Distributed Randomized Kaczmarz and Applications to Seismic Imaging in Sensor Network

Many real-world wireless sensor network applications such as environmental monitoring, structural health monitoring, and smart grid can be formulated as a least-squares problem. In distributed Cyber-Physical System (CPS), each sensor node observes partial phenomena due to spatial and temporal restriction and is able to form only partial rows of least-squares. Traditionally, these partial measurements were gathered at a centralized location. However, with the increase in sensors and their measurements, aggregation is becoming challenging and infeasible. In this paper, we propose distributed randomized kaczmarz that performs in-network computation to solve least-squares over the network by avoiding costly communication. As a case study, we present a volcano monitoring application on a distributed CORE emulator and use real data from Mt. St. Helens to evaluate our proposed method.

[1]  P. Hansen.  Discrete Inverse Problems: Insight and Algorithms , 2010 .

[2]  Léon Bottou,et al.  Large-Scale Machine Learning with Stochastic Gradient Descent , 2010, COMPSTAT.

[3]  Qing Ling,et al.  EXTRA: An Exact First-Order Algorithm for Decentralized Consensus Optimization , 2014, 1404.6264.

[4]  Renjie Huang,et al.  Air-dropped sensor network for real-time high-fidelity volcano monitoring , 2009, MobiSys '09.

[5]  Rosemary A. Renaut,et al.  A parallel multisplitting solution of the least squares problem , 1998, Numer. Linear Algebra Appl..

[6]  D. Needell.  Randomized Kaczmarz solver for noisy linear systems , 2009, 0902.0958.

[7]  Gianluigi Ferrari,et al.  Wireless Sensor Networks for Structural Health Monitoring , 2015, Int. J. Distributed Sens. Networks.

[8]  Gabor T. Herman,et al.  On the Bayesian Approach to Image Reconstruction , 1979, Inf. Control..

[9]  Robbert van Renesse,et al.  Astrolabe: A robust and scalable technology for distributed system monitoring, management, and data mining , 2003, TOCS.

[10]  Rosemary A. Renaut.  A Parallel Multisplitting Solution of the Least Squares Problem , 1998 .

[11]  Yair Censor,et al.  Component averaging: An efficient iterative parallel algorithm for large and sparse unstructured problems , 2001, Parallel Comput..

[12]  John Anderson,et al.  Wireless sensor networks for habitat monitoring , 2002, WSNA '02.

[13]  Anand D. Sarwate,et al.  Broadcast Gossip Algorithms for Consensus , 2009, IEEE Transactions on Signal Processing.

[14]  R. Vershynin,et al.  A Randomized Kaczmarz Algorithm with Exponential Convergence , 2007, math/0702226.

[15]  Deborah Estrin,et al.  An Empirical Study of Epidemic Algorithms in Large Scale Multihop Wireless Networks , 2002 .

[16]  Wilfried N. Gansterer,et al.  Distributed QR Factorization Based on Randomized Algorithms , 2011, PPAM.

[17]  Qing Ling,et al.  On the Convergence of Decentralized Gradient Descent , 2013, SIAM J. Optim..

[18]  Brian Adamson,et al.  Integration of the CORE and EMANE Network Emulators , 2011, 2011 - MILCOM 2011 Military Communications Conference.

[19]  Johannes Gehrke,et al.  Gossip-based computation of aggregate information , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[20]  Yong Meng Teo,et al.  Sensor grid: integration of wireless sensor networks and the grid , 2005, The IEEE Conference on Local Computer Networks 30th Anniversary (LCN'05)l.

[21]  Wei Hong,et al.  Proceedings of the 5th Symposium on Operating Systems Design and Implementation Tag: a Tiny Aggregation Service for Ad-hoc Sensor Networks , 2022 .

[22]  Wilfried N. Gansterer,et al.  Analysis and Comparison of Truly Distributed Solvers for Linear Least Squares Problems on Wireless Sensor Networks , 2014, Euro-Par.

[23]  Lei Shi,et al.  Component-Average Based Distributed Seismic Tomography in Sensor Networks , 2013, 2013 IEEE International Conference on Distributed Computing in Sensor Systems.

[24]  Yair Censor,et al.  On Diagonally Relaxed Orthogonal Projection Methods , 2007, SIAM J. Sci. Comput..

[25]  Stephen P. Boyd,et al.  Randomized gossip algorithms , 2006, IEEE Transactions on Information Theory.

[26]  Jonathan M. Lees,et al.  The magma system of Mount St. Helens: non-linear high-resolution P-wave tomography , 1992 .

[27]  H. Vincent Poor,et al.  Robust Distributed Least-Squares Estimation in Sensor Networks with Node Failures , 2011, 2011 IEEE Global Telecommunications Conference - GLOBECOM 2011.

[28]  José M. F. Moura,et al.  Fast Distributed Gradient Methods , 2011, IEEE Transactions on Automatic Control.

[29]  Jun Ye Yu,et al.  Performance comparison of randomized gossip, broadcast gossip and collection tree protocol for distributed averaging , 2013, 2013 5th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).

[30]  Nikolaos V. Sahinidis,et al.  GPU computing with Kaczmarz's and other iterative algorithms for linear systems , 2010, Parallel Comput..

[31]  Wen-Zhan Song,et al.  Volcanic earthquake timing using wireless sensor networks , 2013, 2013 ACM/IEEE International Conference on Information Processing in Sensor Networks (IPSN).

[32]  James Demmel,et al.  Wireless sensor networks for structural health monitoring , 2006, SenSys '06.

[33]  Lei Shi,et al.  Imaging Seismic Tomography in Sensor Network , 2013, DCOSS.

[34]  David L. Johnson,et al.  Simple pragmatic approach to mesh routing using BATMAN , 2008 .

[35]  Gabor T. Herman,et al.  Image reconstruction from projections : the fundamentals of computerized tomography , 1980 .