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Published in: BIT Numerical Mathematics 2/2015

01-06-2015

A damped semismooth Newton method for the Brugnano–Casulli piecewise linear system

Authors: Zhe Sun, Lei Wu, Zhe Liu

Published in: BIT Numerical Mathematics | Issue 2/2015

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Abstract

The piecewise linear system is a nonsmooth but semismooth equation. In this paper, a damped semismooth Newton method is presented for solving a class of piecewise linear systems. Under appropriate conditions, both monotone convergence and finite termination properties are investigated for the proposed method.

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Literature
1.
go back to reference Bear, J., Verruijt, A.: Modeling Groundwater Flow and Pollution. D. Reidel Publ. Co., Dordrecht (1987)CrossRef Bear, J., Verruijt, A.: Modeling Groundwater Flow and Pollution. D. Reidel Publ. Co., Dordrecht (1987)CrossRef
3.
go back to reference Brugnano, L., Casulli, V.: Iterative solution of piecewise linear systems and applications to flows in porous media. SIAM J. Sci. Comput. 31, 1858–1873 (2009)CrossRefMATHMathSciNet Brugnano, L., Casulli, V.: Iterative solution of piecewise linear systems and applications to flows in porous media. SIAM J. Sci. Comput. 31, 1858–1873 (2009)CrossRefMATHMathSciNet
4.
go back to reference Brugnano, L., Sestini, A.: Iterative solution of piecewise linear systems for the numerical solution of obstacle problems. J. Numer. Anal. Ind. Appl. Math. 6(3–4), 67–82 (2011)MathSciNet Brugnano, L., Sestini, A.: Iterative solution of piecewise linear systems for the numerical solution of obstacle problems. J. Numer. Anal. Ind. Appl. Math. 6(3–4), 67–82 (2011)MathSciNet
5.
go back to reference Casulli, V.: Semi-implicit finite difference methods for the two-dimensional shallow water equations. J. Comput. Phys. 86, 56–74 (1990)CrossRefMATHMathSciNet Casulli, V.: Semi-implicit finite difference methods for the two-dimensional shallow water equations. J. Comput. Phys. 86, 56–74 (1990)CrossRefMATHMathSciNet
6.
go back to reference Casulli, V.: A high resolution wetting and drying algorithm for free-surface hydrodynamics. Int. J. Numer. Methods Fluids 60, 391–408 (2009)CrossRefMATHMathSciNet Casulli, V.: A high resolution wetting and drying algorithm for free-surface hydrodynamics. Int. J. Numer. Methods Fluids 60, 391–408 (2009)CrossRefMATHMathSciNet
8.
go back to reference Chen, C.X., Hu, L.T., Wang, X.S.: Analysis of steady ground water flow toward wells in a confined-unconfined aquifer. Ground Water 44, 609–612 (2006)CrossRef Chen, C.X., Hu, L.T., Wang, X.S.: Analysis of steady ground water flow toward wells in a confined-unconfined aquifer. Ground Water 44, 609–612 (2006)CrossRef
9.
10.
go back to reference Chen, X., Nashed, Z., Qi, L.: Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38, 1200–1216 (2000)CrossRefMATHMathSciNet Chen, X., Nashed, Z., Qi, L.: Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38, 1200–1216 (2000)CrossRefMATHMathSciNet
11.
go back to reference Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)MATH Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)MATH
12.
go back to reference Hackbusch, W.: Iterative Solution of Large Sparse Systems of Equations. Springer, New York (1994) Hackbusch, W.: Iterative Solution of Large Sparse Systems of Equations. Springer, New York (1994)
13.
go back to reference Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semi-smooth Newton method. SIAM J. Optim. 13, 865–888 (2003)CrossRefMATH Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semi-smooth Newton method. SIAM J. Optim. 13, 865–888 (2003)CrossRefMATH
15.
go back to reference Kärkkäinen, T., Kunisch, K., Tarvainen, P.: Augmented Lagrangian active set methods for obstacle problems. J. Optim. Theory Appl. 119, 499–533 (2003)CrossRefMATHMathSciNet Kärkkäinen, T., Kunisch, K., Tarvainen, P.: Augmented Lagrangian active set methods for obstacle problems. J. Optim. Theory Appl. 119, 499–533 (2003)CrossRefMATHMathSciNet
16.
go back to reference Li, D.H., Fukushima, M.: Smoothing Newton and quasi-Newton methods for mixed complementarity problems. Comput. Optim. Appl. 17, 203–230 (2000)CrossRefMATHMathSciNet Li, D.H., Fukushima, M.: Smoothing Newton and quasi-Newton methods for mixed complementarity problems. Comput. Optim. Appl. 17, 203–230 (2000)CrossRefMATHMathSciNet
17.
go back to reference Li, L., Lockington, D.A., Barry, D.A., Parlange, J.-Y., Perrochet, P.: Confined-unconfined flow in a horizontal aquifer. J. Hydrol. 271, 150–155 (2003)CrossRef Li, L., Lockington, D.A., Barry, D.A., Parlange, J.-Y., Perrochet, P.: Confined-unconfined flow in a horizontal aquifer. J. Hydrol. 271, 150–155 (2003)CrossRef
18.
go back to reference Lin, Y., Cryer, C.W.: An alternating direction implicit algorithm for the solution of linear complementarity problems arising from free boundary value problems. Appl. Math. Optim. 13, 1–17 (1985)CrossRefMATHMathSciNet Lin, Y., Cryer, C.W.: An alternating direction implicit algorithm for the solution of linear complementarity problems arising from free boundary value problems. Appl. Math. Optim. 13, 1–17 (1985)CrossRefMATHMathSciNet
19.
20.
go back to reference Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, San Diego (1970) [reprinted by SIAM, Philadelphia (2000)] Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, San Diego (1970) [reprinted by SIAM, Philadelphia (2000)]
22.
go back to reference Scholtes, S.: Introduction to Piecewise Differentiable Equations. Springer, Heidelberg (2012)CrossRefMATH Scholtes, S.: Introduction to Piecewise Differentiable Equations. Springer, Heidelberg (2012)CrossRefMATH
23.
go back to reference Sun, Z., Zeng, J.P., Li, D.H.: Semismooth Newton Schwarz iterative methods for the linear complementarity problem. BIT Numer. Math. 50, 425–449 (2010)CrossRefMATHMathSciNet Sun, Z., Zeng, J.P., Li, D.H.: Semismooth Newton Schwarz iterative methods for the linear complementarity problem. BIT Numer. Math. 50, 425–449 (2010)CrossRefMATHMathSciNet
24.
go back to reference Sun, Z., Zeng, J.P.: A damped semismooth Newton method for mixed linear complementarity problems. Optim. Methods Softw. 26, 187–205 (2011)CrossRefMATHMathSciNet Sun, Z., Zeng, J.P.: A damped semismooth Newton method for mixed linear complementarity problems. Optim. Methods Softw. 26, 187–205 (2011)CrossRefMATHMathSciNet
Metadata
Title
A damped semismooth Newton method for the Brugnano–Casulli piecewise linear system
Authors
Zhe Sun
Lei Wu
Zhe Liu
Publication date
01-06-2015
Publisher
Springer Netherlands
Published in
BIT Numerical Mathematics / Issue 2/2015
Print ISSN: 0006-3835
Electronic ISSN: 1572-9125
DOI
https://doi.org/10.1007/s10543-014-0514-0

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