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A Probabilistic Linear Solver Based on a Multilevel Monte Carlo Method

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Abstract

We describe a new Monte Carlo method based on a multilevel method for computing the action of the resolvent matrix over a vector. The method is based on the numerical evaluation of the Laplace transform of the matrix exponential, which is computed efficiently using a multilevel Monte Carlo method. Essentially, it requires generating suitable random paths which evolve through the indices of the matrix according to the probability law of a continuous-time Markov chain governed by the associated Laplacian matrix. The convergence of the proposed multilevel method has been discussed, and several numerical examples were run to test the performance of the algorithm. These examples concern the computation of some metrics of interest in the analysis of complex networks, and the numerical solution of a boundary-value problem for an elliptic partial differential equation. In addition, the algorithm was conveniently parallelized, and the scalability analyzed and compared with the results of other existing Monte Carlo method for solving linear algebra systems.

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References

  1. Forsythe, G., Leibler, R.: Matrix inversion by a Monte Carlo method. Math. Tables Other Aids Comput. 4, 127–129 (1950)

    Article  MathSciNet  Google Scholar 

  2. Dimov, I.T., Dimov, T.T., Gurov, T.V.: A new iterative Monte Carlo approach for inverse matrix problem. J. Comput. Appl. Math. 92, 15–35 (1998)

    Article  MathSciNet  Google Scholar 

  3. Dimov, I.T., Alexandrov, V.N., Karaivanova, A.: Parallel resolvent Monte Carlo algorithms for linear algebra problems. Math. Comput. Simul. 55, 25–35 (2001)

    Article  MathSciNet  Google Scholar 

  4. Dimov, I.T.: Monte Carlo Methods for Applied Scientists. World Scientific, Singapore (2008)

    MATH  Google Scholar 

  5. Ökten, G.: Solving linear equations by Monte Carlo simulation. SIAM J. Sci. Comput. 27, 511–531 (2005)

    Article  MathSciNet  Google Scholar 

  6. Ji, H., Mascagni, M., Li, Y.: Convergence analysis of Markov Chain Monte Carlo linear solvers using Ulam-von Neumann algorithm. SIAM J. Numer. Anal. 51, 2107–2122 (2013)

    Article  MathSciNet  Google Scholar 

  7. Dimov, I., Maire, S., Sellier, J.M.: A new Walk on Equations Monte Carlo method for solving systems of linear algebraic equations. Appl. Math. Model. 39, 4494–4510 (2015)

    Article  MathSciNet  Google Scholar 

  8. Benzi, M., Evans, T.M., Hamilton, S.P., Pasini, M.L., Slattery, S.R.: Analysis of Monte Carlo accelerated iterative methods for sparse linear systems. Numerical Linear Algebra Appl. 24, e2088 (2017)

    Article  MathSciNet  Google Scholar 

  9. Evans, T.M., Mosher, S.W., Slattery, S.R., Hamilton, S.P.: A Monte Carlo synthetic-acceleration method for solving the thermal radiation diffusion equation. J. Comput. Phys. 258, 338–358 (2014)

    Article  MathSciNet  Google Scholar 

  10. Higham, N.J., Al-Mohy, A.H.: Computing matrix functions. Acta Numerica 19, 159–208 (2010)

    Article  MathSciNet  Google Scholar 

  11. Higham, N.J., Al-Mohy, A.H.: Functions of matrices: Theory and Computation. SIAM, Philadelphia (2008)

    Book  Google Scholar 

  12. Acebrón, J.A.: A Monte Carlo method for computing the action of a matrix exponential on a vector. Appl. Math. Comput. 362, 124545 (2019)

    MathSciNet  MATH  Google Scholar 

  13. Acebrón, J.A., Herrero, J.R., Monteiro, J.: A highly parallel algorithm for computing the action of a matrix exponential on a vector based on a multilevel Monte Carlo method. Submitted (2019). arXiv:1904.12754

  14. Giles, M.B.: Multilevel Monte Carlo path simulation. Oper. Res. 56, 607–617 (2008)

    Article  MathSciNet  Google Scholar 

  15. Giles, M.B.: Multilevel Monte Carlo methods. Acta Numerica 24, 259–328 (2015)

    Article  MathSciNet  Google Scholar 

  16. Anderson, D.F., Higham, D.J.: Multilevel Monte Carlo for continuous time Markov chains, with applications in biochemical kinetics. Multiscale Model. Simul. 10, 146–179 (2012)

    Article  MathSciNet  Google Scholar 

  17. Benzi, M., Estrada, E., Klymko, C.: Ranking hubs and authorities using matrix functions. Linear Algebra Appl. 438, 2447–2474 (2013)

    Article  MathSciNet  Google Scholar 

  18. Higham, D.: An introduction to multilevel Monte Carlo for option valuation. Int. J. Comput. Math. 92, 2347–2360 (2015)

    Article  MathSciNet  Google Scholar 

  19. Glasserman, P.: Monte Carlo Methods in Financial Engineering. Springer, Berlin (2004)

    MATH  Google Scholar 

  20. Chung, F., Lu, L.: Complex Graphs and Networks. American Mathematical Society, Providence (2006)

    Book  Google Scholar 

  21. Jahnke, T., Lubich, C.: Error bounds for exponential operator splittings. BIT 40, 735–744 (2000)

    Article  MathSciNet  Google Scholar 

  22. Mascagni, M., Karaivanova, A.: A parallel Quasi-Monte Carlo method for solving systems of linear equations. In: International Conference on Computational Science, pp. 598–608 (2002)

  23. http://www.maths.strath.ac.uk/research/groups/numerical_analysis/contest

  24. Benzi, M., Klymko, C.: Total communicability as a centrality measure. J. Complex Netw. 1, 124–149 (2013)

    Article  Google Scholar 

  25. Katz, L.: A new status index derived from sociometric analysis. Psychometrika 8, 39–43 (1953)

    Article  Google Scholar 

  26. Mattheij, R.M.M., Rienstra, S.W., ten Thije Boonkkamp, J.H.M.: Partial Differential Equations: Modeling, Analysis, Computation. SIAM monographs (2005)

  27. http://www.comsol.com/

  28. Estrada, E., Hatano, N., Benzi, M.: The physics of communicability in complex networks. Phys. Rep. 514, 89–119 (2012)

    Article  MathSciNet  Google Scholar 

  29. http://www.mathe.tu-freiberg.de/guettels/funm kryl/

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Acknowledgements

This work was supported by Fundação para a Ciência e a Tecnologia under Grant No. UIDB/50021/2020. The author gratefully thanks Paula Marques for her suggestions and assistance with this paper.

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Authors and Affiliations

  1. Department of Information Science and Technology, ISCTE-University Institute of Lisbon, Lisbon, Portugal

    Juan A. Acebrón

  2. INESC-ID, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal

    Juan A. Acebrón

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  1. Juan A. Acebrón
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Correspondence to Juan A. Acebrón.

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Acebrón, J.A. A Probabilistic Linear Solver Based on a Multilevel Monte Carlo Method. J Sci Comput 82, 65 (2020). https://doi.org/10.1007/s10915-020-01168-2

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  • DOI: https://doi.org/10.1007/s10915-020-01168-2

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