Rank-Adaptive Time Integration of Tree Tensor Networks

A rank-adaptive integrator for the approximate solution of high-order tensor differential equations by tree networks is proposed and analyzed. In a recursion from leaves to root, updates bases then evolves connection tensors Galerkin method in augmented subspace spanned new old bases. This followed rank truncation within specified error tolerance. The memory requirements are linear order maximal mode dimension. robust small singular values matricizations tensors. Up error, which controlled given tolerance, preserves norm energy Schrödinger equations, it dissipates gradient systems. Numerical experiments with basic quantum spin system illustrate behavior algorithm.