A High-Order Rank-Adaptive Implicit Algorithm for Solving High Dimensional Diffusion Equations using the Hierarchical Tucker Decomposition

Abstract

This paper presents a high-order rank-adaptive implicit integrator for the tensor solution of high-dimensional diffusion equations. We extend the 3D version of this method from the Tucker decomposition to higher dimensions using the hierarchical Tucker (HT) decomposition, since the storage complexity for the Tucker decomposition increases exponentially with the number of dimensions $d>3$. The HT format avoids this issue by decomposing the solution according to a binary tree consisting of bases for each dimension and core tensors which connect the bases. Spectral methods are considered for spatial discretization, and diagonally implicit Runge-Kutta methods are considered for time discretization. At each stage of the Runge-Kutta method, the bases computed at the previous stages are augmented to predict the upcoming basis and construct projection subspaces. By projecting onto these enriched subspaces, the bases and cores can be updated in a sequential manner going up the tree from leaf-to-root. Unlike the 3D Tucker method which has a single core tensor, the HT method also updates the intermediate core tensors. Numerical experiments demonstrate that the method observes high-order accuracy, and test how well the integrator captures the solution rank for various sets of time-dependent diffusion coefficients.