Linear discrete inverse problems arise in many areas of science and engineering, from medical imaging and geophysics to atmospheric modelling. Their numerical solution often relies on iterative algorithms, particularly Krylov subspace methods, that can efficiently handle large-scale, ill-posed systems. In many practical settings, however, exact computations of matrix–vector products, preconditioners, or right-hand sides are either infeasible or unnecessary, leading to inexact iterations. This talk explores the interplay between inexactness and the regularizing behaviour of Krylov subspace methods for inverse problems. We discuss how approximate computations influence the regularization effect inherent in early iterations, as well as  semiconvergence, and how controlled inexactness may be exploited to improve computational efficiency. The aim is to provide a broad perspective on recent insights and open questions at the interface of inverse problems, iterative solvers, and computational inexactness.

I will explain how the irreversibility of the renormalization group together with anomalies, including anomalies in the space of coupling constants, can be used to constrain the IR phases of defects in familiar quantum field theories. As an example, I will use these techniques to provide evidence for a conjectural "spin-flux duality" which describes how certain line operators are mapped across particle/vortex duality in 2+1d.