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A PROJECTION APPROACH TO THE NUMERICAL ANALYSIS OF LIMIT LOAD PROBLEMS

MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES 2011年第6期21卷 1291-1316页

作者： Carlier, Guillaume Comte, Myriam Ionescu, Ioan Peyre, Gabriel Univ Paris 09 CEREMADE F-75775 Paris 16 France Univ Paris 06 Lab Jacques Louis Lions F-75252 Paris 05 France Univ Paris 13 LPMTM Inst Galiles F-93430 Villetaneuse France

This paper proposes a numerical scheme to approximate the solution of (vectorial) limit load problems. The method makes use of a strictly convex perturbation of the problem, which corresponds to a projection of the deformation field under bounded deformation and incompressibility constraints. The discretized formulation of this perturbation converges to the solution of the original landslide problem when the amplitude of the perturbation tends to zero. The projection is computed numerically with a multi-step gradient descent on the dual formulation of the problem.

关键词： Limit load analysis functions of bounded deformation penalizations nesterov algorithm

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Fast gradient descent algorithm for image classification with neural networks

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SIGNAL IMAGE AND VIDEO PROCESSING 2020年第8期14卷 1565-1572页

作者： El Mouatasim, Abdelkrim Ibn Zohr Univ FPO BP 284 Ouarzazate 45800 Morocco

Any optimization of gradient descent methods involves selecting a learning rate. Tuning the learning rate can quickly become repetitive with deeper models of image classification, does not necessarily lead to optimal convergence. We proposed in this paper, a modification of the gradient descent algorithm in which the Nestrove step is added, and the learning rate is update in each epoch. Instead, we learn learning rate itself, either by Armijo rule, or by control step. Our algorithm called fast gradient descent (FGD) for solving image classification with neural networks problems, the quadratic convergence rate o(k(2)) of FGD algorithm are proved. FGD algorithm are applicate to a MNIST dataset. The numerical experiment, show that our approach FGD algorithm is faster than gradient descent algorithms.

关键词： Gradient algorithm nesterov algorithm Learning rate control Image classification Neural networks

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Single-image super-resolution with total generalised variation and Shearlet regularisations

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IET IMAGE PROCESSING 2014年第12期8卷 833-845页

作者： Feng, Wensen Lei, Hong Univ Sci & Technol Beijing Sch Automat & Elect Engn Beijing 100083 Peoples R China Chinese Acad Sci Inst Elect Beijing 100190 Peoples R China

In this study, the authors proposed a novel regularisation model for resolution enhancement of clean or noisy single image based on the total generalised variation (TGV) and Shearlet transform. The proposed model has two main contributions. Firstly, different from models with total variation regularisation, which assume that images consist of piecewise-constant areas, the author's TGV-based model is aware of higher-order smoothness, thus eliminates the staircase-like artefacts. Secondly, various image features including edges and fine details can be preserved by their model. This is nature since the Shearlets mathematically provide an optimally sparse approximation for the class of piecewise-smooth functions with rich geometric information. Moreover, to solve the proposed model, an efficient numerical scheme is explicitly developed based on the nesterov's algorithm. A series of numerical experiments validate the effectiveness of the proposed method.

关键词： image resolution image enhancement piecewise constant techniques approximation theory transforms single-image super-resolution total generalised variation Shearlet regularisations resolution enhancement clean single image noisy single image Shearlet transform total variation regularisation piecewise-constant areas TGV-based model higher-order smoothness staircase-like artefacts optimally sparse approximation piecewise-smooth functions geometric information nesterov algorithm

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Synthetic aperture radar image despeckling via total generalised variation approach

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IET IMAGE PROCESSING 2015年第3期9卷 236-248页

作者： Feng, Wensen Lei, Hong Qiao, Hong Univ Sci & Technol Beijing Sch Automat & Elect Engn Beijing 100083 Peoples R China Chinese Acad Sci Inst Elect Beijing 100190 Peoples R China Chinese Acad Sci Inst Automat Beijing 100190 Peoples R China

Speckle reduction is an important task in synthetic aperture radar. One extensively used approach is based on total variation (TV) regularisation, which can realise significantly sharp edges, but on the other hand brings in the undesirable staircasing artefacts. In essence, the TV-based methods tend to create piecewise-constant images even in regions with smooth transitions. In this study, a new method is proposed for speckle reduction via total generalised variation (TGV) penalty. This is reasonable from the fact that the TGV-based model can reduce the staircasing artefacts of TV by being aware of higher-order smoothness. An efficient numerical scheme based on the nesterov's algorithm is also developed for solving the TGV-based optimisation problem. Monte Carlo experiments show that the proposed scheme yields state-of-the-art results in terms of both performance and speed. Especially when the image has some higher-order smoothness, the authors' scheme outperforms the TV-based methods.

关键词： synthetic aperture radar radar imaging image denoising speckle variational techniques piecewise constant techniques smoothing methods higher order statistics optimisation Monte Carlo methods synthetic aperture radar image despeckling total generalised variation approach speckle reduction total variation regularisation piecewise constant image smooth transition TGV-based model staircasing artefact reduction higher order smoothness numerical scheme nesterov algorithm TGV-based optimisation problem Monte Carlo method

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Control subgradient algorithm for image l₁ regularization

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SIGNAL IMAGE AND VIDEO PROCESSING 2015年第1期9卷 275-283页

作者： El Mouatasim, Abdelkrim Wakrim, Mohammed Univ Ibn Zohr Fac Polydisplinaire Ouarzazate 45800 Morocco Univ Ibn Zohr Fac Sci Dept Math Agadir Morocco

We consider the class of subgradient methods for solving minimization of a nonsmooth convex function regularized by the discretized l(1) norm models arising in image processing. This class of methods is attractive due to its simplicity;however, they are also known to converge quite slowly. In this paper, we present a control subgradient algorithm (CSA) which preserves the computational simplicity of subgradient methods, but with a convergence rate which is proven to be significantly better, both theoretically and practically. Initial promising numerical results for wavelet-based image deblurring demonstrate the capabilities of CSA.

关键词： Subgradient algorithm nesterov algorithm Step-size control l(1) Regularization Image deblurring

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Control proximal gradient algorithm for image l1 regularization

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SIGNAL IMAGE AND VIDEO PROCESSING 2019年第6期13卷 1113-1121页

作者： El Mouatasim, Abdelkrim Univ Ibn Zohr Fac Polydisplinaire BP 284 Ouarzazate 45800 Morocco

We consider a control proximal gradient algorithm (CPGA) for solving the minimization of a nonsmooth convex function. In particular, the convex function is an l(1) regularized least squares function derived by the discretized l(1) norm models arising in image processing. This proximal gradient algorithm with the control step size is attractive due to its simplicity. However, this method is also known to converge quite slowly. In this paper, we present a fast control proximal gradient algorithm by adding nesterov step. It preserves the computational simplicity of proximal gradient methods with a convergence rate 1/k(2). It is proven to be better than the existing methods both theoretically and practically. The initial numerical results for performing the image deblurring demonstrate the efficiency of CPGA when it is compared to the existing fast iterative shrinkage-thresholding algorithms.

关键词： Proximal gradient algorithm nesterov algorithm Step-size control l(1) Regularization Image deblurring FISTA

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