Speaker
Description
Tensor equations of the form $L(X)=B$, with $L$ a sum of Kronecker products, arise across scientific computing, from high-dimensional PDEs to large-scale inverse problems. When the tensor order is moderate but mode sizes are very large, the Tucker format is an attractive choice — yet using it inside iterative solvers is notoriously hard, as operator applications, orthogonalization, and linear combinations all trigger rank growth and force expensive truncations of large intermediate tensors.
In this talk, I will present a line of work that addresses these bottlenecks through randomized sketching applied directly to Tucker factors. The first part introduces sketching-based methods for Tucker tensor summation that exploit the structure of Khatri–Rao and Kronecker products to perform compressed arithmetic without ever forming dense intermediates. Building on this foundation, the second part develops a randomized Tucker-sketched GMRES solver in two variants: a robust method based on the randomized HOSVD with adaptive accuracy control, and a faster one built on the multilinear Nyström approximation, which — at the price of a prescribed maximal rank — accelerates orthogonalization, avoids storing the Krylov basis, and assembles the solution at negligible cost. Numerical experiments on image deblurring, a parameter-dependent elliptic problem, a transport equation, and a convection–diffusion problem illustrate the efficiency and robustness of the framework.