On the Dirac-Frenkel Variational Principle on Tensor Banach Spaces

TL;DR

This paper extends the Dirac-Frenkel Variational Principle to tensor Banach spaces, providing a geometric framework that facilitates the algorithmic treatment of high-dimensional PDEs and minimization problems.

Contribution

It introduces a novel geometric approach by modeling tensor product components as Banach manifolds and extends the variational principle to this setting.

Findings

Tensor product of normed spaces forms a union of disjoint connected components.

Each component with fixed Tucker rank is a Banach manifold with local charts.

Connected components can be immersed in ambient Banach spaces, enabling the extension of the variational principle.

Abstract

The main goal of this paper is to extend the so-called Dirac-Frenkel Variational Principle in the framework of tensor Banach spaces. To this end we observe that a tensor product of normed spaces can be described as a union of disjoint connected components. Then we show that each of these connected components, composed by tensors in Tucker format with a fixed rank, is a Banach manifold modelled in a particular Banach space, for which we provide local charts. The description of the local charts of these manifolds is crucial for an algorithmic treatment of high-dimensional partial differential equations and minimization problems. In order to describe the relationship between these manifolds and the natural ambient space we prove under natural conditions that each connected component can be immersed in a particular ambient Banach space. This fact allows us to finally extend the Dirac-Frenkel variational principle in the framework of topological tensor spaces.