Abstract
We present a class of iterative fully distributed fixed point methods to solve a system of linear equations, such that each agent in the network holds one or several of the equations of the system. Under a generic directed, strongly connected network, we prove a convergence result analogous to the one for fixed point methods in the classical, centralized, framework: the proposed method converges to the solution of the system of linear equations at a linear rate. We further explicitly quantify the rate in terms of the linear system and network parameters. Next, we show that the algorithm provably works under time-varying directed networks provided that the underlying graph is connected over bounded iteration intervals, and we establish a linear convergence rate for this setting as well. A set of numerical results is presented, demonstrating practical benefits of the method over existing alternatives.
Highlights
The problem we consider is Ay = b (1)where A = [aij] ∈ Rn×n and b = [bi] ∈ Rn are given, and y ∈ Rn is the vector of the unknowns
When the semivariogram γ(h) of the random field has a sill, it can be assumed that there is a range hover which the covariance Cov(Z(s), Z(s + h)) = 0, when |h| > h. In this case the matrix A of the Ordinary Kriging linear system becomes sparse, since its elements are the estimates of γ(h) and each sampled node of the random field Z needs to memorize only the information brought by its neighbours at a distance lower than hto estimate the model parameters
We proposed a class of novel, iterative, distributed methods for the solution of linear systems of equations, derived upon classical fixed point methods
Summary
Major bottlenecks include waiting for the slowest node to complete an iteration, or latency incurred by the time to communicate a message For this reason asynchronous methods, which allow for latency in communication and nonuniform distribution of computational work, are considered, [7]. The framework we consider in this paper for solving systems of linear equations, in more detail, assumes a network of computational nodes which communicate through a generic directed graph, which can depend on time. Any system of linear equation (1) with symmetric matrix A can be considered as the first order optimality condition of an unconstrained optimization problem with cost function xtAx. It is of interest to compare the approach of solving (1) applying some distributed optimization method [11,16,18].
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