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Non-smooth optimization problems based on $L^1$ norms are investigated for smoothing of signals with noise or functions with sharp gradients. The use of $L^1$ norms allows to reduce the blurring introduced by methods based on $L^2$ norms. Numerical methods based on over-relaxation and augmented Lagrangian algorithms are proposed. Applications to free surface flows and image denoising are presented.
}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/772.html} }Non-smooth optimization problems based on $L^1$ norms are investigated for smoothing of signals with noise or functions with sharp gradients. The use of $L^1$ norms allows to reduce the blurring introduced by methods based on $L^2$ norms. Numerical methods based on over-relaxation and augmented Lagrangian algorithms are proposed. Applications to free surface flows and image denoising are presented.