Coordinate-Adaptive Integration of PDEs on Tensor Manifolds

Abstract

We introduce a new tensor integration method for time-dependent partial differential equations (PDEs) that controls the tensor rank of the PDE solution via time-dependent smooth coordinate transformations. Such coordinate transformations are obtained by solving a sequence of convex optimization problems that minimize the component of the PDE operator responsible for increasing the tensor rank of the PDE solution. The new algorithm improves upon the non-convex algorithm we recently proposed in Dektor and Venturi (2023) which has no guarantee of producing globally optimal rank-reducing coordinate transformations. Numerical applications demonstrating the effectiveness of the new coordinate-adaptive tensor integration method are presented and discussed for prototype Liouville and Fokker-Planck equations.

Keywords

Tensor train / Crvilinear coordinates / Step-truncation tensor methods / High-dimensional PDEs / Dynamic tensor approximation / Mathematical Sciences / Numerical and Computational Mathematics

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Funding

Air Force Office of Scientific Research(FA9550-20-1-0174)

Army Research Office(W911NF-18-1-0309)

RIGHTS & PERMISSIONS

The Author(s)