Mathematics

Linear inverse problems on infinite-dimensional Hilbert space are ubiquitous in physical systems; from applications in describing fluid flows to the field of quantum mechanics. For a bounded linear operator, we consider , , for the unknown . A popular family of numerical algorithms used to solve for the unknown in these problems are known as Krylov subspace methods. These methods construct a sequence of iterates , that approximate the exact solution, within a sequence of finite-dimensional increasing nested subspaces known as -th order Krylov spaces. In an infinite-dimensional setting, the Krylov space is constructed by taking the union of all these nested subspaces.

This presentation covers the notion of the associated infinite-dimensional Krylov subspace and highlights necessary and sufficient conditions for the Krylov-solvability of the considered linear inverse problem. The discussion is based on theoretical results together with a series of model examples, and it is corroborated by specific numerical experiments.

This is a joint work with Prof. Alessandro Michelangeli and Prof. Paolo Novati.