Software for Exascale Computing - SPPEXA 2016-2019

This open access book summarizes the research done and results obtained in the second funding phase of the Priority Program 1648 "Software for Exascale Computing" (SPPEXA) of the German Research Foundation (DFG) presented at the SPPEXA Symposium in Dresden during October 21-23, 2019. In that respect, it both represents a continuation of Vol. 113 in Springer’s series Lecture Notes in Computational Science and Engineering, the corresponding report of SPPEXA’s first funding phase, and provides an overview of SPPEXA’s contributions towards exascale computing in today's sumpercomputer technology. The individual chapters address one or more of the research directions (1) computational algorithms, (2) system software, (3) application software, (4) data management and exploration, (5) programming, and (6) software tools. The book has an interdisciplinary appeal: scholars from computational sub-fields in computer science, mathematics, physics, or engineering will find it of particular interest

A Distributed and Parallel Asynchronous Unite and Conquer Method to Solve Large Scale Non-Hermitian Linear Systems with Multiple Right-hand Sides

International audienceMany problems in the field of science and engineering often require to solve simultaneously large-scale non-Hermitian sparse linear systems with multiple right-hand sides (RHSs). Efficiently solving such problems on extreme-scale platforms requires the minimization of global communications, reduction of synchronization points and promotion of asynchronous communications. We develop an extension of the Unite and Conquer GMRES/LS-ERAM (UCGLE) method [1] by combining it with Block GMRES method to solve non-Hermitian linear systems with multiple RHSs. UCGLE is a hybrid method consisting of three computing algorithms with asynchronous communication that allow the use of approximate eigenvalues to accelerate to solve of linear systems and to improve their fault tolerance. In this paper, the variant of UCGLE with novel components and manager engine implementations is introduced. This engine is capable of allocating multiple Block GMRES at the same time, each Block GMRES solving the linear systems with a subset of RHSs and accelerating the convergence using the eigenvalues approximated by other eigensolvers. Dividing the entire linear system with multiple RHSs into subsets and solving them simultaneously with different allocated linear solvers allow localizing calculations, reducing global communication, and improving parallel performance. Meanwhile, the asynchronous preconditioning using eigenvalues is able to speed up the convergence and improve the fault tolerance and reusability. Numerical experiments using different test matrices on supercomputer ROMEO indicate that the proposed method achieves a substantial decrease in both computation time and iterative steps with good scaling performance

Clustering and expansion improvement for well-clustered graphs

A well-clustered graph is a collection of densely connected components (clusters) such that the vertices inside each cluster are better connected than those between different clusters. Well-clustered graphs constitute one of the most important families of graphs occurring in various practical domains of scientific disciplines, including data science, social network analysis and bioinformatics. This thesis conducts algorithmic studies of well-clustered graphs and, by focusing on the clustering and expansion improvement problems, the thesis presents three algorithms which not only broaden our theoretical understanding but also perform well in practice. Specifically, the main contributions are summarised as follows: Firstly, we consider the hierarchical clustering problem on well-clustered graphs. Hierarchical clustering studies a recursive representation of a data set into clusters of increasingly smaller size via a binary tree. Based on the cost function introduced by Dasgupta, we present a polynomial time $O(1)$-approximation algorithm that computes a hierarchical representation of a well-clustered graph. This algorithm is based on our linear time $O(1)$-approximation algorithm for graphs of high expansion, whose design bypasses complicated routines known in the literature. While constructing $O(1)$-approximate hierarchical trees for general graphs is NP-hard under the Small Set Expansion Hypothesis, our result shows that constructing such trees is tractable for well-clustered graphs. Secondly, we consider the scenario when the input graph represents a network distributed among many sites. The design of most graph clustering algorithms is based on complicated techniques which are inapplicable in the distributed setting. We present a novel distributed algorithm for graph clustering that works for well-clustered graphs with clusters of arbitrary sizes, and the approximation guarantee of our algorithm is with respect to every individual cluster. In addition, our algorithm is easy to implement, and only requires a poly-logarithmic number of synchronous rounds for many input graphs. Thirdly, we study the class of well-clustered graphs from the perspective of improving the overall expansion. The objective of the problem of improving the expansion is to add a certain number of external edges to our input, such that the resulting graph is very well connected. We present a fast algorithm that, given a set of suitable candidate edges, finds a small subset of edges which drastically improve the overall expansion if added to the input graph. Overall, the thesis presents three results that improve the state-of-the-art for different problems concerning well-clustered graphs. We believe that these results together with the new techniques introduced would inspire future related studies