Multilevel Adaptive-Rank Methods for Linear and Nonlinear Systems in the Hierarchical Tucker Format
It addresses the challenge of solving high-dimensional PDEs by extending low-rank tensor methods with multilevel preconditioning, offering efficient solvers for computational scientists.
The paper develops multilevel adaptive-rank iterative methods for solving high-dimensional PDEs in the hierarchical Tucker format, achieving robust and scalable preconditioners for both linear and nonlinear systems.
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.