Jorge R. Leis
Abstract
The methodology for the simultaneous solution of ordinary differential equations and the associated first-order parametric sensitivity equations is presented, and a detailed description of its implementation as a modification of a widely disseminated implicit ODE solver is given. The error control strategy ensures that local error criteria are independently satisfied by both the model and sensitivity solutions. The internal logic effectuated by this implementation is detailed. Numerical testing of the algorithm is reported; results indicate that greater reliability and improved efficiency is offered over other sensitivity analysis methods.
- 1 ATHERTON, R. W., SCHAINKER, R. B., AND DUCOT, E.R. On the statistical sensitivity analysis of models for chemical kinetics. AIChE J. 21, 3 (May 1975), 441-448.]]Google Scholar
- 2 BARD, Y. Nonlinear Parameter Estimation. Academic Press, New York, 1974.]]Google Scholar
- 3 BEVERIDGE, G. S. G., AND SCHECHTER, R.S. Optimization: Theory and Practice. McGraw-Hill, New York, 1970.]]Google Scholar
- 4 CARACOTSIOS, M., AND STEWART, W.E. Sensitivity and analysis of initial value problems with mixed ODE's and algebraic equations. Comput. Chem. Eng. 9, 4 (1985), 359-365.]]Google Scholar
- 5 DE JONGH, D. C.J. Structural parameter sensitivity of the 'limits of growth' world model. Appl. Math. Model. 2 (June 1978), 77-80.]]Google Scholar
- 6 DICKINSON, R. P., AND GELINAS, R.J. Sensitivity analysis of ordinary differential equation systems--a direct method. J. Comput. Phys. 21 (1976), 123-143.]]Google Scholar
- 7 DOUGHERTY, E. P., AND RABITZ, H. A computational algorithm for the Green's function method of sensitivity analysis in chemical kinetics. Int. J. Chem. Kinet. XI (1979), 1237-1248.]]Google Scholar
- 8 DOUGHERTY, E. P., AND RABITZ, H. Computational kinetics and sensitivity analysis of hydrogen-oxygen combustion. J. Chem. Phys. 72, 12 (June 1980), 6571-6586.]]Google Scholar
- 9 DUNKER, A.M. The decoupled direct method for calculating sensitivity coefficients in chemical kinetics. J. Chem. Phys. 81 (1984), 2385-2393.]]Google Scholar
- 10 ENo, L., BEUMEE, J. G. B., AND RABITZ, H. Sensitivity analysis of experimental data. Appl. Math. Comput. 16 (1985), 153-163.]] Google Scholar
- 11 GEAR, C.W. The automatic integration of ordinary differential equations. Commun. ACM 14 (March 1971), 176-179.]] Google Scholar
- 12 GEAR, C.W. Numerical Initial Value Problems in Ordinary Differential Equations. Prentice- Hall, Englewood Cliffs, N.J., 1971.]] Google Scholar
- 13 HINDMARSH, A.C. LSODE and LSODI, two new initial value ordinary differential equation solvers. ACM-SIGNUM Newsl. 15, 4 (1980), 10-11.]] Google Scholar
- 14 HOLLAND, C. D., AND LIAPIS, A.I. Computer Methods for Solving Dynamic Separation Problems. McGraw-Hill, New York, 1983.]]Google Scholar
- 15 HWANG, J. T., DOUGHERTY, E. P., RABITZ, S., AND RABITZ, H. The Green's function method of sensitivity analysis in chemical kinetics. J. Chem. Phys. 69, 11 (Dec. 1978), 5180-5191.]]Google Scholar
- 16 KRAMER, M. A., CALO, J. M., AND RABITZ, H. An improved computational method for sensitivity analysis: Green's function method with 'AIM'. AppL Math. Model 5 (Dec. 1981), 432-441.]]Google Scholar
- 17 KRAMER, M. A., RABITZ, H., CALO, J. M., AND KEE, R.J. Sensitivity analysis in chemical kinetics: Recent developments and computational comparisons. Int. J. Chem. Kinet. 16 (1984), 559-578.]]Google Scholar
- 18 LAYOKUN, S. K., AND SLATER, D.H. Mechanism and kinetics of propane pyrolysis. Ind. Eng. Chem. Process Des. Dev. 18, 2 (1979), 232-236.]]Google Scholar
- 19 LEHMAN, S. L., AND STARK, L.W. Three algorithms for interpreting models consisting of ordinary differential equations: Sensitivity coefficients, sensitivity functions, global optimization. Math. Biosciences 62 (1982), 107-122.]]Google Scholar
- 20 LEIS, J. R., AND KRAMER, M.A. Algorithm 658: ODESSA--An ordinary differential equation solver with explicit simultaneous sensitivity analysis. A CM Trans. Math. Software. This issue, pp. 61-67.]] Google Scholar
- 21 LEIS, J. R., AND KRAMER, M.A. Sensitivity analysis in process design. ILP 4-15, Massachusetts Institute of Technology, Dept. of Chemical Engineering, Feb. 1985.]]Google Scholar
- 22 LEIS, J. R., AND KRAMER, M.A. Sensitivity analysis of general differential systems. AIChE National Meeting, San Francisco, Cal., paper 27c (Nov. 1984).]]Google Scholar
- 23 LEXS, J. R., AND KRAMER, M.A. Sensitivity analysis of systems of differential and algebraic equations. Comput. Chem. Eng. 9, 1 (1985), 93-96.]]Google Scholar
- 24 LOJEK, B. Sensitivity analysis of non-linear circuits. IEE Proceedings 129, Pt. G. (1982), 85-88.]]Google Scholar
- 25 MAGNUS, W. On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math., 7 (1954) 649.]]Google Scholar
- 26 NORDSIECK, A. On numerical integration of ordinary differential equations. Math. Comp. 16, 1 (Jan. 1962), 22-49.]]Google Scholar
- 27 PETZOLD, L.R. A description of DASSL: A differefitial/algebraic system solver. Sandia Report SAND82-8637, Sandia, Sept. 1982.]]Google Scholar
- 28 TOMOVIC, R., AND VOKOBRATOVIC, M. General Sensitivity Theory. Elsevier, New York, 1972.]]Google Scholar
- 29 YETTER, R., DR~ER, F., AND R~BITZ, H. Some interpretive aspects of elementary sensitivity gradients in combustion kinetics modelling. Combustion and Flame 59, 2 (Feb. 1985), 107-133.]]Google Scholar
Index Terms
- The simultaneous solution and sensitivity analysis of systems described by ordinary differential equations
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Reviews
Ian Gladwell
This paper describes a code for first-order (linearized) sensitivity analysis for the ordinary differential equation (ODE) initial value problem y? = f( t,y,p), y(0) = y 0, where p denotes given parameters. The paper begins with a survey of previous methods for this problem, then describes ODESSA, a modification of the well-known stiff solver LSODE. The technique used in ODESSA is to integrate the sensitivity (error) equations along with the ODE given above. When integrating the sensitivity equations, the software exploits the availability of the Jacobian f- y- used by LSODE. In ODESSA, the authors have chosen to control the local error in both y and the sensitivity variables, although one could choose to control the local error just in y, thus controlling it indirectly in the sensitivity variables. The numerical results demonstrate the efficiency of this procedure in comparison with earlier methods.
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