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arXiv · 2611.03859

Factorized approximate inverse preconditioners in the tensor train format

Pau Torrente Badia·Daniel Pérez García·Artur Garcia-Saez·Stefano Carignano
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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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Pau Torrente Badia, Daniel Pérez García, Artur Garcia-Saez, Stefano Carignano. 2026-11-02. Factorized approximate inverse preconditioners in the tensor train format. https://arxiv.org/abs/2611.03859

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