Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

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Iterative Methods for Sparse Linear Systems

This paper provides a theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link/node failures, time-delays, and performance guarantees. An overview of basic concepts of information consensus in networks and methods of convergence and performance analysis for the algorithms are provided. Our analysis framework is based on tools from matrix theory, algebraic graph theory, and control theory. We discuss the connections between consensus problems in networked dynamic systems and diverse applications including synchronization of coupled oscillators, flocking, formation control, fast consensus in small-world networks, Markov processes and gossip-based algorithms, load balancing in networks, rendezvous in space, distributed sensor fusion in sensor networks, and belief propagation. We establish direct connections between spectral and structural properties of complex networks and the speed of information diffusion of consensus algorithms. A brief introduction is provided on networked systems with nonlocal information flow that are considerably faster than distributed systems with lattice-type nearest neighbor interactions. Simulation results are presented that demonstrate the role of small-world effects on the speed of consensus algorithms and cooperative control of multivehicle formations

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Consensus and Cooperation in Networked Multi-Agent Systems

We study a distributed computation model for optimizing a sum of convex objective functions corresponding to multiple agents. For solving this (not necessarily smooth) optimization problem, we consider a subgradient method that is distributed among the agents. The method involves every agent minimizing his/her own objective function while exchanging information locally with other agents in the network over a time-varying topology. We provide convergence results and convergence rate estimates for the subgradient method. Our convergence rate results explicitly characterize the tradeoff between a desired accuracy of the generated approximate optimal solutions and the number of iterations needed to achieve the accuracy.

http://www.ifp.illinois.edu/~angelia/distributed_journal_final.pdf

Distributed Subgradient Methods for Multi-Agent Optimization

In cellular wireless communication systems, transmitted power is regulated to provide each user an acceptable connection by limiting the interference caused by other users. Several models have been considered including: (1) fixed base station assignment where the assignment of users to base stations is fixed, (2) minimum power assignment where a user is iteratively assigned to the base station at which its signal to interference ratio is highest, and (3) diversity reception where a user's signal is combined from several or perhaps all base stations. For the above models, the uplink power control problem can be reduced to finding a vector p of users' transmitter powers satisfying p/spl ges/I(p) where the jth constraint p/sub j//spl ges/I/sub j/(p) describes the interference that user j must overcome to achieve an acceptable connection. This work unifies results found for these systems by identifying common properties of the interference constraints. It is also shown that systems in which transmitter powers are subject to maximum power limitations share these common properties. These properties permit a general proof of the synchronous and totally asynchronous convergence of the iteration p(t+1)=I(p(t)) to a unique fixed point at which total transmitted power is minimized. >

A framework for uplink power control in cellular radio systems

We propose an optimization approach to flow control where the objective is to maximize the aggregate source utility over their transmission rates. We view network links and sources as processors of a distributed computation system to solve the dual problem using a gradient projection algorithm. In this system, sources select transmission rates that maximize their own benefits, utility minus bandwidth cost, and network links adjust bandwidth prices to coordinate the sources' decisions. We allow feedback delays to be different, substantial, and time varying, and links and sources to update at different times and with different frequencies. We provide asynchronous distributed algorithms and prove their convergence in a static environment. We present measurements obtained from a preliminary prototype to illustrate the convergence of the algorithm in a slowly time-varying environment. We discuss its fairness property.

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Optimization flow control—I: basic algorithm and convergence

This book focuses on numerical algorithms suited for parallelization for solving systems of equations and optimization problems Emphasis on relaxation methods of the Jacobi and Gauss-Seidel type, and issues of communication and synchronization Topics covered include: Algorithms for systems of linear equations and matrix inversion; Herative methods for nonlinear problems; and Shortest paths and dynamic programming

Alexander N. Sennikov

Papers

Parallel and distributed computation