In this paper, we propose a new stabilized linear finite element method for solving reaction-convection-diffusion equations with arbitrary magnitudes of reaction and diffusion. The key feature of the new method is that the test function in the stabilization term is taken in the adjoint-operator-like form $-\varepsilon \Delta v- ({\mbf a}\cdot\nabla v)/\gamma+\sigma v$, where the parameter $\gamma$ is appropriately designed to adjust the convection strength to achieve high accuracy and stability. We derive the stability estimates for the finite element solutions and establish the explicit dependence of $L^2$ and $H^1$ error bounds on the diffusivity, modulus of the convection field, reaction coefficient and the mesh size. The analysis shows that the proposed method is suitable for a wide range of mesh P\'{e}clet numbers and mesh Damk\"{o}hler numbers. More specifically, if the diffusivity $\varepsilon$ is sufficiently small with $\varepsilon < \Vert{\mbf a}\Vert h$ and the reaction coefficient $\sigma$ is large enough such that $\Vert{\mbf a}\Vert < \sigma h$, then the method exhibits optimal convergence rates in both $L^2$ and $H^1$ norms. However, for a small reaction coefficient satisfying $\Vert{\mbf a}\Vert \ge \sigma h$, the method behaves like the well-known streamline upwind/Petrov-Galerkin formulation of Brooks and Hughes. Several numerical examples exhibiting boundary or interior layers are given to demonstrate the high performance of the proposed method. Moreover, we apply the developed method to time-dependent reaction-convection-diffusion problems and simulation results show the efficiency of the approach.

This paper focuses on coordinate update methods, which are useful for solving problems involving large or high-dimensional datasets. They decompose a problem into simple subproblems, where each updates one, or a small block of, variables while fixing others. These methods can deal with linear and nonlinear mappings, smooth and nonsmooth functions, as well as convex and nonconvex problems. In addition, they are easy to parallelize. The great performance of coordinate update methods depends on solving simple sub-problems. To derive simple subproblems for several new classes of applications, this paper systematically studies coordinate-friendly operators that perform low-cost coordinate updates. Based on the discovered coordinate friendly operators, as well as operator splitting techniques, we obtain new coordinate update algorithms for a variety of problems in machine learning, image processing, as well as sub-areas of optimization. Several problems are treated with coordinate update for the first time in history. The obtained algorithms are scalable to large instances through parallel and even asynchronous computing. We present numerical examples to illustrate how effective these algorithms are.

In this paper, we propose a novel and simple technique to construct effective difference schemes for solving systems of singularly perturbed convection-diffusionreaction equations, whose solutions may display boundary or interior layers. We illustrate the technique by taking the Il’in-Allen-Southwell scheme for 1-D scalar equations as a basis to derive a formally second-order scheme for 1-D coupled systems and then extend the scheme to 2-D case by employing an alternating direction approach. Numerical examples are given to demonstrate the high performance of the obtained scheme on uniform meshes as well as piecewise-uniform Shishkin meshes.

This article studies the problem of image restoration of observed images corrupted by impulse noise and mixed Gaussian impulse noise. Since the pixels damaged by impulse noise contain no information about the true image, how to find this set correctly is a very important problem. We propose two methods based on blind inpainting and $\ell_0$ minimization that can simultaneously find the damaged pixels and restore the image. By iteratively restoring the image and updating the set of damaged pixels, these methods have better performance than other methods, as shown in the experiments. In addition, we provide convergence analysis for these methods; these algorithms will converge to coordinatewise minimum points. In addition, they will converge to local minimum points (or with probability one) with some modifications in the algorithms.

Finding a fixed point to a nonexpansive operator, i.e., x = Tx, abstracts many problems in numerical linear algebra, optimization, and other areas of data sciences. To solve xed- point problems, we propose ARock, an algorithmic framework in which multiple agents (machines, processors, or cores) update x in an asynchronous parallel fashion. Asynchrony is crucial to parallel computing since it reduces synchronization wait, relaxes communication bottleneck, and thus speeds up computing significantly. At each step of ARock, an agent updates a randomly selected coordinate xi based on possibly out-of-date information on x. The agents share x through either global memory or communication. If writing xi is atomic, the agents can read and write x without memory locks. We prove that if the nonexpansive operator T has a fixed point, then with probability one, ARock generates a sequence that converges to a fixed point of T. Our conditions on T and step sizes are weaker than comparable work. Linear convergence is obtained under suitable assumptions. We propose special cases of ARock for linear systems, convex optimization, machine learning, as well as distributed and decentralized consensus problems. Numerical experiments of solving sparse logistic regression problems are presented.