Catherine Powell (The University of Manchester)
Stochastic Galerkin (SG) methods provide a surrogate modelling technique for facilitating forward UQ in PDEs with uncertain inputs. Unlike conventional sampling methods, SG schemes yield approximations that are functions of the random input variables so that all realisations of the PDE solution are essentially approximated simultaneously. Since they use simple tensor product spaces, standard SG schemes give rise to huge linear systems with coefficient matrices with a characteristic Kronecker product structure. Solving these systems is a bottleneck when working on standard desktop computers.
There are two potential remedies. The first is to learn a lower-dimensional Galerkin approximation space using a bespoke a posteriori error estimator. The second is to retain large tensor product spaces, recast the associated Kronecker system as a matrix equation and then appeal to low-rank iterative methods. A wide range of such methods are now available for symmetric and positive definite problems, but there is still work to be done for other types of matrix equations. In this talk, we will briefly outline a new class of short-term recurrences for structured non-symmetric matrix equations [1] that combines rank truncation strategies and randomization procedures to limit memory consumption.
[1] A class of low-rank short recurrences for non-symmetric linear matrix equations, D. Palitta, C.E. Powell, V. Simoncini, arXiv preprint arXiv:2605.01276 (2026).
