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.