/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

/pdf/numerical-solution-of-saddle-point-problems-l583iob5c2.pdf

Numerical solution of saddle point problems

Convex Analysis: (pms-28)

Yair Censor and Stavros A. Zenios, Oxford University Press, New York, 1997, 539 pp., ISBN 0-19-510062-X, $85.00

Parallel Optimization: Theory, Algorithms and Applications

A large number of imaging problems reduce to the optimization of a cost function , with typical structural properties. The aim of this paper is to describe the state of the art in continuous optimization methods for such problems, and present the most successful approaches and their interconnections. We place particular emphasis on optimal first-order schemes that can deal with typical non-smooth and large-scale objective functions used in imaging problems. We illustrate and compare the different algorithms using classical non-smooth problems in imaging, such as denoising and deblurring. Moreover, we present applications of the algorithms to more advanced problems, such as magnetic resonance imaging, multilabel image segmentation, optical flow estimation, stereo matching, and classification.

/pdf/an-introduction-to-continuous-optimization-for-imaging-3lzw7asbyh.pdf

An introduction to continuous optimization for imaging

On the steplength selection in gradient methods for unconstrained optimization

In this paper we establish the convergence of a general primal–dual method for nonsmooth convex optimization problems whose structure is typical in the imaging framework, as, for example, in the Total Variation image restoration problems. When the steplength parameters are a priori selected sequences, the convergence of the scheme is proved by showing that it can be considered as an e-subgradient method on the primal formulation of the variational problem. Our scheme includes as special case the method recently proposed by Zhu and Chan for Total Variation image restoration from data degraded by Gaussian noise. Furthermore, the convergence hypotheses enable us to apply the same scheme also to other restoration problems, as the denoising and deblurring of images corrupted by Poisson noise, where the data fidelity function is defined as the generalized Kullback–Leibler divergence or the edge preserving removal of impulse noise. The numerical experience shows that the proposed scheme with a suitable choice of the steplength sequences performs well with respect to state-of-the-art methods, especially for Poisson denoising problems, and it exhibits fast initial and asymptotic convergence.

/pdf/on-the-convergence-of-primal-dual-hybrid-gradient-algorithms-1j3f6hju19.pdf

On the Convergence of Primal–Dual Hybrid Gradient Algorithms for Total Variation Image Restoration

Variational models are a valid tool for edge-preserving image restoration from data affected by Poisson noise. This paper deals with total variation and hypersurface regularization in combination with the Kullbach Leibler divergence as a data fidelity function. We propose an iterative method, based on an alternating extragradient scheme, which is able to solve, in a numerically stable way, the primal–dual formulation of both total variation and hypersurface regularization problems. In this method, tailored for general smooth saddle-point problems, the stepsize parameter can be adaptively computed so that the convergence of the scheme is proved under mild assumptions. In the numerical experience, we focus the attention on the artificial smoothing parameter that makes different the total variation and hypersurface regularization. A set of experiments on image denoising and deblurring problems is performed in order to evaluate the influence of this smoothing parameter on the stability of the proposed method and on the features of the restored images.

https://iopscience.iop.org/article/10.1088/0266-5611/27/9/095001/pdf

An alternating extragradient method for total variation-based image restoration from Poisson data

Indefinitely preconditioned conjugate gradient method for large sparse equality and inequality constrained quadratic problems

In this paper we consider thearithmetic mean method for solving large sparse systems of linear equations. This iterative method converges for systems with coefficient matrices that are symmetric positive definite or positive real or irreducible L-matrices with a strong diagonal dominance. The method is very suitable for parallel implementation on a multiprocessor system, such as the CRAY X-MP. Some numerical experiments on systems resulting from the discretization, by means of the usual 5-point difference formulae, of an elliptic partial differential equation are presented.

Valeria Ruggiero

Papers

On the steplength selection in gradient methods for unconstrained optimization

On the Convergence of Primal–Dual Hybrid Gradient Algorithms for Total Variation Image Restoration

An alternating extragradient method for total variation-based image restoration from Poisson data

Indefinitely preconditioned conjugate gradient method for large sparse equality and inequality constrained quadratic problems

An iterative method for large sparse linear systems on a vector computer