Abstract: Group invariants are used in high energy physics to define quantum field theory interactions. In this paper, we present the parallel algebraic computation of special invariants called symplectic and focus on one particular invariant that finds recent interest in physics. Our results will export to other invariants. The cost of performing basic computations on the multivariate polynomials evolves during the computation, as the polynomials get larger and/or have increasing numbers of terms. Interestingly, in some cases, they stay small. Traditionally, high-performance software is optimized by running experiments with sample data sets in order to profile and optimize expected behavior of workloads in practice. Since the (communication and computation) costs depend on the changing behavior of the symplectic invariant calculations, the standard optimization approach is insufficient. Thus, it is necessary to implement online performance tuning methods that can track the algorithm’s progress and state, evaluate performance data in situ, and control the parallel resources during execution.