We present and analyze a new a posteriori error estimator for lowest order conforming finite elements. It is based on Raviart-Thomas finite elements and can be obtained locally by a postprocessing technique involving for each vertex a local subproblem associated with a dual mesh. Under certain regularity assumptions on the right-hand side, we obtain an error estimator where the constant in the upper bound for the true error tends to one. Replacing the conforming finite element solution by a postprocessed one, the error estimator is asymptotically exact. The local equivalence between our estimator and the standard residual-based error estimator is established. Numerical results illustrate the performance of the error estimator.

The key challenge of time-resolved Raman spectroscopy is the identification of the constituent species and the analysis of the kinetics of the underlying reaction network. In this work we present an integral approach that allows for determining both the component spectra and the rate constants simultaneously from a series of vibrational spectra. It is based on an algorithm for non-negative matrix factorization which is applied to the experimental data set following a few pre-processing steps. As a prerequisite for physically unambiguous solutions, each component spectrum must include one vibrational band that does not significantly interfere with vibrational bands of other species. The approach is applied to synthetic "experimental" spectra derived from model systems comprising a set of species with component spectra differing with respect to their degree of spectral interferences and signal-to-noise ratios. In each case, the species involved are connected via monomolecular reaction pathways. The potential and limitations of the approach for recovering the respective rate constants and component spectra are discussed.

A nonnegative matrix factorization (NMF) can be computed efficiently under the , which asserts that all the columns of the given input data matrix belong to the cone generated by a (small) subset of them. The provably most robust methods to identify these are based on nonnegative sparse regression and self-dictionaries, and require the solution of large-scale convex optimization problems. In this paper, we study a particular nonnegative sparse regression model with self-dictionary. As opposed to previously proposed models, this model yields a smooth optimization problem, where the sparsity is enforced through linear constraints. We show that the Euclidean projection on the polyhedron defined by these constraints can be computed efficiently, and propose a fast gradient method to solve our model. We compare our algorithm with several state-of-the-art methods on synthetic data sets and real-world hyperspectral images.A nonnegative matrix factorization (NMF) can be computed efficiently under the , which asserts that all the columns of the given input data matrix belong to the cone generated by a (small) subset of them. The provably most robust methods to identify these are based on nonnegative sparse regression and self-dictionaries, and require the solution of large-scale convex optimization problems. In this paper, we study a particular nonnegative sparse regression model with self-dictionary. As opposed to previously proposed models, this model yields a smooth optimization problem, where the sparsity is enforced through linear constraints. We show that the Euclidean projection on the polyhedron defined by these constraints can be computed efficiently, and propose a fast gradient method to solve our model. We compare our algorithm with several state-of-the-art methods on synthetic data sets and real-world hyperspectral images.

Starting from an n-point circular gravitational lens having 3n + 1 images, Rhie (2003) used a perturbation argument to construct an (n + 1)-point lens producing 5n images. In this work we give a concise proof of Rhie's result, and we extend the range of parameters in Rhie's model for which maximal lensing occurs.We also study a slightly different construction given by Bayer and Dyer (2007) arising from the (3n + 1)-point lens. In particular, we extend their results and give sharp parameter bounds for their lens model. By a substitution of variables and parameters we show that both models are equivalent in a certain sense.

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