Abstract
We develop an optimization-based approach for additive decomposition and reconnection of algebraic problems arising from discretizations of partial differential equations (PDEs). Application to a scalar convection–diffusion PDE illustrates the new approach. In particular, we derive a robust iterative solver for convection–dominated problems using standard multilevel solvers for the Poisson equation.
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References
Bank, R.E., Wan, J.W.L., Qu, Z.: Kernel preserving multigrid methods for convection-diffusion equations. SIAM J. Matrix Anal. Appl. 27(4), 1150–1171 (2006)
Bochev, P., Ridzal, D.: An optimization-based approach for the design of robust solution algorithms. SIAM J. Numer. Anal. (submitted)
Brezina, M., Falgout, R., MacLachlan, S., Manteuffel, T., McCormick, S., Ruge, J.: Adaptive smoothed aggregation (αsa) multigrid. SIAM Review 47(2), 317–346 (2005)
Du, Q., Gunzburger, M.: A gradient method approach to optimization-based multidisciplinary simulations and nonoverlapping domain decomposition algorithms. SIAM J. Numer. Anal. 37, 1513–1541 (2000)
Elman, H., Silvester, D., Wathen, A.: Finite Elements and Fast Iterative Solvers. Oxford University Press, Oxford (2005)
Gee, M.W., Siefert, C.M., Hu, J.J., Tuminaro, R.S., Sala, M.G.: ML 5.0 Smoothed Aggregation User’s Guide. Technical Report SAND2006-2649, Sandia National Laboratories, Albuquerque, NM (2006)
Golub, G.H., van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)
Gunzburger, M., Lee, H.K.: An optimization-based domain decomposition method for the Navier-Stokes equations. SIAM J. Numer. Anal. 37, 1455–1480 (2000)
Henson, V.E., Meier Yang, U.: Boomeramg: a parallel algebraic multigrid solver and preconditioner. Appl. Numer. Math. 41(1), 155–177 (2002)
Hughes, T.J.R., Brooks, A.: A theoretical framework for Petrov-Galerkin methods with discontinuous weighting functions: Application to the streamline-upwind procedure. In: Rao, R.H.G., et al. (eds.) Finite Elements in Fluids, vol. 4, pp. 47–65. J. Wiley & Sons, New York (1982)
Lions, J.L.: Virtual and effective control for distributed systems and decomposition of everything. Journal d’Analyse Mathématique 80, 257–297 (2000)
Nocedal, J., Wright, S.J.: Numerical Optimization, 1st edn. Springer, Heidelberg (1999)
Notay, Y.: Aggregation-based algebraic multilevel preconditioning. SIAM J. Matrix Anal. Appl. 27(4), 998–1018 (2006)
Sala, M., Tuminaro, R.S.: A new Petrov–Galerkin smoothed aggregation preconditioner for nonsymmetric linear systems. SIAM J. Sci. Comp. 31(1), 143–166 (2008)
Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506–517 (1968)
Stüben, K.: A review of algebraic multigrid. J. Comp. Appl. Math. 128(1-2), 281–309 (2001)
Toselli, A., Widlund, O.: Domain Decomposition Methods - Algorithms and Theory. Springer, New York (2005)
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Bochev, P., Ridzal, D. (2010). Additive Operator Decomposition and Optimization–Based Reconnection with Applications. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2009. Lecture Notes in Computer Science, vol 5910. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12535-5_77
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DOI: https://doi.org/10.1007/978-3-642-12535-5_77
Publisher Name: Springer, Berlin, Heidelberg
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