The Mumford–Shah model is one of the most important image segmentation models and has been studied extensively in the last twenty years. In this paper, we propose a two-stage segmentation method based on the Mumford–Shah model. The first stage of our method is to find a smooth solution g to a convex variant of the Mumford–Shah model. Once g is obtained, then in the second stage the segmentation is done by thresholding g into different phases. The thresholds can be given by the users or can be obtained automatically using any clustering methods. Because of the convexity of the model, g can be solved efficiently by techniques like the split-Bregman algorithm or the Chambolle–Pock method. We prove that our method is convergent and that the solution g is always unique. In our method, there is no need to specify the number of segments K (K ≥ 2) before finding g. We can obtain any K-phase segmentations by choosing (K − 1) thresholds after g is found in the first stage, and in the second stage there is no need to recompute g if the thresholds are changed to reveal different segmentation features in the image. Experimental results show that our two-stage method performs better than many standard two-phase or multiphase segmentation methods for very general images, including antimass, tubular, MRI, noisy, and blurry images.

We propose an efficient threshold dynamics method for topology optimization for fluids modeled with the Stokes equation. The proposed algorithm is based on minimization of an objective energy function that consists of the dissipation power in the fluid and the perimeter approximated by nonlocal energy, subject to a fluid volume constraint and the incompressibility condition. We show that the minimization problem can be solved with an iterative scheme in which the Stokes equation is approximated by a Brinkman equation. The indicator functions of the fluid-solid regions are then updated according to simple convolutions followed by a thresholding step. We demonstrate mathematically that the iterative algorithm has the total energy decaying property. The proposed algorithm is simple and easy to implement. A simple adaptive time strategy is also used to accelerate the convergence of the iteration. Extensive numerical experiments in both two and three dimensions show that the proposed iteration algorithm is quite robust and converges in much fewer iterations and is more efficient than many existing methods.

A fundamental and very well studied region of the Erdős–Rényi process is the phase transition at$m$∼$n$/2 edges in which a giant component suddenly appears. We examine the process beginning with an initial graph. We further examine the Bohman–Frieze process in which edges between isolated vertices are more likely. While the positions of the phase transitions vary, the three processes belong, roughly speaking, to the same universality class. In particular, the growth of the giant component in the barely supercritical region is linear in all cases.

We study the generalized trace regression with a near low-rank regression coefficient matrix, which extends notion of sparsity for regression coefficient vectors. Specifically, given a matrix covariate X, the probability density function of the response Y is f (Y| X)= c (Y) exp ( 1 Y + b ()), where = tr ( T X). This model accommodates various types of responses and embraces many important problem setups such as reduced-rank regression, matrix regression that accommodates a panel of regressors, matrix completion, among others. We estimate through minimizing empirical negative log-likelihood plus nuclear norm penalty. We first establish a general theory and then for each specific problem, we derive explicitly the statistical rate of the proposed estimator. They all match the minimax rates in the linear trace regression up to logarithmic factors. Numerical studies confirm the rates we established and

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