We establish an algorithm framework for solving optimization problems with orthogonality constraints. The new framework combines a function value reduction step with a multiplier correction step. The multiplier correction step minimizes the objective function in the range space of the current iterate. We also propose three algorithms which are called gradient reflection (GR), gradient projection (GP) and columnwise block coordinate descent (CBCD), respectively. Preliminary numerical experiments demonstrate that our new framework is of great potential.
joint work with Yaxiang Yuan
We present a stochastic trust region scheme in which the trust region radius is directly related to the probabilistic models. Especially, we show a specific algorithm STRME in which the trust region radius is selected as depending linearly on model gradient. Moreover, the complexity of STRME in nonconvex, convex and strongly convex settings has all been analyzed. Finally, some numerical experiments are carried out to reveal the benefits of the proposed methods compared to the existing stochastic trust region methods and other relevant stochastic gradient methods.
With the CT image segmentation, the images of shale micropores can be obtained, especially the pore type, shape, size, spatial distribution and connectivity. In this work, based on the reconstructed synchrotron radiation X-ray shale CT data, we develop a neural network image segmentation technique with deep learning based on multi-energy CT image data in order to obtain the 3D structural characteristics and spatial distribution of shale. Since it is an ill-posed inverse problem, regularization and optimization issues are discussed.