Speaker
Description
Inverse problems focus on reconstructing hidden objects from indirect and often noisy measurements and are prevalent in numerous scientific and engineering disciplines. However, these reconstructions are typically highly sensitive to perturbations such as measurement errors, making regularization essential to obtain meaningful approximations.
Krylov subspace methods are a class of very popular projection methods with regularization properties, which typically construct stable bases for Krylov subspaces related to the original linear system using orthogonalization. However, there are scenarios where the inner products required in this process can hinder the usability of the solvers. For example, in low-precision arithmetic, standard Krylov solvers might break too early, or there can be over or under-flows in the computation of norms. Moreover, inner-products can be a limiting factor for high-performance computing, since they require global communication. On the other hand, the most used inner-product free solvers, e.g. Chebyshev semi-iterations, can show very poor convergence. In this work, I present a family of solvers which leverage the fast converge of Krylov methods while being inherently inner-product free, and I show that these can be used to tackle large-scale linear inverse problems efficiently. In particular, I revise the changing minimal residual Hessenberg method (CMRH) in the context of inverse problems, showing that it has regularization properties; and I introduce a new method, the least squares LU (LSLU). Both methods rely on the (possibly modified) Hessenberg iterative algorithm and are based off implicit LU factorizations of the Krylov basis. Moreover, this framework is extended to include Tikhonov regularization, in the fashion of hybrid regularization, so that the regularization parameters can be chosen on-the-fly. Theoretical results and extensive numerical experiments suggest that inner-product free variants exhibit comparable performance to the established methods.
