Advertisement

Mathematical Programming

, Volume 108, Issue 1, pp 135–158 | Cite as

Bounds on linear PDEs via semidefinite optimization

  • Dimitris BertsimasEmail author
  • Constantine Caramanis
Article

Abstract

Using recent progress on moment problems, and their connections with semidefinite optimization, we present in this paper a new methodology based on semidefinite optimization, to obtain a hierarchy of upper and lower bounds on linear functionals defined on solutions of linear partial differential equations. We apply the proposed method to examples of PDEs in one and two dimensions, with very encouraging results. We pay particular attention to a PDE with oblique derivative conditions, commonly arising in queueing theory. We also provide computational evidence that the semidefinite constraints are critically important in improving the quality of the bounds, that is, without them the bounds are weak.

Keywords

Differential Equation Partial Differential Equation Lower Bound Mathematical Method Recent Progress 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Akhiezer, N.: The classical moment problem, Eng. Ed., Oliver and Boyd, London 1965Google Scholar
  2. 2.
    Bertsimas, D.: The achievable region method in the optimal control of queueing systems; formulations, bounds and policies. Queueing Systems and Applications 21 (3–4), 337–389 (1995)Google Scholar
  3. 3.
    Bertsimas, D., Paschalidis, I., Tsitsiklis, J.: Optimization of multiclass queueing networks: polyhedral and nonlinear characterizations of achievable performance. Ann. Appl. Prob. 4 (2), 43–75 (1994)MathSciNetGoogle Scholar
  4. 4.
    Bertsimas, D., Popescu, I.: Optimal inequalities in probability theory: a convex optimization approach. SIAM Journal of Optimization 15, 780–804 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bertsimas, D., Popescu, I.: On the relation between option and stock prices: a convex optimization approach. Operations Research 50 (2), 358–374 (2002)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Bochnak, J., Coste, M., Roy, M.: Real algebraic geometry, Springer–Verlag, Berlin Heidelberg, 1998Google Scholar
  7. 7.
    Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer–Verlag, New York, 1991Google Scholar
  8. 8.
    Fujisawa, K., Kojima, M., Nakata, K.: SDPA (Semidefinite Programming Algorithm) User's Manual, Version 4.10, Research Report on Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, Japan, 1998Google Scholar
  9. 9.
    Haviland, E.K.: On the momentum problem for distribution functions in more than one dimension II. Amer. J. Math. 58, 164–168 (1936)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Harrison, J.M.: The diffusion approximation for tandem queues in heavy traffic. Adv. Appl. Prob. 10, 886–905 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Harrison, J.M., Landau, H.J., Shepp, L.A.: The stationary distribution of reflected Brownian motion in a planar region. Ann. Prob. 13, 744–757 (1985)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Kumar, S., Kumar, P.R.: Performance bounds for queueing networks and scheduling policies. IEEE Trans. on Aut. Control 39, 1600–1611 (1994)CrossRefzbMATHGoogle Scholar
  13. 13.
    Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Lasserre, J.B.: Bounds on measures satisfying moment conditions. Annals Appl. Prob. 12, 1114–1137 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Paraschivoiu, M., Patera, A.: A Hierarchical Duality Approach to Bounds for the Outputs of Partial Differential Equations. Comput. Methods Appl. Eng. 158, 389–407 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Paraschivoiu, M., Peraire, J., Patera, A.: A Posteriori Finite Element Bounds for Linear–Functional Outputs of Elliptic Partial Differential Equations. Comput. Meth. Appl. Mech. Eng. 150, 289–312 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Parrilo, P.: Semidefinite programming relaxations for semi-algebraic problems. Math. Programming Ser. B 96, 293–320 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Peraire, J., Patera, A.: Bounds for linear-functional outputs of coercive partial differential equations: Local indicators and adaptive refinement. New Advances in Adaptive Computational Methods in Mechanics, available at http://raphael.mit.edu/peraire/, 1997
  19. 19.
    Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations, Springer–Verlag, Berlin, 1997Google Scholar
  20. 20.
    Reznick, B.: Some concrete aspects of Hilbert's 17th problem. In: Delzell, C.N., Madden, J.J. (eds.), Real Algebraic Geometry and Ordered Structures. Cont. Math. 2000, p 253Google Scholar
  21. 21.
    Schwerer, E.: A Linear Programming Approach to the Steady–State Analysis of Markov Processes. Palo Alto, CA: Ph.D. Thesis, Stanford University, 1996Google Scholar
  22. 22.
    Schmüdgen, K.: The K–moment problem for compact semi–algebraic sets. Math. Ann. 289, 203–206 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Strang, G., Fix, G.: An Analysis of the Finite Element Method, Prentice–Hall, Englewood Cliffs, NJ, 1973Google Scholar
  24. 24.
    Sturm, J.: SeDuMi Semidefinite Software, http://www/unimaas.nl/sturm
  25. 25.
    Trefethen, L.N., Williams, R.J.: Conformal mapping solution of Laplace's equation on a polygon with oblique derivative boundary conditions.'' J. Comp. and App. Math. 14, 227–249 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Vandenberghe, L., Boyd, S.: Semidefinite Programming. SIAM Review 38, 49–95 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Varadhan, S.R.S., Williams, R.J.: Brownian motion in a wedge with oblique reflection. Comm. Pure. Appl. Math. 38, 405–443 (1985)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd Edition, The Macmillan Company, Cambridge, 1994Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Boeing Professor of Operations Research, Sloan School of Management and Operations Research CenterMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Laboratory for Information and Decision SystemsMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations