2018/06/18 by Silvia Gazzola, Gazzola, Silvia, Silvia Noschese +5 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Electromagnetic Scattering and Analysis #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1806.06599
openalex publication_date 2018/06/18 · openalex created_date 2022/10/03 · openalex updated_date 2026/08/01
GMRES is one of the most popular iterative methods for the solution of large\nlinear systems of equations that arise from the discretization of linear\nwell-posed problems, such as Dirichlet boundary value problems for elliptic\npartial differential equations. The method is also applied to iteratively solve\nlinear systems of equations that are obtained by discretizing linear ill-posed\nproblems, such as many inverse problems. However, GMRES does not always perform\nwell when applied to the latter kind of problems. This paper seeks to shed some\nlight on reasons for the poor performance of GMRES in certain situations, and\ndiscusses some remedies based on specific kinds of preconditioning. The\nstandard implementation of GMRES is based on the Arnoldi process, which also\ncan be used to define a solution subspace for Tikhonov or TSVD regularization,\ngiving rise to the Arnoldi-Tikhonov and Arnoldi-TSVD methods, respectively. The\nperformance of the GMRES, the Arnoldi-Tikhonov, and the Arnoldi-TSVD methods is\ndiscussed. Numerical examples illustrate properties of these methods.\n