https://web.stanford.edu/~boyd/papers/pdf/lms_consensus.pdf

Distributed average consensus with least-mean-square deviation

The Trilinos Project is an effort to facilitate the design, development, integration, and ongoing support of mathematical software libraries within an object-oriented framework for the solution of large-scale, complex multiphysics engineering and scientific problems. Trilinos addresses two fundamental issues of developing software for these problems: (i) providing a streamlined process and set of tools for development of new algorithmic implementations and (ii) promoting interoperability of independently developed software.Trilinos uses a two-level software structure designed around collections of packages. A Trilinos package is an integral unit usually developed by a small team of experts in a particular algorithms area such as algebraic preconditioners, nonlinear solvers, etc. Packages exist underneath the Trilinos top level, which provides a common look-and-feel, including configuration, documentation, licensing, and bug-tracking.Here we present the overall Trilinos design, describing our use of abstract interfaces and default concrete implementations. We discuss the services that Trilinos provides to a prospective package and how these services are used by various packages. We also illustrate how packages can be combined to rapidly develop new algorithms. Finally, we discuss how Trilinos facilitates high-quality software engineering practices that are increasingly required from simulation software.

/pdf/an-overview-of-the-trilinos-project-72l6jq6lsz.pdf

An overview of the Trilinos project

Graph layout problems are a particular class of combinatorial optimization problems whose goal is to find a linear layout of an input graph in such way that a certain objective cost is optimized. This survey considers their motivation, complexity, approximation properties, upper and lower bounds, heuristics and probabilistic analysis on random graphs. The result is a complete view of the current state of the art with respect to layout problems from an algorithmic point of view.

A survey of graph layout problems

We survey recent trends in practical algorithms for balanced graph partitioning, point to applications and discuss future research directions.

/pdf/recent-advances-in-graph-partitioning-4eois71ejs.pdf

Recent Advances in Graph Partitioning

This paper presents a new approach for analysing the structural properties of time series from complex systems. Starting from the concept of recurrences in phase space, the recurrence matrix of a time series is interpreted as the adjacency matrix of an associated complex network, which links different points in time if the considered states are closely neighboured in phase space. In comparison with similar network-based techniques the new approach has important conceptual advantages, and can be considered as a unifying framework for transforming time series into complex networks that also includes other existing methods as special cases. It has been demonstrated here that there are fundamental relationships between many topological properties of recurrence networks and different nontrivial statistical properties of the phase space density of the underlying dynamical system. Hence, this novel interpretation of the recurrence matrix yields new quantitative characteristics (such as average path length, clustering coefficient, or centrality measures of the recurrence network) related to the dynamical complexity of a time series, most of which are not yet provided by other existing methods of nonlinear time series analysis.

https://iopscience.iop.org/article/10.1088/1367-2630/12/3/033025/pdf

Recurrence networks—a novel paradigm for nonlinear time series analysis

A new approximation algorithm for maximum weighted matching in general edge-weighted graphs is presented. It calculates a matching with an edge weight of at least 1/2 of the edge weight of a maximum weighted matching. Its time complexity is O(|E|), with |E| being the number of edges in the graph. This improves over the previously known 1/2 -approximation algorithms for maximum weighted matching which require O(|E| ċ log(|V|)) steps, where |V| is the number of vertices.

Linear time 1/2 -approximation algorithm for maximum weighted matching in general graphs

A new approximation algorithm for maximum weighted matching in general edge-weighted graphs is presented. It calculates a matching with an edge weight of at least of the edge weight of a maximum weighted matching. Its time complexity is O(|E|), with |E| being the number of edges in the graph. This improves over the previously known -approximation algorithms for maximum weighted matching which require O(|E| log(|V|)) steps, where |V| is the number of vertices.

Linear Time 1/2-Approximation Algorithm for Maximum Weighted Matching in General Graphs

We present a dynamic distributed load balancing algorithm for parallel, adaptive Finite Element simulations in which we use preconditioned Conjugate Gradient solvers based on domain-decomposition The load balancing is designed to maintain good partition aspect ratio and we show that cut size is not always the appropriate measure in load balancing Furthermore, we attempt to answer the question why the aspect ratio of partitions plays an important role for certain solvers We define and rate different kinds of aspect ratio and present a new center-based partitioning method of calculating the initial distribution which implicitly optimizes this measure During the adaptive simulation, the load balancer calculates a balancing flow using different versions of the diffusion algorithm and a variant of breadth first search Elements to be migrated are chosen according to a cost function aiming at the optimization of subdomain shapes Experimental results for Bramble's preconditioner and comparisons to state-of-the-art load balancers show the benefits of the construction

Shape-optimized mesh partitioning and load balancing for parallel adaptive FEM

Several different diffusion schemes have previously been developed for load balancing on homogeneous processor networks. We generalize existing schemes, in order to deal with heterogeneous networks.

Diffusion Schemes for Load Balancing on Heterogeneous Networks

Multilevel strategies have proven to be very powerful approaches in order to partition graphs efficiently. Their efficiency is dominated by two parts; the coarsening and the local improvement strategies. Several methods have been developed to solve these problems, but their efficiency has only been proven on an experimental basis. In this paper, we present new and efficient methods for both problems, while satisfying certain quality measurements. For the coarsening part we develop a new approximation algorithm for maximum weighted matching in general edge-weighted graphs. It calculates a matching with an edge weight of at least 1 2 of the edge weight of a maximum weighted matching. Its time complexity is O(|E|), with |E| being the number of edges in the graph. Furthermore, we use the Helpful-Set strategy for the local improvement of partitions. For partitioning graphs with a regular degree of 2k into two parts, it guarantees an upper bound of ((k−1)/2)|V|+1 on the cut size of the partition, with |V| being the number of vertices. These quality methods used for the two parts of the multilevel approach lead to an efficient graph-partitioning concept.

Quality matching and local improvement for multilevel graph-partitioning

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