×

zbMATH — the first resource for mathematics

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.

MSC:
65H10 Numerical computation of solutions to systems of equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.