Nanosimulations present a big HPC challenge as they present increasing performance demands in heterogeneous execution environments. In this paper, we present our optimization methodology for BigDFT, a nanosimulation software using Density Functional Theory. We explore autotuning possibilities for BigDFT's 3D convolutions by studying optimization techniques for several architectures. Namely, we focus on processors with vector units and on GPU acceleration. We report on the portability and the performance gains of our approach (speedup x2 on CPU, x5 on GPU) and discuss the relation between algorithmic specifics, architecture and performance.