Skip to main content
SpringerLink
Log in

A Quasi Monte Carlo Method for Large-Scale Inverse Problems

  • Conference paper
  • First Online:
Monte Carlo and Quasi-Monte Carlo Methods 2010

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 23))

Abstract

We consider large-scale linear inverse problems with a simulation-based algorithm that approximates the solution within a low-dimensional subspace. The algorithm uses Tikhonov regularization, regression, and low-dimensional linear algebra calculations and storage. For sampling efficiency, we implement importance sampling schemes, specially tailored to the structure of inverse problems. We emphasize various alternative methods for approximating the optimal sampling distribution and we demonstrate their impact on the reduction of simulation noise. The performance of our algorithm is tested on a practical inverse problem arising from Fredholm integral equations of the first kind.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Unless otherwise stated, from now on we deal exclusively with G lq . A simplified analysis applies to c l .

References

  1. Asmussen, S., and Glynn, P. W.: Stochastic Simulation. Springer, New York (2007)

    Google Scholar 

  2. Bertero, M., and Boccacci, P.: Introduction to Inverse Problems in Imaging. IoP, Bristol (2002)

    Google Scholar 

  3. Bertsekas, D.P., and Tsitsiklis, J.: Neuro-Dynamic Programming. Athena Scientific (1996)

    Google Scholar 

  4. Bertsekas D.P., and Yu, H.: Projected Equation Methods for Approximate Solution of Large Linear Systems. J. Comp. Appl. Math., 227, 27–50 (2009)

    Google Scholar 

  5. Biegler, L., Biros, G., Ghattas, O., Heinkenschloss, M., Keyes, D., Mallick, B., Marzouk, Y., Tenorio, L., van Blomen Waanders, B., and Willcox, K. (eds.): Large-Scale Inverse Problems and Quantification of Uncertainty. Wiley, Chichester (2011)

    Google Scholar 

  6. Curtiss, J. H.: Monte Carlo Methods for the Iteration of Linear Operators. J. Math. Phys., 32(4), 209–232 (1953)

    Google Scholar 

  7. Drineas, P., Kannan, R., and Mahoney, M.W.: Fast Monte Carlo Algorithms for Matrices I: Approximating Matrix Multiplication. SIAM J. Comput. 36, 132–157 (2006)

    Google Scholar 

  8. Drineas, P., Kannan, R., and Mahoney M.W.: Fast Monte Carlo Algorithms for Matrices II: Computing a Low-Rank Approximation to a Matrix. SIAM J. Comput. 36, 158–183 (2006)

    Google Scholar 

  9. Drineas, P., Kannan, R., and Mahoney M.W.:, Fast Monte Carlo Algorithms for Matrices III: Computing a Compressed Approximate Matrix Decomposition. SIAM J. Comput. 36, 184–206 (2006)

    Google Scholar 

  10. Forsythe, G.E., and Leibler, R.A.: Matrix Inversion by a Monte Carlo Method. Math. Tabl. Aids to Comp., 6(38), 78–81 (1950)

    Google Scholar 

  11. Groetsch, C.W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, London (1984)

    Google Scholar 

  12. Halton, J.H.: A Retrospective and Prospective Survey of the Monte Carlo Method. SIAM Review, 12(1) (1970)

    Google Scholar 

  13. Hansen, P.C.: Discrete Inverse Problems: Insight and Algorithms. SIAM, Philadelphia (2010)

    Google Scholar 

  14. Kaipio, J., and Somersalo, E.: Statistical and Computational Inverse Problems. Springer, New York (2004)

    Google Scholar 

  15. Lemieux, C.: Monte Carlo and Quasi-Monte Carlo Sampling, Springer, New York (2009)

    Google Scholar 

  16. Matlab, The Mathworks Ltd

    Google Scholar 

  17. O’Leary, D.P.: Near-optimal Parameters for Tikhonov and Other Regularization Methods. SIAM J. on Scientific Computing, 23(4), 1161–1171 (2001)

    Google Scholar 

  18. Sutton, R.S., and Barto, A.G.: Reinforcement Learning: An Introduction. The MIT Press, Boston (1998)

    Google Scholar 

  19. Wang, M., Polydorides, N., and Bertsekas, D.P.: Approximate Simulation-Based Solution of Large-Scale Least Squares Problems, Report LIDS-P-2819, MIT (2009)

    Google Scholar 

Download references

Acknowledgements

Research is supported by the Cyprus Program at MIT Energy Initiative, the LANL Information of Science and Technology Institute, and by NSF Grant ECCS-0801549.

Author information

Authors and Affiliations

  1. EEWRC, The Cyprus Institute, Nicosia, Cyprus

    Nick Polydorides

  2. LIDS, MIT, Cambridge, MA, USA

    Mengdi Wang & Dimitri P. Bertsekas

Authors
  1. Nick Polydorides
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mengdi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dimitri P. Bertsekas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nick Polydorides .

Editor information

Editors and Affiliations

  1. , Institute of Applied Mathematics, University of Warsaw, Banacha 2, Warsaw, 02-097, Poland

    Leszek Plaskota

  2. , Institute of Applied Mathematics, University of Warsaw, Banacha 2, Warsaw, 02-097, Poland

    Henryk Woźniakowski

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Polydorides, N., Wang, M., Bertsekas, D.P. (2012). A Quasi Monte Carlo Method for Large-Scale Inverse Problems. In: Plaskota, L., Woźniakowski, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2010. Springer Proceedings in Mathematics & Statistics, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27440-4_36

Download citation

Publish with us

Policies and ethics

Search

Navigation