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Methods based on the many-body Green's function are generally accepted as the path forward beyond Kohn-Sham based density functional theory, in order to compute from first principles electronic structure of materials with strong correlations and excited-state properties in nano- and materials science. Here we present an efficient method to compute the screened Coulomb interaction W, the crucial and computationally most demanding ingredient in the GW method, within the framework of the all-electron Linearized Augmented Plane Wave method. We use the method to compute from first principles, within the constrained random phase approximation (c-RPA), the frequency-dependent screened Hubbard U-matrix defined for a Wannier basis in which we downfold the many-body Hamiltonian for La2CuO4, the canonical parent compound of several cuprate high-temperature superconductors. These results were computed at scale on the Cray XT5 at ORNL, sustaining 1.30 petaflop. We discuss the details of the algorithm and its implementation that allowed us to reach high efficiency and short time to solution on today's petaflop supercomputers.