Flexible GMRES for total variation regularization

Abstract

This paper presents a novel approach to the regularization of linear problems involving total variation (TV) penalization, with a particular emphasis on image deblurring applications. The starting point of the new strategy is an approximation of the nondifferentiable TV regularization term by a sequence of quadratic terms, expressed as iteratively reweighted 2-norms of the gradient of the solution. The resulting problem is then reformulated as a Tikhonov regularization problem in standard form, and solved by an efficient Krylov subspace method. Namely, flexible GMRES is considered in order to incorporate new weights into the solution subspace as soon as a new approximate solution is computed. The new method is dubbed TV-FGMRES. Theoretical insight is given, and computational details are carefully unfolded. Numerical experiments and comparisons with other algorithms for TV image deblurring, as well as other algorithms based on Krylov subspace methods, are provided to validate TV-FGMRES.

A stacked frequency approach for inhomogeneous time-dependent MRE: an inverse problem for the elastic shear modulus

“…where B ∈ R n×n (modelling the process) and c ∈ R n (modelling the output) are known, and the aim is compute an accurate approximation of the 'input' z ∈ R n . As noted in Gazzola et al (2015) (see also Chung & Gazzola (2019) and Gazzola & Sabaté Landman (2019)), a typical scenario is for the right-hand side c to be affected by noise, but the ill-conditioned matrix B is essentially noise-free with singular values that decay extremely rapidly. If its singular value decomposition (SVD) is B = U Σ V T for orthogonal U, V and diagonal Σ = diag(σ 1 , .…”

Manifold-Based Nonlocal Second-Order Regularization for Hyperspectral Image Inpainting

“…However, the whole equation is asymmetric due to the projection operator t   . Thus we choose the generalized minimal residual method (GMRES) to solve the system [34]. The optimization procedures for solving WNSR under the WNLL discretization framework is presented in Algorithm 1.…”

Krylov methods for inverse problems: Surveying classical, and introducing new, algorithmic approaches

“…Here, the lower bidiagonal matrix B k defined in (18) can also be regarded as the Cholesky factor of the symmetric positive definite tridiagonal matrix T k = B k B T k obtained after k iterations of the symmetric Lanczos algorithm applied to AA T with initial vector b. Moreover, multiplying the first expression in (17) from the left with A T , and using again the second equation in (17), one obtains…”

“…In all these situations, the only computationally viable approach to recover a solution of () is to apply an iterative linear solver that, as far as A is concerned, only requires matrix‐vector products with A and (possibly) A T (these are commonly referred to as “matrix‐free” methods). Note that, for problems originally expressed in general Tikhonov form whose regularization matrix L does not have an exploitable structure, must also be computed iteratively; this is the case especially for 2D or 3D problems, when, for example, L is a reweighted finite differences approximation of a derivative operator (see, eg, [18]). Although sometimes it may be more desirable to regularize () by applying an iterative solver to (), in some situations just applying an iterative solver to () can lead to a regularized solution.…”