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.

20 pages, 5 figures

References in corpus (10)

• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data
• Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations
• Adaptive Leja sparse grid constructions for stochastic collocation and high-dimensional approximation
• Unnormalized Optimal Transport
• The Numerical Approximation of Nonlinear Functionals and Functional Differential Equations
• A Riemannian trust-region method for low-rank tensor completion
• Optimal randomized multilevel algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition
• Parallel numerical tensor methods for high-dimensional PDEs
• Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs

Cited by in corpus (6)

• A Low Rank Tensor Representation of Linear Transport and Nonlinear Vlasov Solutions and Their Associated Flow Maps
• A new scalable algorithm for computational optimal control under uncertainty
• Tensor methods for the Boltzmann-BGK equation
• Dynamic tensor approximation of high-dimensional nonlinear PDEs
• Tensor rank reduction via coordinate flows
• Statistics for stochastic differential equations and approximations of resolvent