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A quantitative metric for robustness of nonlinear algebraic equation solvers. (English) Zbl 1252.65094
Summary: Practitioners in the area of dynamic simulation of technical systems report difficulties at times with steady-state initialization of models developed using general declarative modeling languages. These difficulties are analyzed in detail in this work and a rigorous approach to quantify robustness in the context of systems of nonlinear algebraic equations is presented. This tool is then utilized in a study of six state of the art gradient-based iterative solvers on a set of industrial test problems. Finally, conclusions are drawn on the observed solver robustness in general, and it is argued whether the reported difficulties with steady-state initialization can be supported using the proposed quantitative metric.

65H10 Numerical computation of solutions to systems of equations
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[1] Allgower, E.L.; Georg, K., Introduction to numerical continuation methods, (2003), SIAM Classics in Applied Mathematics · Zbl 1036.65047
[2] Baharev, A.; Rév, E., Reliable computation of equilibrium cascades with affine arithmetic, Aiche j., 54, 1782-1797, (2008)
[3] A. Baharev, E. Rév, A complete nonlinear system solver using affine arithmetic, in: Interval Analysis, Constraint Propagation for Applications, (IntCP 2009), Workshop Held in Conjunction with the 15th International Conference on Principles, Practice of Constraint Programming, Lisbon, Portugal.
[4] Bell, B.M.; Burke, J.V., Algorithmic differentiation of implicit functions and optimal values, (), 67-77 · Zbl 1152.65434
[5] Bouaricha, A.; Schnabel, R.B., Tensolve: a software package for solving systems of nonlinear equations and nonlinear least-squares problems using tensor methods (algorithm 768), ACM. trans. math. software., 23, 174-195, (1997) · Zbl 0887.65051
[6] Cayley, A., The newton – fourier imaginary problem, Am. J. math., 2, 97, (1879) · JFM 11.0260.02
[7] M. Dempsey, personal communication, 2009.
[8] Dennis, J.E.; Schnabel, R.B., Numerical methods for unconstrained optimization and nonlinear equations, (1996), SIAM Classics in Applied Mathematics · Zbl 0847.65038
[9] D. Dent, M. Paprzycki, A. Kucaba-Pietal, Comparing solvers for large systems of nonlinear algebraic equations, in: Proceedings of the Southern Conference on Computing.
[10] D. Dent, M. Paprzycki, A. Kucaba-Pietal, Solvers for systems of nonlinear algebraic equations – their sensitivity to starting vectors, in: L. Vulkov, P. Yalamov, J. Wasniewski (Eds.), Numerical Analysis and Its Applications, vol. 1988 of Lecture Notes in Computer Science, Springer Verlag, 2001, pp. 230-237. · Zbl 0978.65042
[11] Deuflhard, P., Newton methods for nonlinear problems. affine invariance and adaptive algorithms, (2004), Springer Verlag · Zbl 1056.65051
[12] Dolan, E.D.; Moré, J.J., Benchmarking optimization software with performance profiles, Math. program., 91, 201-213, (2002) · Zbl 1049.90004
[13] Elmqvist, H.; Brück, D.; Otter, M., Dymola users’ manual, (1995), Dynasim AB, Research Park Ideon Lund, Sweden
[14] H. Elmqvist, M. Otter, Methods for tearing systems of equations in object-oriented modeling, Proceedings of the European Simulation Multiconference, Barcelona, Spain, 326-332.
[15] Griffin, J.D.; Kolda, T.G., Nonlinearly-constrained optimization using heuristic penalty methods and asynchronous parallel generating set search, Appl. math. res. express, 25, 36-62, (2010) · Zbl 1187.90324
[16] Güttinger, T.E.; Dorn, C.; Morari, M., Experimental study of multiple steady states in homogeneous azeotropic distillation, Ind. eng. chem. res., 36, 794-802, (1997)
[17] Honkala, M.; Roos, J.; Karanko, V., On nonlinear iteration methods for dc analysis of industrial circuits, (), 144-148 · Zbl 1272.94102
[18] IEEE Computer Society, IEEE standard VHDL analog and mixed-signal extensions, IEEE 1076.1-1, 1999.
[19] Issa, R.I., Solution of the implicitly discretised fluid flow equations by operator-splitting, J. comput. phys., 62, 40-65, (1986) · Zbl 0619.76024
[20] Kelley, C.T., Solving nonlinear equations with newton’s method, (2003), SIAM Classics in Applied Mathematics · Zbl 1031.65069
[21] T. Young Lee, J. Kyung Shim, Elimination-based solution method for the forward kinematics of the General Stewart-Gough Platform, in: Proceedings of the Computational Kinematics Conference, Seoul. · Zbl 1137.93393
[22] L. Luksan, J. Vleck, Sparse and partially separable test problems for unconstrained and equality constrained optimization, Technical Report Research Report 767, Institute of Computer Science, Academy of Sciences of the Czech Republic, 1999.
[23] J.P. Merlet, The Coprin examples page, http://www-sop.inria.fr/coprin/logiciels/ALIAS/Benches/benches.html, 2006.
[24] Metropolis, N.; Ulam, S., The Monte Carlo method, J. am. stat. assoc., 44, 335-341, (1949) · Zbl 0033.28807
[25] J.J. Moré, B.S. Garbow, K.E. Hillstrom, User Guide for MINPACK-1, Technical Report ANL-80-74, Argonne National Laboratory, 1980.
[26] Moré, J.J.; Wild, S.M., Benchmarking derivative-free optimization algorithms, SIAM J. opt., 20, 172-191, (2009) · Zbl 1187.90319
[27] Neumaier, A., Interval methods for systems of equations, (1990), Cambridge University Press · Zbl 0706.15009
[28] U. Nowak, L. Weimann, A family of Newton codes for systems of highly nonlinear equations, Technical Report TR-91-10, Konrad-Zuse-Zentrum für Infomrationstechnik, Berlin, 1991.
[29] Oh, M.; Pantelides, C.C., A modelling and simulation language for combined lumped and distributed parameter systems, Comput. chem. eng., 20, 611-633, (1996)
[30] Ortega, J.M.; Rheinboldt, W.C., Iterative solution of nonlinear equations in several variables, (1970), SIAM Classics in Applied Mathematics · Zbl 0241.65046
[31] Patankar, S.; Spalding, D., A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Int. J. heat mass transfer, 15, 1787-1806, (1972) · Zbl 0246.76080
[32] Shacham, M.; Brauner, N.; Pozin, M., Comparing software for interactive solution of systems of nonlinear algebraic equations, Comput. chem. eng., 22, (1998)
[33] Wagner, W.; Kretzschmar, H.J., International steam tables – properties of water and steam based on the industrial formulation IAPWS-IF97, (1997), Springer
[34] www.modelica.org, 2010.
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