×

zbMATH — the first resource for mathematics

Numerical methods for computing sensitivities for ODEs and DDEs. (English) Zbl 1362.65070
Summary: We investigate the performance of the adjoint approach and the variational approach for computing the sensitivities of the least squares objective function commonly used when fitting models to observations. We note that the discrete nature of the objective function makes the cost of the adjoint approach for computing the sensitivities dependent on the number of observations. In the case of ordinary differential equations (ODEs), this dependence is due to having to interrupt the computation at each observation point during numerical solution of the adjoint equations. Each observation introduces a jump discontinuity in the solution of the adjoint differential equations. These discontinuities are propagated in the case of delay differential equations (DDEs), making the performance of the adjoint approach even more sensitive to the number of observations for DDEs. We quantify this cost and suggest ways to make the adjoint approach scale better with the number of observations. In numerical experiments, we compare the adjoint approach with the variational approach for computing the sensitivities.

MSC:
65L05 Numerical methods for initial value problems
65L03 Numerical methods for functional-differential equations
34A34 Nonlinear ordinary differential equations and systems, general theory
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
Software:
ADIC
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alexe, M; Sandu, A, Forward and adjoint sensitivity analysis with continuous explicit Runge-Kutta schemes, Appl. Math. Comput., 208, 328-346, (2009) · Zbl 1159.65071
[2] Baker, CT; Paul, CA, Pitfalls in parameter estimation for delay differential equations, SIAM J. Sci. Comput., 18, 305-314, (1997) · Zbl 0867.65032
[3] Bischof, CH; Hovland, PD; Norris, B, On the implementation of automatic differentiation tools, Higher Order Symbol. Comput., 3, 311-331, (2008) · Zbl 1168.65324
[4] Cao, Y; Li, S; Petzold, L; Serban, R, Adjoint sensitivity analysis for differential algebraic equations: the adjoint DAE system and its numerical solution, SIAM J. Sci. Comput., 3, 1076-1089, (2003) · Zbl 1034.65066
[5] Hutchinson, GE, Circular causal systems in ecology, Ann. New York Acad. Sci., 50, 221-246, (1948)
[6] Kermack, W., McKendrick, A.: A contribution to the mathematical theory of epidemics. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 115, pp. 700-721. The Royal Society (1927) · JFM 53.0517.01
[7] Kuang, Y.: Delay Differential Equations: with Applications in Population Dynamics. Academic Press (1993) · Zbl 0777.34002
[8] Lenz, S; Schlder, J; Bock, H, Numerical computation of derivatives in systems of delay differential equations, Math. Comput. Simul., 96, 124-156, (2014)
[9] Petzold, L; Li, S; Cao, Y; Serban, R, Sensitivity analysis of differential-algebraic equations and partial differential equations, Comput. Chem. Eng., 30, 1553-1559, (2006)
[10] Sengupta, B; Friston, K; Penny, W, Efficient gradient computation for dynamical models, NeuroImage, 98, 521-527, (2014)
[11] Shakeri, F; Dehghan, M, Solution of delay differential equations via a homotopy perturbation method, Math. Comput. Modell., 48, 486-498, (2008) · Zbl 1145.34353
[12] Varah, J, A spline least squares method for numerical parameter estimation in differential equations, SIAM J. Sci. Stat. Comput., 3, 28-46, (1982) · Zbl 0481.65050
[13] Wang, B.: Parameter Estimation for Odes using a Cross-Entropy Approach. University of Toronto, Master’s thesis (2012)
[14] Zivari-Piran, H.: Efficient Simulation, Accurate Sensitivity Analysis and Reliable Parameter Estimation for Delay Differential Equations. Ph.D. thesis, Univeristy of Toronto (2009)
[15] Zivari-Piran, H., Enright, W.: Accurate First-Order Sensitivity Analysis for Delay Differential Equations: Part II: the Adjoint Approach. preprint, Department of Computer Science, University of Toronto (2009) · Zbl 1259.65108
[16] Zivari-Piran, H; Enright, WH, Accurate first-order sensitivity analysis for delay differential equations, SIAM J. Sci. Comput., 34, a2704-a2717, (2012) · Zbl 1259.65108
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.