Fast Alternating Direction Multipliers Method by Generalized Krylov Subspaces

Buccini, Alessandro

doi:10.1007/s10915-021-01727-1

Cited by 3 publications

(1 citation statement)

References 41 publications

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“…While various iterative techniques for the solution of (2.14) have been discussed in the literature, including a generalized Krylov approach, e.g., [10], a standard approach is to use the LSQR algorithm, also based on a standard Krylov iteration [55]. This is our method of choice for comparison with our new projected algorithm, described in section 4.…”

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confidence: 99%

The Variable Projected Augmented Lagrangian Method

Chung¹,

Renaut²

2022

Preprint

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Inference by means of mathematical modeling from a collection of observations remains a crucial tool for scientific discovery and is ubiquitous in application areas such as signal compression, imaging restoration, and supervised machine learning. The inference problems may be solved using variational formulations that provide theoretically proven methods and algorithms. With ever-increasing model complexities and growing data size, new specially designed methods are urgently needed to recover meaningful quantifies of interest. We consider the broad spectrum of linear inverse problems where the aim is to reconstruct quantities with a sparse representation on some vector space; often solved using the (generalized) least absolute shrinkage and selection operator (lasso). The associated optimization problems have received significant attention, in particular in the early 2000's, because of their connection to compressed sensing and the reconstruction of solutions with favorable sparsity properties using augmented Lagrangians, alternating directions and splitting methods. We provide a new perspective on the underlying 1 regularized inverse problem by exploring the generalized lasso problem through variable projection methods. We arrive at our proposed variable projected augmented Lagrangian (vpal) method. We analyze this method and provide an approach for automatic regularization parameter selection based on a degrees of freedom argument. Further, we provide numerical examples demonstrating the computational efficiency for various imaging problems.

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confidence: 99%

The Variable Projected Augmented Lagrangian Method

Chung¹,

Renaut²

2022

Preprint

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An Efficient Implementation of the Gauss–Newton Method Via Generalized Krylov Subspaces

Buccini,

de Alba,

Pes

et al. 2023

J Sci Comput

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The solution of nonlinear inverse problems is a challenging task in numerical analysis. In most cases, this kind of problems is solved by iterative procedures that, at each iteration, linearize the problem in a neighborhood of the currently available approximation of the solution. The linearized problem is then solved by a direct or iterative method. Among this class of solution methods, the Gauss–Newton method is one of the most popular ones. We propose an efficient implementation of this method for large-scale problems. Our implementation is based on projecting the nonlinear problem into a sequence of nested subspaces, referred to as Generalized Krylov Subspaces, whose dimension increases with the number of iterations, except for when restarts are carried out. When the computation of the Jacobian matrix is expensive, we combine our iterative method with secant (Broyden) updates to further reduce the computational cost. We show convergence of the proposed solution methods and provide a few numerical examples that illustrate their performance.

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A variable projection method for large-scale inverse problems with ℓ1 regularization

Chung

Renaut

2023

Applied Numerical Mathematics

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Fast Alternating Direction Multipliers Method by Generalized Krylov Subspaces

Cited by 3 publications

References 41 publications

The Variable Projected Augmented Lagrangian Method

The Variable Projected Augmented Lagrangian Method

An Efficient Implementation of the Gauss–Newton Method Via Generalized Krylov Subspaces

A variable projection method for large-scale inverse problems with ℓ1 regularization

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