William A. Sands

Nanyi Zheng, William A. Sands, Jing-Mei Qiu

We propose a semi-Lagrangian adaptive-rank (SLAR) method that combines the large time-step capability of semi-Lagrangian schemes with the efficiency of adaptive-rank tensor representations while simultaneously enforcing local conservation laws for mass, momentum, and energy. The method builds on the high-dimensional SLAR framework introduced in our previous work and achieves high-order accuracy in both space and time. To address the loss of conservation in long-time simulations, we extend the implicit local macroscopic conservative (LoMaC) correction technique for the BGK equation to the high-dimensional Vlasov--Poisson (VP) system. The implicit macroscopic system is discretized using backward differentiation formulas and solved with a Jacobian-free Newton-Krylov method. This approach enables a consistent coupling with semi-Lagrangian methods which are capable of taking large time steps. A novel component of the proposed method is a unified adaptive-weight projection technique that eliminates the ad hoc parameter tuning required by previous LoMaC approaches. These weights capture problem-dependent velocity space structures and are constructed from the low-rank velocity bases of the solution. The local semi-Lagrangian method used in this work reconstructs the solution at the feet of the characteristics using efficient tensor contractions. To the best of our knowledge, this is the first successful implementation of an implicit LoMaC method for the VP system up to the 2D--2V setting. Numerical experiments on several classical benchmark problems demonstrate the accuracy and efficiency of the proposed method, as well as its ability to preserve conservation laws in VP simulations.

William A. Sands, Pierson T. Guthrey, Nathan V. Roberts

We develop multilevel adaptive-rank iterative methods for the solution of linear and nonlinear systems arising from high-dimensional partial differential equations. Our contributions are threefold. First, we extend the projection method of Ballani and Grasedyck [6] to enable flexible preconditioning of high-dimensional linear systems in low-rank tensor formats. Second, we construct multilevel preconditioning strategies by adapting geometric multigrid methods to the low-rank setting. In contrast to prior work, which primarily employs multigrid as a standalone solver, we emphasize its role as an efficient and robust preconditioner. Third, we integrate these techniques within an inexact Newton framework for the solution of nonlinear systems. The proposed methods are evaluated on a range of model problems, including both linear and nonlinear equations, to assess their convergence behavior and computational efficiency. The results demonstrate that multilevel adaptive-rank strategies yield robust and scalable preconditioners, providing effective solvers for high-dimensional problems in low-rank formats.