Towards the Use of Anderson Acceleration in Coupled Transport-Gyrokinetic Turbulence Simulations

Abstract

Predicting the behavior of a magnetically confined fusion plasma over long time periods requires methods that can bridge the difference between transport and turbulent time scales. The nonlinear transport solver, Tango, enables simulations of very long times, in particular to steady state, by advancing each process independently with different time step sizes and couples them through a relaxed iteration scheme. We examine the use of Anderson Acceleration (AA) to reduce the total number of coupling iterations required by interfacing Tango with the AA implementation, including several extensions to AA, provided by the KINSOL nonlinear solver package in SUNDIALS. The ability to easily enable and adjust algorithmic options through KINSOL allows for rapid experimentation to evaluate different approaches with minimal effort. Additionally, we leverage the GPTune library to automate the optimization of algorithmic parameters within KINSOL. We show that AA can enable faster convergence in stiff and very stiff tests cases without noise present and in all cases, including with noisy fluxes, increases robustness and reduces sensitivity to the choice of relaxation strength.

Figures (7)

• Figure 1: In Algorithm \ref{['alg:lodestro']}, the
• Figure 2: Heat map of the number of iterations to reach a residual of $10^{-11}$. Empty cells correspond to failed solves. Without a delay, $m \geq 2$ is needed to see a benefit in iteration counts from AA. With a one iteration delay, AA consistently reduces the number of iterations and decreases sensitivity to the value of $\beta$.
• Figure 3: Residual history for autotuned configurations. The default (faded circles) and one setup from Fig. \ref{['fig:heatmap_delay']} (faded stars) are included for reference. The autotuned results align well with the parameter sweep in Fig. \ref{['fig:heatmap']}.
• Figure 4: Residual history for autotuned configurations with adaptive $\beta$ and/or $m$. The adaptive results closely follow the autotuned fixed parameter results in Fig. \ref{['fig:stiff_case_optimized']} (faded triangles and squares included for reference).
• Figure 5: Residual (left axis) and parameter history (right axis) for adaptive $\beta$ and/or $m$ configurations. Adapting $m$ automatically selects the AA delay, and adapting $\beta$ alone leads to values near the max allowed. Adapting $\beta$ and $m$ together leads to large, frequent changes in $\beta$ slightly degrading performance compared to other setups.