A new Walk on Equations Monte Carlo method for solving systems of linear algebraic equations

Dimov, Ivan; Maire, Sylvain; Sellier, Jean Michel

doi:10.1016/j.apm.2014.12.018

Cited by 34 publications

(34 citation statements)

References 26 publications

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“…The Ulam von Neumann algorithm is a Monte Carlo method which obtains an estimate for a single component x i by sampling random walks starting at the vertex i. It interprets the Neumann series representation of the solution x as a sum over weighted walks on G(G), and obtains an estimate by sampling random walks starting from vertex i over G(G) and appropriately reweighting to obtain an unbiased estimator [14], [15], [16], [17]. The challenge is to control the variance of this estimator.…”

Section: Local Algorithmsmentioning

confidence: 99%

Asynchronous Approximation of a Single Component of the Solution to a Linear System

Ozdaglar

Shah

2020

IEEE Trans. Netw. Sci. Eng.

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We present a distributed asynchronous algorithm for approximating a single component of the solution to a system of linear equations Ax = b, where A is a positive definite real matrix and b ∈ R n . This can equivalently be formulated as solving for x i in x = Gx + z for some G and z such that the spectral radius of G is less than 1. Our algorithm relies on the Neumann series characterization of the component x i , and is based on residual updates. We analyze our algorithm within the context of a cloud computation model motivated by frameworks such as Apache Spark, in which the computation is split into small update tasks performed by small processors with shared access to a distributed file system. We prove a robust asymptotic convergence result when the spectral radius ρ(|G|) < 1, regardless of the precise order and frequency in which the update tasks are performed. We provide convergence rate bounds which depend on the order of update tasks performed, analyzing both deterministic update rules via counting weighted random walks, as well as probabilistic update rules via concentration bounds. The probabilistic analysis requires analyzing the product of random matrices which are drawn from distributions that are time and path dependent. We specifically consider the setting where n is large, yet G is sparse, e.g., each row has at most d nonzero entries. This is motivated by applications in which G is derived from the edge structure of an underlying graph. Our results prove that if the local neighborhood of the graph does not grow too quickly as a function of n, our algorithm can provide significant reduction in computation cost as opposed to any algorithm which computes the global solution vector x. Our algorithm obtains an x 2 additive approximation for x i in constant time with respect to the size of the matrix when the maximum row sparsity d = O(1) and 1/(1 − G 2 ) = O(1), where G 2 is the induced matrix operator 2-norm.Index Terms-linear system of equations, local computation, asynchronous randomized algorithms, distributed algorithms ! arXiv:1411.2647v4 [cs.DS] 21 Jan 2019 1. By optimal, we would like to minimize ρ(G), which maximizes the convergence rate of the algorithm.

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Section: Local Algorithmsmentioning

confidence: 99%

Asynchronous Approximation of a Single Component of the Solution to a Linear System

Ozdaglar

Shah

2020

IEEE Trans. Netw. Sci. Eng.

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“…In order to apply the Monte Carlo method, existing work [8,16,17] assumes H < 1 (for the infinity norm H = max i j |H i,j |), which suffices to show Var[X] < ∞, but which is a stronger condition than ρ(H) < 1. Although it is possible Var[X] < ∞ when H ≥ 1, there is no easy way to check.…”

Section: Our Contributionsmentioning

confidence: 99%

“…To tackle this problem, we propose a multi-way Markov random walk which uses multiple transition matrices. At each step of the random walk, the transition matrix is constructed in a way akin to the Monte Carlo Almost Optimal (MAO) framework [8,12]. We prove that under this type of random walk, the new method always converges when ρ(H + ) < 1, where H + is the nonnegative matrix as H + ij = |H ij |.…”

Section: Our Contributionsmentioning

confidence: 99%

Multiway Monte Carlo Method for Linear Systems

Wu¹,

Gleich²

2019

SIAM J. Sci. Comput.

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We study the Monte Carlo method for solving a linear system of the form x = Hx + b. A sufficient condition for the method to work is H < 1, which greatly limits the usability of this method. We improve this condition by proposing a new multi-way Markov random walk, which is a generalization of the standard Markov random walk. Under our new framework we prove that the necessary and sufficient condition for our method to work is the spectral radius ρ(H + ) < 1, which is a weaker requirement than H < 1. In addition to solving more problems, our new method can work faster than the standard algorithm. In numerical experiments on both synthetic and real world matrices, we demonstrate the effectiveness of our new method.

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“…The system of linear equations plasy a crucial role in the mathematical models of real-world issues such as economic, physics and engineering [1][2][3][4][5]. Due to the uncertain parameters appearing in substantive problems, people often use fuzzy number in a real or complex form to describe fuzzy factors of the complex or real linear system, thus discovering new and easy-implemented schemes that would suitably deal with the fuzzy linear systems (FLS) and solve them, and enabling an expansion to a hotspot exploration [6][7][8][9][10][11][12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

A Novel Weak Fuzzy Solution for Fuzzy Linear System

Salahshour

Ahmadian

Ismail

et al. 2016

Entropy

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This article proposes a novel weak fuzzy solution for the fuzzy linear system. As a matter of fact, we define the right-hand side column of the fuzzy linear system as a piecewise fuzzy function to overcome the related shortcoming, which exists in the previous findings. The strong point of this proposal is that the weak fuzzy solution is always a fuzzy number vector. Two complex and non-complex linear systems under uncertainty are tested to validate the effectiveness and correctness of the presented method.

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A new Walk on Equations Monte Carlo method for solving systems of linear algebraic equations

Cited by 34 publications

References 26 publications

Asynchronous Approximation of a Single Component of the Solution to a Linear System

Asynchronous Approximation of a Single Component of the Solution to a Linear System

Multiway Monte Carlo Method for Linear Systems

A Novel Weak Fuzzy Solution for Fuzzy Linear System

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