Recently, a number of researchers have investigated a class of graph partitioning algorithms that reduce the size of the graph by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph [Bui and Jones, Proc. of the 6th SIAM Conference on Parallel Processing for Scientific Computing, 1993, 445--452; Hendrickson and Leland, A Multilevel Algorithm for Partitioning Graphs, Tech. report SAND 93-1301, Sandia National Laboratories, Albuquerque, NM, 1993]. From the early work it was clear that multilevel techniques held great promise; however, it was not known if they can be made to consistently produce high quality partitions for graphs arising in a wide range of application domains. We investigate the effectiveness of many different choices for all three phases: coarsening, partition of the coarsest graph, and refinement. In particular, we present a new coarsening heuristic (called heavy-edge heuristic) for which the size of the partition of the coarse graph is within a small factor of the size of the final partition obtained after multilevel refinement. We also present a much faster variation of the Kernighan--Lin (KL) algorithm for refining during uncoarsening. We test our scheme on a large number of graphs arising in various domains including finite element methods, linear programming, VLSI, and transportation. Our experiments show that our scheme produces partitions that are consistently better than those produced by spectral partitioning schemes in substantially smaller time. Also, when our scheme is used to compute fill-reducing orderings for sparse matrices, it produces orderings that have substantially smaller fill than the widely used multiple minimum degree algorithm.

/pdf/a-fast-and-high-quality-multilevel-scheme-for-partitioning-2i240yved4.pdf

A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs

Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions, two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a "global" cost to every such transport, using the "local" consideration of how much it costs to move a grain of sand from one place to another. Recent years have witnessed the spread of OT in several fields, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.

Computational Optimal Transport

http://www.mathcs.emory.edu/%7Ebenzi/Web_papers/survey.pdf

Preconditioning techniques for large linear systems: a survey

The purpose of this work is to present the MOSEK optimizer intended for solution of large-scale sparse linear programs. The optimizer is based on the homogeneous interior-point algorithm which in contrast to the primal-dual algorithm detects a possible primal or dual infeasibility reliably. It employs advanced (parallelized) linear algebra, it handles dense columns in the constraint matrix efficiently, and it has a basis identification procedure.

The Mosek Interior Point Optimizer for Linear Programming: An Implementation of the Homogeneous Algorithm

An approximate minimum degree (AMD) ordering algorithm for preordering a symmetric sparse matrix prior to numerical factorization is presented. We use techniques based on the quotient graph for matrix factorization that allow us to obtain computationally cheap bounds for the minimum degree. We show that these bounds are often equal to the actual degree. The resulting algorithm is typically much faster than previous minimum degree ordering algorithms and produces results that are comparable in quality with the best orderings from other minimum degree algorithms.

An Approximate Minimum Degree Ordering Algorithm

Over the past fifteen years, the implementation of the minimum degree algorithm has received much study, and many important enhancements have been made to it. This paper describes these various enhancements, their historical development, and some experiments showing how very effective they are in improving the execution time of the algorithm. A shortcoming is also presented that exists in all of the widely used implementations of the algorithm, namely, that the quality of the ordering provided by the implementations is surprisingly sensitive to the initial ordering. For example, changing the input ordering can lead to an increase (or decrease) of as much as a factor of three in the cost of the subsequent numerical factorization. This sensitivity is caused by the lack of an effective tie-breaking strategy, and the authors’ experiments illustrate the importance of developing such a strategy

The evolution of the minimum degree ordering algorithm

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The evolution of the minimum degree ordering algorithm