World Scientific
  • Search
Skip main navigation

Cookies Notification

We use cookies on this site to enhance your user experience. By continuing to browse the site, you consent to the use of our cookies. Learn More
×
Our website is made possible by displaying certain online content using javascript.
In order to view the full content, please disable your ad blocker or whitelist our website www.worldscientific.com.

System Upgrade on Tue, Oct 25th, 2022 at 2am (EDT)

Existing users will be able to log into the site and access content. However, E-commerce and registration of new users may not be available for up to 12 hours.
For online purchase, please visit us again. Contact us at [email protected] for any enquiries.

Parallel Nesterov Domain Decomposition Method for Elliptic Partial Differential Equations

    We study a parallel non-overlapping domain decomposition method, based on the Nesterov accelerated gradient descent, for the numerical approximation of elliptic partial differential equations. The problem is reformulated as a constrained (convex) minimization problem with the interface continuity conditions as constraints. The resulting domain decomposition method is an accelerated projected gradient descent with convergence rate O(1/k2). At each iteration, the proposed method needs only one matrix/vector multiplication. Numerical experiments show that significant (standard and scaled) speed-ups can be obtained.

    References

    • 1. A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci. 2(1) (2009) 183–202. Crossref, ISIGoogle Scholar
    • 2. (Eds.) T. F. Chanet al., in Proceedings of the 12th International Conference on Domain Decomposition Methods, Chiba, Japan, 2001, ddm.org. Google Scholar
    • 3. (Eds.) N. Debitet al., in Proceedings of the 13th International Conference on Domain Decomposition Methods, Lyon, France, 9–12 October 2000 (CIMNE, Barcelona, Spain, 2002). Google Scholar
    • 4. P. Giselsson and S. Boyd, Monotonicity and restart in fast gradient methods, in IEEE Conference on Decision and Control, 2014, pp. 5058–5063. Google Scholar
    • 5. M. D. Gunzburger, M. Heinkenschloss and H. K. Lee, Solution of elliptic partial differential equations by an optimization-based domain decomposition method, Appl. Math. Comput. 113 (2000) 111–139. ISIGoogle Scholar
    • 6. (Eds.) I. Herreraet al., in Proceedings of the 14th International Conference on Domain Decomposition Methods in Cocoyoc, Mexico, 6–11 January 2002 (UNAM Press, 2003). Google Scholar
    • 7. J. Koko and F. A. Okoubi, Parallel Nesterov’s method for large-scale miimization of partially separable functions, Optimisation Letters (2016). https://doi.org/10.1007/s11590-016-1020-x Google Scholar
    • 8. J. Koko, F. A. Okoubi and T. Sassi, Domain decomposition with Nesterov’s method, in eds. J. Erhelet al., Domain Decomposition in Sciences and Engineering XXI, Lectures Notes in Computational Science and Engineering, Vol. 98 (Springer, 2014), pp. 947–954. CrossrefGoogle Scholar
    • 9. (Eds.) R. Kornhuberet al., in Proceedings of the 15th International Conference on Domain Decomposition Methods, Berlin, Germany, 15–21 July 2003, Lectures Notes in Computational Science and Engineering, Vol. 40 (Springer, 2004). Google Scholar
    • 10. Y. Nesterov, A method for solving the convex programming problem with convergence rate 0(1/k2), Dokl. Akad. Nauk. SSSR 269(3) (1983) 543–547. Google Scholar
    • 11. Y. Nesterov, Smooth minimization of non-smooth functions, Mathematical Programming (A) 103 (2005) 127–152. Crossref, ISIGoogle Scholar
    • 12. B. O’Donoghue and E. Candes, Adaptative restart for accelerated gradient schemes, Found. Comput. Math. 15 (2015) 715–732. Crossref, ISIGoogle Scholar
    • 13. A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations (Oxford University Press, 1999). Google Scholar
    • 14. B. F. Smith, P. Bjørstad and W. D. Gropp, Domain Decomposition: Parallel Multilevel Algorithms for Elliptic Partial Differential Equations (Cambridge University Press, New York, 1996). Google Scholar
    • 15. A. Toselli and O. B. Widlund, Domain decomposition methods: Algorithms and theory, in Domain Decomposition Methods: Algorithms and Theory, Series in Computational Mathematics, Vol. 34 (Springer, 2005). CrossrefGoogle Scholar
    • 16. P. Weiss, L. Blanc-Ferrand and G. Aubert, Efficient schemes for total variation minimization under constraints in image processing, SIAM J. Scientific Computing 31 (2009) 2047–2080. Crossref, ISIGoogle Scholar