage over other ab initio methods. It demonstrates its petascale computing capability in the ab initio calculation of magnetic and crystal structure phase transition temperatures. The chapter discusses its potential applications at exascale. The unit cell consists of the constituent atoms in a predetermined proportion and in a real-space distribution to mimic the atomic composition and spatial arrangement in the actual material. The construction of the local interaction zone itself requires a loop over all atom sites in our naive implementation. As significant amount of computational effort of ocally self-consistent multiple-scattering is spent in inverting the multiple-scattering matrix, a non-Hermitian complex matrix, this formed the main focus for porting to graphics processing units and the experience is guiding the portability to future platforms.