Time integration of tensor trains

Christian Lubich, Ivan V. Oseledets, Bart Vandereycken

    Research output: Contribution to journalArticlepeer-review

    86 Citations (Scopus)


    A robust and efficient time integrator for dynamical tensor approximation in the tensor train or matrix product state format is presented. The method is based on splitting the projector onto the tangent space of the tensor manifold. The algorithm can be used for updating time-dependent tensors in the given data-sparse tensor train/matrix product state format and for computing an approximate solution to high-dimensional tensor differential equations within this data-sparse format. The formulation, implementation, and theoretical properties of the proposed integrator are studied, and numerical experiments with problems from quantum molecular dynamics and with iterative processes in the tensor train format are included.

    Original languageEnglish
    Pages (from-to)917-941
    Number of pages25
    JournalSIAM Journal on Numerical Analysis
    Issue number2
    Publication statusPublished - 2015


    • Low-rank approximation
    • Matrix product state
    • Splitting integrator
    • Tensor differential equations
    • Tensor train
    • Time-varying tensors


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