27th Conference of the International Linear Algebra Society (ILAS 2026)

Randomized Generalized Error Minimizing Method for Linear Ill-Posed Problem

May 20, 2026, 10:45 AM

25m

Torgersen Hall 1030 (Virginia Tech)

Minisymposium Talk Inverse Problems and Uncertainty Quantification through the Lens of Numerical Linear Algebra Inverse Problems and Uncertainty Quantification through the Lens of Numerical Linear Algebra

Speaker

Prof. Ning Zheng (Tongji University)

Description

For solving noisy linear ill--posed problems arising from the practical applications, the residual based iterative methods may suffer semi-convergence phenomenon, where the iterates initially get closer to the desired solution but then degrade as the iteration continues. Building upon the randomized Gram--Schmidt algorithm, a random sketching technique known to reduce inner product computational costs over classical Gram--Schmidt and numerical stability comparable to modified Gram--Schmidt, we develop a novel randomized generalized error minimizing (GMERR) Krylov subspace method. This process extends the successful application of randomized Gram--Schmidt in methods such as randomized GMRES and LSQR. We further introduce a block variant, resulting in a block randomized Arnoldi process and a block GMERR method for large-scale ill-posed problems. A theoretical analysis of the regularization properties and numerical stability of the proposed methods is provided, leveraging random projection theory. Numerical experiments demonstrate the efficacy of the new algorithms.