In this paper, we propose distributed solvers for systems of linear equations given by symmetric diagonally dominant M-matrices based on the parallel solver of Spielman and Peng. We propose two versions of the solvers, where in the first, full communication in the network is required, while in the second communication is restricted to the R-Hop neighborhood between nodes for some $R \geq 1$. We rigorously analyze the convergence and convergence rates of our solvers, showing that our methods are capable of outperforming state-of-the-art techniques.
Having developed such solvers, we then contribute by proposing an accurate distributed Newton method for network flow optimization. Exploiting the sparsity pattern of the dual Hessian, we propose a Newton method for network flow optimization that is both faster and more accurate than state-of-the-art techniques. Our method utilizes the distributed SDDM solvers for determining the Newton direction up to any arbitrary precision $ε>0$. We analyze the properties of our algorithm and show superlinear convergence within a neighborhood of the optimal. Finally, in a set of experiments conducted on randomly generated and barbell networks, we demonstrate that our approach is capable of significantly outperforming state-of-the-art techniques.
Computational acceleration strategies for large-scale energy system optimization: a comparative study of GPU-accelerated and distributed-memory solvers
Freie Universität Berlin（柏林自由大学）；Deutsches Zentrum für Luft- und Raumfahrt e.V.（德国航空航天中心）；Zuse Institute Berlin（祖斯研究所柏林）；GAMS Software GmbH（GAMS软件有限公司）
CLAIRE: A distributed-memory solver for constrained large deformation diffeomorphic image registration
University of Houston（休斯顿大学）；University of California, Berkeley（加州大学伯克利分校）；University of Pennsylvania（宾夕法尼亚大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
N2N: A Parallel Framework for Large-Scale MILP under Distributed Memory
Shenzhen Research Institute of Big Data（深圳大数据研究 institute）；Theory Lab, Central Research Institute, 2012 Labs, Huawei Technologies Co. Ltd.（华为技术有限公司理论实验室，中央研究所，2012实验室）；Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
Finite-Time Distributed Linear Equation Solver for Minimum $l_1$ Norm Solutions
School of Aeronautics and Astronautics, Purdue University（普渡大学航空航天学院）；Hangzhou Dianzi University（杭州电子科技大学）；The Australian National University（澳大利亚国立大学）；Data-61 CSIRO (formerly NICTA)（CSIRO数据六部（原NICTA））
Overview of Bayesian Solvers in EEG Distributed Source Models: Prior Selection, Algorithmic Implementation, and Depth Bias Reduction
Faculty of Information Technology and Communication Sciences, Tampere University（坦佩雷大学信息与通信科学学院）；Institute for Mathematical Innovation, University of Bath（巴斯大学数学创新研究所）
