[NA] Melven Röhrig-Zöllner: Performance of low-rank linear solvers in tensor-train format
21 April 2023 12:30 till 13:15 - Location: van der Poelzaal (EWI, LB 01.220) | Add to my calendar
In this talk we discuss the problem of efficiently computing a low-rank solution of high-dimensional linear systems. More specifically, we discuss several methods for linear systems in the tensor-train format, also known as matrix-product-states (MPS) in physics. In particular, we consider global approaches like TT-GMRES and local approaches like TT-(M)ALS and TT-AMEn and look at suitable preconditioners and some algorithmic variants for non-symmetric operators. Overall, we focus on the computational complexity and on the performance on today's multi-core CPUs: The considered algorithms are composed of tensor contractions and of dense linear algebra operations like QR-decompositions and singular value decompositions (SVDs). We show significant speedup by carefully choosing suitable combinations of building blocks (e.g. using a tall-skinny QR + SVD). In addition, we show how to exploit orthogonalities from previous steps to speed-up tensor-train truncations. We illustrate the different effects in numerical experiments for simple Laplace- and convection-diffusion equations.