News
Geneva Chalk Talks - Bart Vandereycken
In this inaugural edition of Geneva Chalk Talks, Bart Vandereycken invites us to explore the mathematics behind his blackboard.
The blackboard contains a schematic explanation of the dynamical low-rank algorithm (DLRA) for approximating differential equations that involve large matrices and tensors. It is also known as the time-dependent variational principle (TDVP) in theoretical physics when approximating ground states of spin systems. The main idea of the method is to project the equations of motions onto the approximation manifold of low rank matrices or tensors. This allows us to obtain an approximation that is quasi-optimal and of low rank.
Together with colleagues from numerical analysis, Christian Lubich and Ivan Oseledets, we developed a discrete integrator for DLRA with tensors. We proved how it is the natural extension of the Lie-Trotter splitting of an earlier development of the integrator for matrices. In addition, the discrete integrator preserves invariants like energy and norm when applied to Hamiltonian problems.
With colleagues from theoretical physics, Jutho Haegeman and Frank Verstraete, we also presented a version more amenable for the physics community with matrix product states (MPS). On the right side of the blackboard, the key steps of the method are explained in terms of operations on the tensor network.
Recent work with PhD students in Geneva focused on how to provably apply DLRA to stiff problems using exponential integrators and how to use randomized methods from numerical linear algebra to speed up computations.
is an associate professor in the numerical analysis group. Prior to joining the university of Geneva, he was an instructor of mathematics at Princeton AV短视频 from September 2012 to January 2015. Before that, he was a post doc at EPF Lausanne and ETH Zurich. He obtained his PhD at KU Leuven in December 2010.
His research is on large-scale and high-dimensional problems that are solved numerically using low-rank matrix and tensor techniques. Examples of such problems are the electronic Schrödinger equation, parametric partial differential equations, and low-rank matrix completion. He tends to focus on practical algorithms that can be formulated on Riemannian matrix manifolds and use techniques from numerical linear algebra and numerical optimization. His other research interests include nonlinear eigenvalue problems, machine learning, and multilevel preconditioning.
Images: Jaime Benicio Neto 23 Nov 2024