Courant Institute of Mathematical Sciences

About: Courant Institute of Mathematical Sciences is a education organization based out in New York, New York, United States. It is known for research contribution in the topics: Nonlinear system & Boundary value problem. The organization has 2414 authors who have published 7759 publications receiving 439773 citations. The organization is also known as: CIMS & New York University Department of Mathematics.

Multilevel Schwarz Methods for Elliptic Problems with Discontinuous Coefficients in Three Dimensions

Abstract: Multilevel Schwarz methods are developed for a conforming finite element approximation of second order elliptic problems. We focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions. We establish a condition number estimate for the iterative operator, which is independent of the coefficients, and grows at most as the square of the number of levels. We also characterize a class of distributions of the coefficients, called quasi-monotone, for which the weighted\(L^{2}\) -projection is stable and for which we can use the standard piecewise linear functions as a coarse space. In this case, we obtain optimal methods, i.e. bounds which are independent of the number of levels and subregions. We also design and analyze multilevel methods with new coarse spaces given by simple explicit formulas. We consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods.

The milne and kramers problems for the boltzmann equation of a hard sphere gas

Abstract: Existence, uniqueness and properties of asymptotic behavior are proved for solutions of the Milne and Kramers problems for the linearized Boltzmann equation for a gas of hard spheres. The proof uses energy-like estimates and follows the analysis of Bardos, Santos and Sentis for the equations of neutron transport.

Scene parsing with Multiscale Feature Learning, Purity Trees, and Optimal Covers

Abstract: Scene parsing consists in labeling each pixel in an image with the category of the object it belongs to. We propose a method that uses a multiscale convolutional network trained from raw pixels to extract dense feature vectors that encode regions of multiple sizes centered on each pixel. The method alleviates the need for engineered features. In parallel to feature extraction, a tree of segments is computed from a graph of pixel dissimilarities. The feature vectors associated with the segments covered by each node in the tree are aggregated and fed to a classifier which produces an estimate of the distribution of object categories contained in the segment. A subset of tree nodes that cover the image are then selected so as to maximize the average "purity" of the class distributions, hence maximizing the overall likelihood that each segment will contain a single object. The system yields record accuracies on the the Sift Flow Dataset (33 classes) and the Barcelona Dataset (170 classes) and near-record accuracy on the Stanford Background Dataset (8 classes), while being an order of magnitude faster than competing approaches, producing a 320 × 240 image labeling in less than 1 second, including feature extraction.