Jan 16 – 20, 2023
Europe/Paris timezone

Christian Lubich: "Time integration of tree tensor networks"

Jan 17, 2023, 11:30 AM
Amphithéâtre Schwartz (IMT)

Amphithéâtre Schwartz


Université Paul Sabatier, 118 Route de Narbonne, 31000 Toulouse France


First I report on recent numerical experiments with time-dependent
tree tensor network algorithms for the approximation of quantum spin
systems. I will then describe the basics in the design of time
integration methods that are robust to the usual presence of small
singular values, that have good structure-preserving properties (norm,
energy conservation or dissipation), and that allow for rank (= bond
dimension) adaptivity and also have some parallelism. This discussion
of basic concepts will be done for the smallest possible type of
tensor network differential equations, namely low-rank matrix
differential equations. Once this simplest case is understood, there
is a systematic path to the extension of the integrators and their
favourable properties to general tree tensor networks.

This talk is based on joint work with many colleagues and former and
present students, among which I wish to single out Othmar Koch for the
first mathematical work on dynamical low-rank approximation (DLRA) in
2007, Ivan Oseledets for jointly finding the first robust DLRA
integrator (the projector-splitting integrator) in 2014, Gianluca
Ceruti for jointly finding the Basis Update & Galerkin (BUG)
integrators in 2021, and him and Hanna Walach and Dominik Sulz for the
recent systematic extension from low-rank matrices to general tree
tensor networks.

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