Time Integration of Tree Tensor Networks

Abstract

Dynamical low-rank approximation by tree tensor networks is studied for the datasparse approximation to large time-dependent data tensors and unknown solutions of tensor differential equations. A time integration method for tree tensor networks of prescribed tree rank is presented and analyzed. It extends the known projector-splitting integrators for dynamical low-rank approximation by matrices and Tucker tensors and is shown to inherit their favorable properties. The integrator is based on recursively applying the Tucker tensor integrator. In every time step, the integrator climbs up and down the tree: it uses a recursion that passes from the root to the leaves of the tree for the construction of initial value problems on subtree tensor networks using appropriate restrictions and prolongations, and another recursion that passes from the leaves to the root for the update of the factors in the tree tensor network. The integrator reproduces given time-dependent tree tensor networks of the specified tree rank exactly and is robust to the typical presence of small singular values in matricizations of the connection tensors, in contrast to standard integrators applied to the differential equations for the factors in the dynamical low-rank approximation by tree tensor networks.

Dynamical Low-Rank Approximation for Burgers' Equation With Uncertainty

“…Thus, we can conclude that DLRA provides an opportunity to battle the curse of dimensionality, since the required memory to achieve a satisfactory solution approximation grows moderately with dimension. When employing tenor approximations as presented in [48], we expect linear instead of exponential growth with respect to the dimension. However, we leave an extension to higher uncertain domains in which this strategy becomes crucial to future work.…”

“…In order to further increase the uncertain dimension efficiently, we aim to perform further splitting of the random domain according to [48]. Here, the unconventional integrator will be of high interest, since it allows for parallel solves of all spatial and uncertain basis functions.…”

“…Note that computational requirements become prohibitively expensive for large p. The increased numerical costs can be reduced by further splitting the uncertain domain [45,46,47,48], which we will leave to future work.…”

Time Evolution of ML-MCTDH Wavefunctions II: Application of the Projector Splitting Integrator

“…The PSI has been shown to be stable with respect to almost or fully rank deficient wavefunction parametrizations in its MCTDH 36 and ML-MCTDH formulations, 31,33,34,39 including the important subcategory of matrix product states (maximally layered trees with physical degrees of freedom at each layer). 32,40,41 The stability of the PSI with respect to nearly rankdeficient wavefunction parametrizations can be attributed to the fact that the EOMs solved are always well-conditioned.…”

Tensor Network States for Vibrational Spectroscopy

“…A notable exception are TTNSs, for which a DMRG-like optimization strategy has been devised. 56 Moreover, the recently developed time-dependent TTNS variant 66,67 has paved the route towards the optimization of TTNS with imaginary-time propagation algorithms. Although these developments could set the ground for new efficient tensor-based methods in the near future, DMRG currently provides the optimal balance between the complexity of the tensor factorization and of its optimization.…”