Speaker
Description
In recent years, tensor network methods have garnered increased attention for modeling high dimensional quantum many body systems and for representing high dimensional functions with structured correlations.
A basic unresolved question is whether standard numerical linear algebra primitives, such as Krylov based iterative solvers, can be realized in matrix product state and matrix product operator form while remaining computationally efficient.
This talk gives a pedagogical overview of the current state of tensor network Krylov methods. I will define what it means to run a Krylov iteration when vectors and operators are represented in the tensor network format, and explain why familiar steps like basis construction and reorthogonalization become nontrivial algorithmic design problems in this setting. I will then show how tensor network structure can be paired with randomized low rank approximation to speed up the tensor network analog of matrix vector products, opening a broader design space for fast high dimensional Krylov solvers. Recent approaches will be surveyed, including tensor network Lanczos variants and sketched tensor network GMRES methods.
I will close with open questions around stability, error control, and tensor network bond dimension growth that are currently shaping the development of new algorithms in this space.