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2015 | 5 | 589--594
Tytuł artykułu

Fast Solvers for Nonsmooth Optimization Problems in Phase Separation

Autorzy
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The phase separation processes are typically modeled by well known Cahn-Hilliard equation with obstacle potential. Solving these equations correspond to a nonsmooth and nonlinear optimization problem. Recently a globally convergent Newton Schur method was proposed for the non-linear Schur complement corresponding to this 2 × 2 non-linear system. The proposed method is similar to an inexact active set method in the sense that the active sets are first identified by solving a quadratic obstacle problem corresponding to the (1, 1) block of the 2 × 2 system, and later solving a reduced linear system by annihilating the rows and columns corresponding to identified active sets. For solving the quadratic obstacle problem, various optimal multigrid like methods have been proposed. However solving the reduced system remains a major bottleneck. In this paper, we explore an effective preconditioner for the reduced linear system that allows solving large scale optimization problem corresponding to Cahn-Hilliard and to possibly similar models.(original abstract)
Słowa kluczowe
PL
Fizyka  
EN
Physics  
Rocznik
Tom
5
Strony
589--594
Opis fizyczny
Twórcy
autor
  • Freie Universitat Berlin
Bibliografia
  • ] J. W. Cahn and J. E. Hilliard, "Free Energy of a Nonuniform System. I. Interfacial Free Energy," The Journal of Chemical Physics, vol. 28, no. 2, 1958. [Online]. Available: http://dx.doi.org/10.1063/1.1744102
  • Y. Oono and S. Puri, "Study of phase-separation dynamics by use of cell dynamical systems. I. Modeling," Physical Review A, vol. 38, no. 1, 1987. [Online]. Available: http://dx.doi.org/10.1103/PhysRevA.38.434
  • J. F. Blowey and C. M. Elliott, "The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy Part I: Numerical analysis," European J. Appl. Math., no. 2, pp. 233-280, 1991. [Online]. Available: http://dx.doi.org/10.1017/S095679250000053X
  • "The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy Part II: Numerical analysis," European J. Appl. Math., no. 3, 1992. [Online]. Available: http://dx.doi.org/10.1017/ S0956792500000759
  • J. Bosch, M. Stoll, and P. Benner, "Fast solution of Cahn-Hilliard variational inequalities using implicit time discretization and finite elements," Journal of Computational Physics, vol. 262, pp. 38-57, 2014. [Online]. Available: http://dx.doi.org/10.1016/j.jcp.2013.12.053
  • C. Graeser and R. Kornhuber, "Nonsmooth newton methods for setvalued saddle point problems," SIAM Journal on Numerical Analysis, vol. 47, no. 2, pp. 1251-1273, 2009.
  • J. Barrett, R. Nurnberg, and V. Styles, "Finite element approximation of a phase field model for void electromigration," SIAM J. Numer. Anal., vol. 42, no. 2, pp. 738-772, 2004. [Online]. Available: http://dx.doi.org/10.1137/S0036142902413421
  • R. Kornhuber, "Monotone multigrid methods for elliptic variational inequalities I," Numerische Mathematik, vol. 69, no. 2, pp. 167-184, 1994.
  • "Monotone multigrid methods for elliptic variational inequalities II," Numerische Mathematik, vol. 72, no. 4, pp. 481-499, 1996.
  • J. Mandel, "A Multilevel lterative Method for Symmetric, Positive Definite Linear Complementarity Problems," Applied Mathematics and Optimization, vol. 11, pp. 77-95, 1984.
  • C. Graser and R. Kornhuber, "Multigrid Methods for Obstacle Problems," Journal of Computational Mathematics, vol. 27, no. 1, pp. 1-44, 2009.
  • C. Graser, "Convex Minimization and Phase Field Models," Ph.D. dissertation, FU Berlin, 2011.
  • P. Kumar, "Aggregation based on graph matching and inexact coarse grid solve for algebraic two grid," International Journal of Computer Mathematics, vol. 91, no. 5, pp. 1061-1081, 2014. [Online]. Available: http://dx.doi.org/10.1080/00207160.2013.821115
  • U. Trottenberg, C. Oosterlee, and A. Schuller, Multigrid. Academic Press, 2001. [Online]. Available: http://www.academicpress.com
  • Y. Notay, "An aggregation-based algebraic multigrid method," Electronic Transactions on Numerical Analysis, vol. 37, pp. 123-146, 2010. [Online]. Available: http://dx.doi.org/10.1109/ISQED.2007.31
  • Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. Philadelphia: SIAM, 2003. [Online]. Available: http://dx.doi.org/10. 1137/1.9780898718003
Typ dokumentu
Bibliografia
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bwmeta1.element.ekon-element-000171422588

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