Imaging and Computing Seminar

Bart Vandereycken, PACM and Math, Princeton

Differential geometry for rank-structured tensors from a numerical analysis perspective

Many practical problems lead to high-dimensional equations with an extremely large number of variables that scale exponentially with the dimension; take for example parametric PDEs and the Schroedinger equation. Low-rank tensor techniques are a popular technique to solve such equations since they can circumvent this exponential scaling, known as the curse of dimensionality.

Certain rank-structured tensors were recently shown to admit smooth structures that turn them into differential manifolds, for example, tensor train (TT) or matrix product states (MPS) in [1,2], and hierarchical Tucker (HT) in [3]. In this talk, I will discuss why treating these tensors as manifolds can be useful from a numerical linear algebra perspective when solving  high-dimensional equations.

In particular, I will explain how the dynamical low-rank algorithm [4] can be used to approximate tensor differential equations in the HT/TT/MPS formats. In this approach, also known as the Dirac-Frenkel variational principle, the time derivative of the tensor to be approximated is projected onto the time-dependent  tangent space of the approximation manifold along the solution trajectory. I will discuss the approximation properties of this approach and introduce an explicit but  surprisingly stable time-stepping scheme that extends the matrix version of [5].

Based on joint work with C. Lubich and I. Oseledets.

[1] S. Holtz, T. Rohwedder, and R. Schneider. On manifolds of tensors of fixed TT-rank. Num. Math., 2012.
[2] J. Haegeman, T. J. Osborne, and F. Verstraete.  Post-matrix product state methods: To tangent space and beyond. Phys. Rev. B, 2013.
[3] A. Uschmajew and B. Vandereycken. The geometry of algorithms using hierarchical tensors. Lin. Alg. Appl., 2013.
[4] C. Lubich, T. Rohwedder, R. Schneider, B. Vandereycken, Dynamical approximation of hierarchical Tucker and tensor-train tensors. SIAM J. Matrix Anal. Appl., 2013.
[5] C. Lubich, I. Oseledets. A projector-splitting integrator for dynamical low-rank approximation. Submitted to BIT, 2013.