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A modified quasi-boundary value method for an abstract ill-posed biparabolic problem

100%
Besma K., Nadjib B., Faouzia R.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1649-1666
EN
In this paper, we are concerned with the problem of approximating a solution of an ill-posed biparabolic problem in the abstract setting. In order to overcome the instability of the original problem, we propose a modified quasi-boundary value method to construct approximate stable solutions for the original ill-posed boundary value problem. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution. Moreover, numerical tests are presented to illustrate the accuracy and efficiency of this method.
2
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Employing different loss functions for the classification of images via supervised learning

76%
Boţ R., Heinrich A., Wanka G.
Open Mathematics
|
2014
|
tom 12
|
nr 2
381-394
EN
Supervised learning methods are powerful techniques to learn a function from a given set of labeled data, the so-called training data. In this paper the support vector machines approach is applied to an image classification task. Starting with the corresponding Tikhonov regularization problem, reformulated as a convex optimization problem, we introduce a conjugate dual problem to it and prove that, whenever strong duality holds, the function to be learned can be expressed via the dual optimal solutions. Corresponding dual problems are then derived for different loss functions. The theoretical results are applied by numerically solving a classification task using high dimensional real-world data in order to obtain optimal classifiers. The results demonstrate the excellent performance of support vector classification for this particular problem.
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