Factorized approximate inverse preconditioners in the tensor train format
Abstract
We present a preconditioning pipeline that accelerates Tensor Train (TT) linear solvers for symmetric positive-definite problems. Extending the Factorized Sparse Approximate Inverse framework to tensor networks, we construct a preconditioner as the product of a TT operator and its transpose. This formulation is positive semidefinite for any bond dimension and truncation, and definite whenever the factor is non-singular, satisfying the mathematical requirements of the Conjugate Gradient method. We detail the construction of this preconditioner and its integration into Density Matrix Renormalization Group (DMRG)-like linear solvers. Performance evaluations show consistent iteration reductions for anisotropic diffusion and Poisson problems across various configurations, leading to net speedups of up to 25% in ill-conditioned settings of time stepping schemes that can reuse the approximate inverses. These benchmarks were performed with the open-source Julia library DifferenTT.jl, which we also introduce in this work to provide out-of-the-box access to these preconditioned pipelines.
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BibTeX
@misc{badia2026factorized,
title = {{Factorized approximate inverse preconditioners in the tensor train format}},
author = {Pau Torrente Badia and Daniel Pérez García and Artur Garcia-Saez and Stefano Carignano},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03859},
eprint = {2611.03859},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03859},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Factorized approximate inverse preconditioners in the tensor train format
AU - Pau Torrente Badia
AU - Daniel Pérez García
AU - Artur Garcia-Saez
AU - Stefano Carignano
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03859
UR - https://arxiv.org/abs/2611.03859
AB - We present a preconditioning pipeline that accelerates Tensor Train (TT) linear solvers for symmetric positive-definite problems. Extending the Factorized Sparse Approximate Inverse framework to tensor networks, we construct a preconditioner as the product of a TT operator and its transpose. This formulation is positive semidefinite for any bond dimension and truncation, and definite whenever the factor is non-singular, satisfying the mathematical requirements of the Conjugate Gradient method. We detail the construction of this preconditioner and its integration into Density Matrix Renormalization Group (DMRG)-like linear solvers. Performance evaluations show consistent iteration reductions for anisotropic diffusion and Poisson problems across various configurations, leading to net speedups of up to 25% in ill-conditioned settings of time stepping schemes that can reuse the approximate inverses. These benchmarks were performed with the open-source Julia library DifferenTT.jl, which we also introduce in this work to provide out-of-the-box access to these preconditioned pipelines.
ER -