Abstract
Many successful variational regularization methods employed to solve linear inverse problems in imaging applications (such as image deblurring, image inpainting, and computed tomography) aim at enhancing edges in the solution, and often involve non-smooth regularization terms (e.g., total variation). Such regularization methods can be treated as iteratively reweighted least squares problems (IRLS), which are usually solved by the repeated application of a Krylov projection method. This approach gives rise to an inner–outer iterative scheme where the outer iterations update the weights and the inner iterations solve a least squares problem with fixed weights. Recently, flexible or generalized Krylov solvers, which avoid inner–outer iterations by incorporating iteration-dependent weights within a single approximation subspace for the solution, have been devised to efficiently handle IRLS problems. Indeed, substantial computational savings are generally possible by avoiding the repeated application of a traditional Krylov solver. This paper aims to extend the available flexible Krylov algorithms in order to handle a variety of edge-enhancing regularization terms, with computationally convenient adaptive regularization parameter choice. In order to tackle both square and rectangular linear systems, flexible Krylov methods based on the so-called flexible Golub–Kahan decomposition are considered. Some theoretical results are presented (including a convergence proof) and numerical comparisons with other edge-enhancing solvers show that the new methods compute solutions of similar or better quality, with increased speedup.
Highlights
In this paper, we consider the solution of large-scale linear systems of the formAxtrue + e = btrue + e = b . (1)We are interested in problems (1) associated with the discretization of linear inverse problems, where b ∈ Rm represents the measured data, A ∈ Rm×n represents the forward mapping, xtrue ∈ Rn is the desired solution, and e ∈ Rm is unknown Gaussian white noise.In this setting, A is typically ill-conditioned with ill-determined rank
This paper aims at introducing new solvers for (2) based on the flexible Golub–Kahan (FGK) decomposition [24], introducing significant elements of novelty with respect to available solvers based on either flexible Krylov subspace (FKS) or GKS methods
The idea of incorporating an edge-enhancing regularizer within a flexible Krylov method based on the flexible Arnoldi algorithm (i.e., FGMRES) was already proposed in [30], this is limited to the case of isotropic total variation (TV) and a square matrix A; it is well-known that iterative solvers based on the Arnoldi algorithm are not general-purpose regularization methods and are only successful for matrices A close to normal or when the generated approximation subspace is favorable for a particular solution; we refer to [31] for more details about GMRES, which can be extended to FGMRES
Summary
We consider the solution of large-scale linear systems of the form. Axtrue + e = btrue + e = b. The idea of incorporating an edge-enhancing regularizer within a flexible Krylov method based on the flexible Arnoldi algorithm (i.e., FGMRES) was already proposed in [30], this is limited to the case of isotropic TV and a square matrix A; it is well-known that iterative solvers based on the Arnoldi algorithm are not general-purpose regularization methods and are only successful for matrices A close to normal or when the generated approximation subspace is favorable for a particular solution; we refer to [31] for more details about GMRES, which can be extended to FGMRES.
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