A spectral analysis of the domain decomposed Monte Carlo method for linear systems

S. R. Slattery; P. P.H. Wilson; T. M. Evans

A spectral analysis of the domain decomposed Monte Carlo method for linear systems

S. R. Slattery
, P. P.H. Wilson
, T. M. Evans

Radiation Transport and HPC Methods

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

1 Scopus citations

Abstract

The domain decomposed behavior of the adjoint Neumann-Ulam Monte Carlo method for solving linear systems is analyzed using the spectral properties of the linear operator. Relationships for the average length of the adjoint random walks, a measure of convergence speed and serial performance, are made with respect to the eigenvalues of the linear operator. In addition, relationships for the effective optical thickness of a domain in the decomposition are presented based on the spectral analysis and diffusion theory. Using the effective optical thickness, the Wigner rational approximation and the mean chord approximation are applied to estimate the leakage fraction of stochastic histories from a domain in the decomposition as a measure of parallel performance and potential communication costs. The one-speed, two-dimensional neutron diffusion equation is used as a model problem to test the models for symmetric operators. In general, the derived approximations show good agreement with measured computational results.

Original language	English
Title of host publication	International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013
Pages	2523-2534
Number of pages	12
State	Published - 2013
Event	International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013 - Sun Valley, ID, United States Duration: May 5 2013 → May 9 2013

Publication series

Name	International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013
Volume	4

Conference

Conference	International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013
Country/Territory	United States
City	Sun Valley, ID
Period	05/5/13 → 05/9/13

Funding

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research program.

Keywords

Domain decomposition
Linear solvers
MCSA
Monte Carlo
Parallel computing

Cite this

Slattery, S. R., Wilson, P. P. H., & Evans, T. M. (2013). A spectral analysis of the domain decomposed Monte Carlo method for linear systems. In International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013 (pp. 2523-2534). (International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013; Vol. 4).

Slattery, S. R. ; Wilson, P. P.H. ; Evans, T. M. / A spectral analysis of the domain decomposed Monte Carlo method for linear systems. International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013. 2013. pp. 2523-2534 (International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013).

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title = "A spectral analysis of the domain decomposed Monte Carlo method for linear systems",

abstract = "The domain decomposed behavior of the adjoint Neumann-Ulam Monte Carlo method for solving linear systems is analyzed using the spectral properties of the linear operator. Relationships for the average length of the adjoint random walks, a measure of convergence speed and serial performance, are made with respect to the eigenvalues of the linear operator. In addition, relationships for the effective optical thickness of a domain in the decomposition are presented based on the spectral analysis and diffusion theory. Using the effective optical thickness, the Wigner rational approximation and the mean chord approximation are applied to estimate the leakage fraction of stochastic histories from a domain in the decomposition as a measure of parallel performance and potential communication costs. The one-speed, two-dimensional neutron diffusion equation is used as a model problem to test the models for symmetric operators. In general, the derived approximations show good agreement with measured computational results.",

keywords = "Domain decomposition, Linear solvers, MCSA, Monte Carlo, Parallel computing",

author = "Slattery, \{S. R.\} and Wilson, \{P. P.H.\} and Evans, \{T. M.\}",

year = "2013",

language = "English",

isbn = "9781627486439",

series = "International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013",

pages = "2523--2534",

booktitle = "International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013",

note = "International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013 ; Conference date: 05-05-2013 Through 09-05-2013",

}

Slattery, SR, Wilson, PPH & Evans, TM 2013, A spectral analysis of the domain decomposed Monte Carlo method for linear systems. in International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013. International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013, vol. 4, pp. 2523-2534, International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013, Sun Valley, ID, United States, 05/5/13.

A spectral analysis of the domain decomposed Monte Carlo method for linear systems. / Slattery, S. R.; Wilson, P. P.H.; Evans, T. M.
International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013. 2013. p. 2523-2534 (International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013; Vol. 4).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

TY - GEN

T1 - A spectral analysis of the domain decomposed Monte Carlo method for linear systems

AU - Slattery, S. R.

AU - Wilson, P. P.H.

AU - Evans, T. M.

PY - 2013

Y1 - 2013

N2 - The domain decomposed behavior of the adjoint Neumann-Ulam Monte Carlo method for solving linear systems is analyzed using the spectral properties of the linear operator. Relationships for the average length of the adjoint random walks, a measure of convergence speed and serial performance, are made with respect to the eigenvalues of the linear operator. In addition, relationships for the effective optical thickness of a domain in the decomposition are presented based on the spectral analysis and diffusion theory. Using the effective optical thickness, the Wigner rational approximation and the mean chord approximation are applied to estimate the leakage fraction of stochastic histories from a domain in the decomposition as a measure of parallel performance and potential communication costs. The one-speed, two-dimensional neutron diffusion equation is used as a model problem to test the models for symmetric operators. In general, the derived approximations show good agreement with measured computational results.

AB - The domain decomposed behavior of the adjoint Neumann-Ulam Monte Carlo method for solving linear systems is analyzed using the spectral properties of the linear operator. Relationships for the average length of the adjoint random walks, a measure of convergence speed and serial performance, are made with respect to the eigenvalues of the linear operator. In addition, relationships for the effective optical thickness of a domain in the decomposition are presented based on the spectral analysis and diffusion theory. Using the effective optical thickness, the Wigner rational approximation and the mean chord approximation are applied to estimate the leakage fraction of stochastic histories from a domain in the decomposition as a measure of parallel performance and potential communication costs. The one-speed, two-dimensional neutron diffusion equation is used as a model problem to test the models for symmetric operators. In general, the derived approximations show good agreement with measured computational results.

KW - Domain decomposition

KW - Linear solvers

KW - MCSA

KW - Monte Carlo

KW - Parallel computing

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M3 - Conference contribution

AN - SCOPUS:84883378768

SN - 9781627486439

T3 - International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013

SP - 2523

EP - 2534

BT - International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013

T2 - International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013

Y2 - 5 May 2013 through 9 May 2013

ER -

Slattery SR, Wilson PPH, Evans TM. A spectral analysis of the domain decomposed Monte Carlo method for linear systems. In International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013. 2013. p. 2523-2534. (International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013).

A spectral analysis of the domain decomposed Monte Carlo method for linear systems

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