Implementations of AO tomography for the next generation of Extremely Large Telescopes (ELTs) is challenging because of the extremely large number of degrees of freedom of such systems, in particular when it comes to the tomographic reconstructor computation, due to its size. The computation of this matrix, via the supervisor module, requires leveraging high performance computing techniques, on shared or distributed memory systems, to comply with the specifications of tomographic AO systems, which prescribe an update rate of the order of few minutes. In the scope of the Green-Flash project, we are exploring several approaches to optimize the execution of this soft real-time supervision pipeline. This includes low-rank techniques to reduce the computational load. We have tested several compression schemes to optimize the linear algebra involved in the tomographic reconstructor as well as the computation of the covariance matrices involved in this process. We present, in this paper, the scalable and portable pipeline we have developed to address these issues. Performance in terms of time to solution and scalability are reported. Additionally, the case of low-rank algorithms is stressed as both an attempt to address the computation challenge of the tomographic reconstructor for the supervisor module, and a way to reduce the computational load (hence the overall RTC system latency) at the level of the real-time data pipeline.