Abstract
An algebraic multigrid method (AMG) for solving convection-diffusion optimality systems is presented. Results of numerical experiments demonstrate robustness of the AMG scheme with respect to changes of the weight of the cost of the control and show that the computational performance of the proposed AMG scheme is comparable to that of AMG applied to single scalar equations.
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Arian, E., Ta’asan, S.: Smoothers for optimization problems. In Seventh Copper Mountain Conference on Multigrid Methods, Vol. CP3339, NASA Conference Publication, NASA, Duane Melson, N., Manteuffel, T.A., McCormick, S.F., Douglas, C.C. (eds.), Hampton, VA, 1995, pp. 15–30
Bertsekas, D.P.: Nonlinear Programming. Belmont: Athena Scientific 1995
Borzì, A., Borzì, G.: An algebraic multigrid method for a class of elliptic differential systems. SIAM J. Sci. Comp. 25(1), 302–323 (2003)
Borzì, A., Kunisch, K.: The numerical solution of the steady state solid fuel ignition model and its optimal control. SIAM J. Sci. Comp. 22(1), 263–284 (2000)
Borzì, A., Kunisch, K., Kwak, D.Y.: Accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system. SIAM J. Control Optim. 41(5), 1477–1497 (2003)
Braess, D.: Towards algebraic multigrid for elliptic problems of second order. Computing 55, 379–393 (1995)
Bramble, J.H., Pasciak, J.E., Xu, J.: The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms. Mathematics of Computation 56, 1–34 (1991)
Bramble, J.H., Pasciak, J.E., Wang, J., Xu, J.: Convergence estimates for multigrid algorithms without regularity assumptions. Mathematics of Computation 57, 23–45 (1992)
Bramble, J.H., Kwak, D.Y., Pasciak, J.E.: Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems. SIAM J. Numer. Anal. 31, 1746–1763 (1994)
Brandt, A.: Algebraic multigrid theory: The symmetric case. Proc. Int. Multigrid Conf., Copper Mountain, Colorado, 1983; Appl. Math. Comp. 19, 23–56 (1986)
Brandt, A.: General highly algebraic coarsening. ETNA 10, 1–20 (2000)
Brezina, M., Cleary, A.J., Falgout, R.D., Henson, V.E., Jones, J.E., Manteuffel, T.A., McCormick, S.F., Ruge, J.W.: Algebraic multigrid based on element interpolation (AMGe). SIAM J. Sci. Comput. 22(5), 1570–1592 (2000)
Dreyer, Th., Maar, B., Schulz, V.: Multigrid optimization in applications. J. Comput. Appl. Math. 120, 67–84 (2000)
Hackbusch, W.: Fast solution of elliptic control problems. Journal of Optimization Theory and Application 31, 565–581 (1980)
Hackbusch, W.: Iterative Solution of Large Sparse Systems of Equations. New York: Springer-Verlag 1994
Hoffmann, K.H., Hoppe, R., Schulz, V. (eds.): Fast Solution of Discretized Optimization Problems. International Series on Numerical Mathematics, Vol. 138, Birkhäuser 2001
Lions, J.-L.: Optimal Control of Systems Governed by Partial Differential Equations. Berlin: Springer 1971
Mandel, J.: Local Approximation Estimators for Algebraic Multigrid. ETNA 15, 56–65 (2003)
Muszynski, P., Rüde, U., Zenger, Chr.: Application of algebraic multigrid (AMG) to constrained quadratic optimization. technical report TUM-I8801, Technische Universität München, 1988
Ruge, J.W., Stüben, K.: Algebraic Multigrid (AMG). In: McCormick, S. (ed.), Multigrid Methods, Frontiers in Applied Mathematics, Vol. 5, Philadelphia: SIAM 1986
Saad, Y.: SPARSKIT: A basic tool kit for sparse matrix computations. Rep. No. 90-20, Research Institute for Advanced Computer Science, NASA Ames Research Center, Moffet Field, CA, 1990
Schulz, V., Wittum, G.: Multigrid optimization methods for stationary parameter identification problems in groundwater flow. In: Hackbusch, W., Wittum, G. (eds.): Multigrid Methods V, Lecture Notes in Computational Science and Engineering 3, pp. 276–288, Springer 1998
Stüben, K.: Algebraic Multigrid (AMG): An Introduction with Applications. GMD Report 53, March 1999
Vanek, P., Brezina, M., Mandel, J.: Convergence of algebraic multigrid based on smoothed aggregation. Numerische Mathematik 88(3), 559–579 (2001)
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Borzì, A., Borzì, G. An efficient algebraic multigrid method for solving optimality systems. Comput. Visual Sci. 7, 183–188 (2004). https://doi.org/10.1007/s00791-004-0148-x
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DOI: https://doi.org/10.1007/s00791-004-0148-x