Antithetic Monte Carlo Linear Solver

Tan, C. J. Kenneth

doi:10.1007/3-540-46080-2_40

Cited by 6 publications

(3 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This iterative scheme is also used to approximately evaluate the matrix inverse. Tan [5] studied the antithetic variates techniques for variance reduction in Monte Carlo linear solvers. Srinivasan and Aggarwal [6] used nondiagonal splitting to improve Monte Carlo linear solvers.…”

Section: Applicabilitymentioning

confidence: 99%

Convergence Analysis of Markov Chain Monte Carlo Linear Solvers Using Ulam--von Neumann Algorithm

Ji¹,

Mascagni²,

Li³

2013

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

The convergence of Markov chain-based Monte Carlo linear solvers using the Ulamvon Neumann algorithm for a linear system of the form x = Hx + b is investigated in this paper. We analyze the convergence of the Monte Carlo solver based on the original Ulam-von Neumann algorithm under the conditions that H < 1 as well as ρ(H) < 1, where ρ(H) is the spectral radius of H. We find that although the Monte Carlo solver is based on sampling the Neumann series, the convergence of Neumann series is not a sufficient condition for the convergence of the Monte Carlo solver. Actually, properties of H are not the only factors determining the convergence of the Monte Carlo solver; the underlying transition probability matrix plays an important role. An improper selection of the transition matrix may result in divergence even though the condition H < 1 holds. However, if the condition H < 1 is satisfied, we show that there always exist certain transition matrices that guarantee convergence of the Monte Carlo solver. On the other hand, if ρ(H) < 1 but H ≥ 1, the Monte Carlo linear solver may or may not converge. In particular, if the row sum N j=1 |H ij | > 1 for every row in H or, more generally, ρ(H +) > 1, where H + is the nonnegative matrix where H + ij = |H ij |, we show that transition matrices leading to convergence of the Monte Carlo solver do not exist. Finally, given H and a transition matrix P , denoting the matrix H * via H * ij = H 2 ij /P ij , we find that ρ(H *) < 1 is a necessary and sufficient condition for convergence of the Markov chain-based Monte Carlo linear solvers using the Ulam-von Neumann algorithm.

show abstract

Section: Applicabilitymentioning

confidence: 99%

Convergence Analysis of Markov Chain Monte Carlo Linear Solvers Using Ulam--von Neumann Algorithm

Ji¹,

Mascagni²,

Li³

2013

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

show abstract

“…Historically, the theory was developed on two seemingly independent tracks, related to the analysis of potential theory [8], [15], [17], [20], [21], [23] and to the solution of systems of linear equations [11], [15], [31], [32], [34]. However, the two applications are closely related and research along each of these tracks has resulted in the development of analogous algorithms, some of which are equivalent.…”

Section: −1mentioning

confidence: 99%

“…Both [11] and [34] have the advantage of being able to compute part of an inverse matrix without solving the whole system, in other words, localizing computation. Over the years, various descendant stochastic solvers have been developed [15], [31], [32], though some of them do not have the locality property.…”

Section: −1mentioning

confidence: 99%

Stochastic Preconditioning for Diagonally Dominant Matrices

Qian¹,

Sapatnekar²

2008

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Abstract. This paper presents a new stochastic preconditioning approach for large sparse matrices. For the class of matrices that are row-wise and column-wise irreducibly diagonally dominant, we prove that an incomplete LDL T factorization in a symmetric case or an incomplete LDU factorization in an asymmetric case can be obtained from random walks, and used as a preconditioner. It is argued that our factor matrices have better quality, i.e., better accuracy-size tradeoffs, than preconditioners produced by existing incomplete factorization methods. Therefore a resulting preconditioned Krylov-subspace iterative solver requires less computation than traditional methods to solve a set of linear equations with the same error tolerance. The advantage increases for larger and denser matrices. These claims are verified by numerical tests, and we provide techniques that can potentially extend the theory to non-diagonally-dominant matrices.

show abstract

Parallel implementations of randomized vector algorithm for solving large systems of linear equations

2023

View full text Add to dashboard Cite

The results of a parallel implementation of a randomized vector algorithm for solving systems of linear equations are presented in the paper. The solution is represented as a Neumann series. The stochastic method computes this series by sampling only random columns, avoiding matrixby-matrix and matrix-by-vector multiplications. We consider the case when the matrix is too large to fit in random-access memory (RAM). We take two approaches to solving this problem. In the first approach, the matrix is divided into parts, which are distributed among the MPI processes and stored in the available RAM of the cluster nodes. In the second approach, the entire matrix is stored on the hard drive of each node, loaded into RAM and processed in parts. Independent Monte Carlo experiments for random column indices are distributed among MPI processes or OpenMP threads for both matrix storage approaches. The efficiency of parallel implementations is analyzed. Results are given for a system governed by dense matrices of size 10 4 and 10 5 .

show abstract

Antithetic Monte Carlo Linear Solver

Cited by 6 publications

References 14 publications

Convergence Analysis of Markov Chain Monte Carlo Linear Solvers Using Ulam--von Neumann Algorithm

Convergence Analysis of Markov Chain Monte Carlo Linear Solvers Using Ulam--von Neumann Algorithm

Stochastic Preconditioning for Diagonally Dominant Matrices

Parallel implementations of randomized vector algorithm for solving large systems of linear equations

Contact Info

Product

Resources

About