In this paper, we study a band constrained nonnegative matrix factorization (band NMF) problem: for a given nonnegative matrix $Y$, decompose it as $Y ≈ AX$ with $A$ a nonnegative matrix and $X$ a nonnegative block band matrix. This factorization model extends a single low rank subspace model to a mixture of several overlapping low rank subspaces, which not only can provide sparse representation, but also can capture significant grouping structure from a dataset. Based on overlapping subspace clustering and the capture of the level of overlap between neighbouring subspaces, two simple and practical algorithms are presented to solve the band NMF problem. Numerical experiments on both synthetic data and real images data show that band NMF enhances the performance of NMF in data representation and processing.

In this paper, we propose an algorithm for solving inequality constrained mini-max optimization problem. In this algorithm, an active set strategy is used together with multiplier method to convert the inequality constrained mini-max optimization problem into unconstrained optimization problem. A trust-region method is a well-accepted technique in constrained optimization to assure global convergence and is more robust when they deal with rounding errors. One of the advantages of trust-region method is that it does not require the objective function of the model to be convex. 

A global convergence analysis for the proposed algorithm is presented under some conditions. To show the efficiency of the algorithm numerical results for a number of test problems are reported.

Over the last ten years, Finite Element Exterior Calculus (FEEC) has been developed as a general framework for linear mixed variational problems, their numerical approximation by mixed methods, and their error analysis. The basic approach in FEEC, pioneered by Arnold, Falk, and Winther in two seminal articles in 2006 and 2010, interprets these problems in the setting of Hilbert complexes, leading to a more general and complete understanding. Over the last five years, the FEEC framework has been extended to a broader set of problems. One such extension, due to Holst and Stern in 2012, was to problems with variational crimes, allowing for the analysis and numerical approximation of linear and geometric elliptic partial differential equations on Riemannian manifolds of arbitrary spatial dimension. Their results substantially generalize the existing surface finite element approximation theory in several respects. In 2014, Gillette, Holst, and Zhu extended FEEC in another direction, namely to parabolic and hyperbolic evolution systems by combining the FEEC framework for elliptic operators with classical approaches for parabolic and hyperbolic operators, by viewing solutions to the evolution problem as lying in Bochner spaces (spaces of Banach-space valued parametrized curves). Related work on developing an FEEC theory for parabolic evolution problems has also been done independently by Arnold and Chen. In this article, we extend the work of Gillette-Holst-Zhu and Arnold-Chen to evolution problems on Riemannian manifolds, through the use of framework developed by Holst and Stern for analyzing variational crimes. We establish a priori error estimates that reduce to the results from earlier work in the flat (non-criminal) setting. Some numerical examples are also presented.

This paper proves the saturation assumption for the nonconforming Morley finite element discretization of the biharmonic equation. This asserts that the error of the Morley approximation under uniform refinement is strictly reduced by a contraction factor smaller than one up to explicit higher-order data approximation terms. The refinement has at least to bisect any edge such as red refinement or 3-bisections on any triangle.

This justifies a hierarchical error estimator for the Morley finite element method, which simply compares the discrete solutions of one mesh and its red-refinement. The related adaptive mesh-refining strategy performs optimally in numerical experiments. A remark for Crouzeix-Raviart nonconforming finite element error control is included.

The electromagnetic wave propagation in the chiral medium is governed by Maxwell's equations together with the Drude-Born-Fedorov (constitutive) equations. The problem is simplified to a two-dimensional scattering problem, and is formulated in a bounded domain by introducing two pairs of transparent boundary conditions. An a posteriori error estimate associated with the truncation of the nonlocal boundary operators is established. Based on the a posteriori error control, a finite element adaptive strategy is presented for computing the diffraction problem. The truncation parameter is determined through sharp a posteriori error estimate. Numerical experiments are included to illustrate the robustness and effectiveness of our error estimate and the proposed adaptive algorithm.

The weak Galerkin (WG) finite element method was first introduced by Wang and Ye for solving second order elliptic equations, with the use of weak functions and their weak gradients. The basis function spaces depend on different combinations of polynomial spaces in the interior subdomains and edges of elements, which makes the WG methods flexible and robust in many applications. Different from the definition of jump in discontinuous Galerkin (DG) methods, we can define a new weaker jump from weak functions defined on edges. Those functions have double values on the interior edges shared by two elements rather than a limit of functions defined in an element tending to its edge. Naturally, the weak jump comes from the difference between two weak functions defined on the same edge. We introduce an over-penalized weak Galerkin (OPWG) method, which has two sets of edge-wise and element-wise shape functions, and adds a penalty term to control weak jumps on the interior edges. Furthermore, optimal a priori error estimates in $H^1$ and $L^2$ norms are established for the finite element ($\mathbb{P}_k(K)$, $\mathbb{P}_k(e)$, $RT_k(K)$). In addition, some numerical experiments are given to validate theoretical results, and an incomplete LU decomposition has been used as a preconditioner to reduce iterations from the GMRES, CG, and BICGSTAB iterative methods.

A new two-level subspace method is proposed for solving the general unconstrained minimization formulations discretized from infinite-dimensional optimization problems. At each iteration, the algorithm executes either a direct step on the current level or a coarse subspace correction step. In the coarse subspace correction step, we augment the traditional coarse grid space by a two-dimensional subspace spanned by the coordinate direction and the gradient direction at the current point. Global convergence is proved and convergence rate is studied under some mild conditions on the discretized functions. Preliminary numerical experiments on a few variational problems show that our two-level subspace method is promising.