Stability analysis of hierarchical tensor methods for time-dependent PDEs

Abstract

In this paper we address the question of whether it is possible to integrate time-dependent high-dimensional PDEs with hierarchical tensor methods and explicit time stepping schemes. To this end, we develop sufficient conditions for stability and convergence of tensor solutions evolving on tensor manifolds with constant rank. We also argue that the applicability of PDE solvers with explicit time-stepping may be limited by time-step restriction dependent on the dimension of the problem. Numerical applications are presented and discussed for variable coefficients linear hyperbolic and parabolic PDEs.

Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs

“…The numerical results we obtained demonstrate that the proposed rank-adaptive tensor method is effective in controlling the temporal integration error, and outperforms previous dynamical tensor methods for PDEs in terms of accuracy, robustness and computational cost. We also proved that the new method is consistent with recently proposed step-truncation algorithms [31,50,49] in the limit of small time steps.…”

“…Step-truncation methods Another methodology to integrate nonlinear PDEs on fixed-rank tensor manifolds M r is step-truncation [31,50,49]. The idea is to integrate the solution off of M r for short time, e.g., by performing one time step of the full equation with a conventional time-stepping scheme, followed by a truncation operation back onto M r .…”

“…Computing the numerical solution to high-dimensional PDEs is an extremely challenging problem which has attracted substantial research efforts in recent years. Techniques such as sparse collocation methods [10,13,6,24,39], highdimensional model representations [34,11,5], deep neural networks [45,46,58], and numerical tensor methods [29,4,49,8,27,31] were proposed to mitigate the exponential growth of the degrees of freedom, the computational cost and the memory requirements.…”

“…In Section 3 we discuss dynamic tensor approximation of nonlinear PDEs of the form (1) and develop robust temporal integration schemes based on operator splitting methods. We also discuss steptruncation algorithms [50,49] and prove that dynamic tensor approximation and step-truncation are at least order one consistent to one another. In Section 4 we develop new rank-adaptive time integrators on rank-structured FTT tensor manifolds and prove that the resulting scheme is consistent.…”

The numerical approximation of nonlinear functionals and functional differential equations

“…We also provided sufficient conditions for consistency, stability and convergence of functional approximation schemes to compute the solution of FDEs, thus extending the well-known Lax-Richtmyer theorem from PDEs to FDEs. As we suggested in [69], these results open the possibility to utilize techniques for highdimensional model representation such as deep neural networks [52,53,79] and numerical tensor methods [17,3,55,7,59,37] to represent nonlinear functionals and compute approximate solutions to functional differential equations. We conclude by emphasizing that the results we obtained in this paper can be extended to real-or complex-valued functionals in compact Banach spaces (see, e.g., [33,65]).…”