The inexact Newton backtracking method as a tool for solving differential-algebraic systems

Paweł Drąg,

Krystyn Styczeń

Abstrakt

The classical inexact Newton method was presented as a tool for solving nonlinear differential algebraic equations (DAEs) in a fully implicit form F(y, y, t) = 0. This is especially in chemical engineering where describing the DAE system in a different form can be difficult or even impossible to realize. The appropriate rewriting of the DAEs using the backward Euler method makes it possible to present the differentialalgebraic system as a large-scale system of nonlinear equations. To solve the obtained system of nonlinear equations, the inexact Newton backtracking method was proposed. Because the convergence of the inexact Newton algorithm is strongly affected by the choice of the forcing terms, new variants of the inexact Newton method were presented and tested on the catalyst mixing problem.

Słowa kluczowe: differential-algebraic equations, systems of nonlinear equations, inexact Newton method
References

An H.-B., Mo Z.-Y., Liu X.-P., A choice of forcing terms in inexact Newton method, Journal of Computational and Applied Mathematics, Vol. 200, 2007, 47-60.

Betts J.T., Practical Methods for Optimal Control and Estimation Using Nonlinear Programming. Second edition, SIAM, Philadelphia 2010.

Biegler L.T., Nonlinear Programming. Concepts, Algorithms, and Applications to

Chemical Processes, SIAM, Philadelphia 2010.

Brenan K.E., Campbell S.L., Petzold L.R., Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, SIAM, Philadelphia 1996.

Brown P.N., Hindmarsh A.C., Petzold L.R., Using Krylov methods in the solution of large-scale differential-algebraic systems, SIAM Journal on Scientific Computing,
Vol. 15, 1994, 1467-1488.

Cai X.-C., Keyes D.E., Nonlinearly Preconditioned Inexact Newton Algorithms, SIAM Journal on Scientific Computing, Vol. 24, 2002, 183-200.

Caracotsis M., Stewart W.E., Sensitivity analysis of Initial Value Problems with mixed ODEs and algebraic equation, Computers and Chemical Engineering, Vol. 9, 1985, 359-365.

Drąg P., Styczeń K., A Two-Step Approach for Optimal Control of Kinetic Batch Reactor with electroneutrality condition, Przegląd Elektrotechniczny (Electrical Review), Vol. 6, 2012, 176-180.

Drąg P., Styczeń K., Inexact Newton method as a tool for solving differential-algebraic system, Proceedings of the 2013 Federated Conference on Computer Science and Information Systems (FedCSIS), 2013, 639-642.

Dembo R.S., Eisenstat S.C., Steihaug T., Inexact Newton Methods, SIAM Journal on Numerical Analysis, Vol. 19, 1982, 400-408.

Dembo R.S., Steihaug T., Truncated-Newton algorithm for large-scale unconstrained optimization, Mathematical Programming, Vol. 26, 1983, 190-212. Eisenstat S.C., Walker H.F., Globally convergent inexact Newton methods, SIAM

Journal on Optimization, Vol. 4, 1994, 393-422.

Eisenstat S.C., Walker H.F., Choosing the forcing terms in an inexact Newton method, SIAM Journal on Scientific Computing, Vol. 17, 1996, 16-32.

Gear C.W., The simultaneous numerical solution of differential-algebraic equations, IEEE Transactions on Circuit Theory, Vol. 18, 1971, 89-95.

Huang Y.J., Reklaitis G.V., Venkatasubramanian V., Model decomposition based model for solving general dynamic optimization problems, Computers and Chemical, Vol. 26, 2002, 863-873.

Knoll D.A., Keyes D.E., Jacobian-free Newton-Krylov methods: a survey of approaches and applications, Journal of Computational Physics, Vol. 193, 2004, 357-397.

Petzold L., Differential/Algebraic Equations are not ODEs, SIAM Journal on Scientific Computing, Vol. 3, 1982, 367-384.

Saad Y., Schultz M.H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., Vol. 7, 1986, 856-869.

Vassiliadis V.S., Sargent R.W.H., Pantelides C.C., Solution of a Class of Multistage Dynamic Optimization Problems. 1. Problems without Path Constraints, Ind. Eng. Chem. Res., Vol. 33, 1994, 2111-2122.

Vassiliadis V.S., Sargent R.W.H., Pantelides C.C., Solution of a Class of Multistage Dynamic Optimization Problems. 2. Problems with Path Constraints, Ind. Eng. Chem. Res., Vol. 33, 1994, 2123-2133.