This was part of Computational Challenges and Optimization in Kinetic Plasma Physics

Quantized tensor trains for solving the Vlasov-Maxwell equations

Erika Ye, Lawrence Berkeley National Laboratory

Thursday, February 22, 2024


The quantized tensor network (QTN) format is a low-rank framework which enables the efficient calculation and manipulation of high-dimensional datasets. For QTNs of rank $D$ representing problems of size $N$, computational costs typically scale like $sim mathcal{O}(text{poly}(D) log(N))$. QTNs have been successfully applied to a wide range of numerical simulation problems such as finding extremal eigenvalues, solving linear systems, and even performing time integration of nonlinear partial differential equations; solving them with reasonable accuracy while significantly reducing computational costs. Here, we discuss the application of QTNs for solving the Vlasov-Maxwell equations, and show results for common test problems in 2D3V. In particular, we find that QTNs of rank $D=64$ appears to be sufficient for capturing the expected physics, despite the simulations using a total of $2^{36}$ grid points and thus requiring $D=2^{18}$ for exact calculations. Additionally, we utilize a QTN time evolution scheme based on the Dirac-Frenkel variational principle, which allows us to use larger time steps than that prescribed by the Courant-Friedrichs-Lewy (CFL) constraint. As such, the QTN format appears to be a promising means of approximately solving the Vlasov-Maxwell equations with significantly reduced cost.