Papers › Stability of Low-Rank Tensor Representations and Structured Multilevel Preconditioning...
Stability of Low-Rank Tensor Representations and Structured Multilevel Preconditioning for Elliptic PDEs
Markus Bachmayr, Vladimir Kazeev
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
Folding grid value vectors of size 2ᴸ into Lth order tensors of mode sizes 2×⋯×2, combined with low-rank representation in the tensor train format, has been shown to lead to highly efficient approximations for various classes of functions. These include solutions of elliptic PDEs on nonsmooth domains or with oscillatory data. This tensor-structured approach is attractive because it leads to highly compressed, adaptive approximations based on simple discretizations. Standard choices of the underlying basis, such as piecewise multilinear finite elements on uniform tensor product grids, entail the well-known matrix ill-conditioning of discrete operators. We demonstrate that for low-rank representations, the use of tensor structure itself additionally introduces representation ill-conditioning, a new effect specific to computations in tensor networks. We analyze the tensor structure of a BPX preconditioner for second-order linear elliptic operators and construct an explicit tensor-structured representation of the preconditioner, with ranks independent of the number L of discretization levels. The straightforward application of the preconditioner yields discrete operators whose matrix conditioning is uniform with respect to the discretization parameter, but in decompositions that suffer from representation ill-conditioning. By additionally eliminating certain redundancies in the representations of the preconditioned discrete operators, we obtain reduced-rank decompositions that are free of both matrix and representation ill-conditioning. For an iterative solver based on soft thresholding of low-rank tensors, we obtain convergence and complexity estimates and demonstrate its reliability and efficiency for discretizations with up to 2⁵⁰ nodes in each dimension.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections