Coordinate-adaptive integration of PDEs on tensor manifolds

2023/08/04 by Dektor, Alec, Venturi, Daniele · 3 citations
#Computational Physics (physics.comp-ph) #FOS: Mathematics #FOS: Physical sciences #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2308.02659

Abstract

We introduce a new tensor integration method for time-dependent PDEs that controls the tensor rank of the PDE solution via time-dependent diffeomorphic coordinate transformations. Such coordinate transformations are generated by minimizing the normal component of the PDE operator relative to the tensor manifold that approximates the PDE solution via a convex functional. The proposed method significantly improves upon and may be used in conjunction with the coordinate-adaptive algorithm we recently proposed in JCP (2023) Vol. 491, 112378, which is based on non-convex relaxations of the rank minimization problem and Riemannian optimization. Numerical applications demonstrating the effectiveness of the proposed coordinate-adaptive tensor integration method are presented and discussed for prototype Liouville and Fokker-Planck equations.

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