This paper deals with gradient methods for minimizing n-dimensional strictly convex quadratic functions. Two new adaptive stepsize selection rules are presented and some key properties are proved. Practical insights on the effectiveness of the proposed techniques are given by a numerical comparison with the Barzilai-Borwein (BB) method, the cyclic/adaptive BB methods and two recent monotone gradient methods.
Gradient projection methods based on the Barzilai-Borwein spectral steplength choices are considered for quadratic programming (QP) problems with simple constraints. Well-known nonmonotone spectral projected gradient methods and variable projection methods are discussed. For both approaches, the behavior of different combinations of the two spectral steplengths is investigated. A new adaptive steplength alternating rule is proposed, which becomes the basis for a generalized version of the variable projection method (GVPM). Convergence results are given for the proposed approach and its effectiveness is shown by means of an extensive computational study on several test problems, including the special quadratic programs arising in training support vector machines (SVMs). Finally, the GVPM behavior as inner QP solver in decomposition techniques for large-scale SVMs is also evaluated
Although deconvolution can improve the quality of any type of microscope, the high computational time required has so far limited its massive spreading. Here we demonstrate the ability of the scaled-gradient-projection (SGP) method to provide accelerated versions of the most used algorithms in microscopy. To achieve further increases in efficiency, we also consider implementations on graphic processing units (GPUs). We test the proposed algorithms both on synthetic and real data of confocal and STED microscopy. Combining the SGP method with the GPU implementation we achieve a speed-up factor from about a factor 25 to 690 (with respect the conventional algorithm). The excellent results obtained on STED microscopy images demonstrate the synergy between super-resolution techniques and image-deconvolution. Further, the real-time processing allows conserving one of the most important property of STED microscopy, i.e the ability to provide fast sub-diffraction resolution recordings.
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