Robust time integration for matrix product states

Bart Vandereycken - Princeton University
Dec 10 2014 - 3:00pm
Event type: 
110 Fine Hall

Matrix product states (MPS) are an effective low-rank approximation format for very large tensors. They have a long tradition in computational physics and chemistry to compute, for example, ground states of spin systems. I will give an introduction to MPS biased towards numerical analysis applications for high-dimensional PDEs such as uncertainty quantification, and mention some open problems.
In the second part, I will explain how to approximate the solution of an ODE directly as a time-dependent MPS. The approach is known under many names and conceptually straightforward: project the original ODE onto the manifold of low-rank MPS and integrate the new and much smaller ODE with your favorite method. In practice, however, these new ODEs are stiff: when the MPS approximation converges to the true solution, standard explicit integrators have a severe step size restriction making them virtually unusable. We will show how a particular splitting of the projected flow leads to an explicit yet efficient Lie-Trotter integrator that has remarkable stability properties. This is based on joint work with J. Haegeman, Ch. Lubich, I. Oseledets, and F. Verstraete.