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Cáceres Juan (Catholic University), Barán Benjamín (Catholic University), Schaerer Christian (National University of Asuncion)
Implementation of a distributed parallel in time scheme using PETSc for a Parabolic Optimal Control Problem
Annals of Computer Science and Information Systems, 2014, vol. 2, s. 577 - 586, rys., tab., bibliogr. 32 poz.
Równania różniczkowe, Algorytmy, Analiza matematyczna
Differential equations, Algorithms, Mathematical analysis
This work presents a parallel implementation of the Parareal method using Portable Extensible Toolkit for Scientific Computation (PETSc). An optimal control problem of a parabolic partial differential equation with known boundary conditions and initial state is solved, where the minimized cost function relates the controller $v$ usage and the approximation of the solution $y$ to an optimal known function $y^*$, measured by $|y|$ and $|y|$, respectively. The equations that model the process are discretized in space using Finite Elements and in time using Finite Differences. After the discretizations, the problem is transformed to a large linear system of algebraic equations, that is solved by the Conjugate Gradient method. A Parareal preconditioner is implemented to speed up the convergence of the Conjugate Gradien.(original abstract)
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  1. Amdahl G. M., "Validity of the single processor approach to achieving large scale computing capabilities," AFIPS spring joint computer conference, 1967.
  2. Balay S., Brown J., Buschelman K., Eijkhout V., Gropp W. D., Kaushik D., Knepley M. G., McInnes L. C., Smith B. F., and Zhang H., "PETSc users manual," Argonne National Laboratory, Tech. Rep. ANL-95/11 - Revision 3.3, 2012.
  3. Balay S., Gropp W. D., McInnes L. C., and Smith B. F., "Efficient management of parallelism in object oriented numerical software libraries," in Modern Software Tools in Scientific Computing, E. Arge, A. M. Bruaset, and H. P. Langtangen, Eds. Birkh¨auser Press, 1997, pp. 163-202.
  4. Bashir O., Willcox1 K., Ghattas O., van Bloemen Waanders B., and Hill J., "Hessian-based model reduction for large-scale systems with initial condition inputs," International Journal for Numerical Methods in Engineering, vol. 73, no. 6, pp. 844-868, 2008.
  5. Benzi M., Golub G. H., and Liesen J., "Numerical solution of saddle point problems," Acta Numerica, no. 14, pp. 1-137, 2005.
  6. Biros G. and Ghattas O., "Parallel Lagrange-Newton-Krylov-Schur methods for PDE-Constrained optimization I. The Krylov-Schur solver," SIAM Journal on Scientific Computing, no. 27, p. 687-713, 2005.
  7. Carlsson J., "Optimal Control of Partial Differential Equations in Optimal Design," PhD Dissertation, KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA, 2008.
  8. Du X., Sarkis M., Schaerer C. E., and Szyld D., "Inexact and truncated Parareal-in-time Krylov subspace methods for parabolic optimal control problems," Electronic Transactions on Numerical Analysis, 2013.
  9. Falgout R. D. and Yang U. M., "hypre: a library of high performance preconditioners," Numerical Solution of Partial Differential Equations on Parallel Computers, 2002.
  10. Fritz J., Partial Differential Equations. Springer, 1982.
  11. Gallier J., "The Schur Complement and Symmetric Positive Semidefnite (and Defnite) Matrices," Penn Engineering, 2010.
  12. Gander M. J. and Vandewalle S., "Analysis of the parareal time-parallel time-integration method," SIAM Journal on Scientific Computing, no. 29, pp. 556-578, 2007.
  13. Gockenbach M. S., Partial differential equations: analytical and numerical methods. SIAM, 2002.
  14. Gustafson J. L., "Reevaluating Amdahl's Law," Communications of the ACM, 1988.
  15. Kaminsky A., BIG CPU, BIG DATA: Solving the World's Toughest Computational Problems with Parallel Computing. Rochester Institute of Technology, 2013.
  16. Lions J. -L., Maday Y., and Turinici G., "R´esolution d'edp par un sch´ema en temps "parar´eel"," C. R. Acad. Sci. Paris S´er. I Math., vol. 332, no. 7, pp. 661-668, 2001.
  17. Lions J. L., Optimal control of systems governed by partial differential equations. Springer, 1971.
  18. Liu E. H. -L., "Fundamental Methods of Numerical Extrapolation With Applications," MIT Open Course Ware, 2006.
  19. Mathew T. P., Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations. Springer, 2008.
  20. Mathew T. P., Sarkis M., and Schaerer C. E., "Analysis of block parareal preconditioners for parabolic optimal control problems," SIAM, no. 32, pp. 1180-1200, 2010.
  21. Neittaanmaki P. and Tiba D., Optimal Control of Nonlinear Parabolic Systems. Marcel Dekker, Inc., 1994.
  22. Rees T., Stoll M., and Wathen A., "All-at-once preconditioning in PDEconstrained optimization," Kybernetika, no. 46, p. 341-360, 2010.
  23. Saad Y., "Krylov subspace methods for solving large unsymmetric linear systems," Mathematics of Computation, 1981.
  24. Saad Y., Iterative methods for sparse linear systems. SIAM, 2003.
  25. Samyono W., "Hessian matrix-free Lagrange-Newton-Krylov-Schur-Schwarz methods for elliptic inverse problems," PhD Dissertation, Old Dominion University, 2006.
  26. Schaerer C. E. and Kaszkurewicz E., "The shooting method for the solution of ordinary differential equations: A control-theoretical perspective," International Journal of Systems Science, vol. 32, no. 8, 2001.
  27. Schaerer C. E., Kaszkurewicz E., and Mangiavacchi N., "A Multileve Schwarz Shooting Method for the solution of the Poisson Equation in Two Dimensional Incompressible Flow Simulations," Applied Mathematics and Computation, vol. 153, no. 3, pp. 803-831, 2004.
  28. Silva J. J. C., "Implementaci´on de un esquema de paralelizaci´on temporal distribuida usando petsc," Diploma thesis, Ciencias y Tecnologıa - Universidad Cat´olica Nuestra Se˜nora de la Asunci´on, feb 2014.
  29. Stoer J. and Bulirsch R., Introduction to Numerical Analysis, 3rd ed. Springer-Verlag, 2002.
  30. Strang G., Introduction to Linear Algebra. Wellesley-Cambridge Press, 1998.
  31. Vogel C., Computational Methods for Inverse Problems. SIAM, 2002.
  32. Zhang F., "The Schur Complement and Its Applications," Springer, 2005.
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