Sensitivity via the complex-step method for delay differential equations with non-smooth initial data
Authors:
H. T. Banks, Kidist Bekele-Maxwell, Lorena Bociu and Chuyue Wang
Journal:
Quart. Appl. Math. 75 (2017), 231-248
MSC (2010):
Primary 34A55, 90C31
DOI:
https://doi.org/10.1090/qam/1458
Published electronically:
November 2, 2016
MathSciNet review:
3614496
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Abstract: In this report, we use the complex-step derivative approximation technique to compute sensitivities for delay differential equations (DDEs) with non-smooth (discontinuous and even distributional) history functions. We compare the results with exact derivatives and with those computed using the classical sensitivity equations whenever possible. Our results demonstrate that the implementation of the complex-step method using the method of steps and the Matlab solver dde23 provides a very good approximation of sensitivities as long as discontinuities in the initial data do not cause loss of smoothness in the solution: that is, even when the underlying smoothness with respect to the initial data for the Cauchy-Riemann derivation of the method does not hold. We conclude with remarks on our findings regarding the complex-step method for computing sensitivities for simpler ordinary differential equation systems in the event of lack of smoothness with respect to parameters.
References
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- G. Evelyn Hutchinson, An introduction to population ecology, Yale University Press, New Haven, Conn., 1978. MR 492532
- Yang Kuang, Delay differential equations with applications in population dynamics, Mathematics in Science and Engineering, vol. 191, Academic Press, Inc., Boston, MA, 1993. MR 1218880
- J. N. Lyness, Numerical algorithms based on the theory of complex variables, Proc. ACM 22nd Nat. Conf., 4 (1967), 124â-134.
- J. N. Lyness and C. B. Moler, Numerical differentiation of analytic functions, SIAM J. Numer. Anal. 4 (1967), 202–210. MR 214285, DOI https://doi.org/10.1137/0704019
- Joaquim R. R. A. Martins, Ilan M. Kroo, and Juan J. Alonso. An automated method for sensitivity analysis using complex variables. AIAA Paper 2000-0689 (Jan.), 2000.
- Joaquim R. R. A. Martins, Peter Sturdza, and Juan J. Alonso, The complex-step derivative approximation, Journal ACM Transactions on Mathematical Software (TOMS), 2003.
- N. Minorsky, Self-excited oscillations in dynamical systems possessing retarded actions, Journal of Applied Mechanics, 9 (1942) A65–A71.
- N. Minorsky, On non-linear phenomenon of self-rolling, Proc. Nat. Acad. Sci. U.S.A. 31 (1945), 346–349. MR 13473, DOI https://doi.org/10.1073/pnas.31.11.346
- Nicolas Minorsky, Nonlinear oscillations, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1962. MR 0137891
- L. F. Shampine and S. Thompson, Solving DDEs in MATLAB, Appl. Numer. Math. 37 (2001), no. 4, 441–458. MR 1831793, DOI https://doi.org/10.1016/S0168-9274%2800%2900055-6
References
- W. Kyle Anderson and Eric J. Nielsen, Sensitivity analysis for Navier-Stokes equations on unstructured meshes using complex variables, AIAA Journal, 39 (2001).
- W. Kyle Anderson, Eric J. Nielsen, and D.L. Whitfield, Multidisciplinary sensitivity derivatives using complex variables, Technical report, Engineering Research Center Report, Mississippi State University, mSSU-COE-ERC-98-08, July, 1998.
- H. T. Banks, Delay systems in biological models: approximation techniques, Nonlinear systems and applications (Proc. Internat. Conf., Univ. Texas, Arlington, Tex., 1976) Academic Press, New York, 1977, pp. 21–38. MR 0489995
- H. T. Banks, Robert Baraldi, Karissa Cross, Kevin Flores, Christina McChesney, Laura Poag, and Emma Thorpe, Uncertainty quantification in modeling HIV viral mechanics, Math. Biosci. Eng. 12 (2015), no. 5, 937–964. MR 3356519, DOI https://doi.org/10.3934/mbe.2015.12.937
- H. T. Banks, K. Bekele-Maxwell, L. Bociu, M. Noorman and K. Tillman, The complex-step method for sensitivity analysis of non-smooth problems arising in biology, Eurasian Journal of Mathematical and Computer Applications, 3 (2015), 16–68.
- H. T. Banks and D. M. Bortz, A parameter sensitivity methodology in the context of HIV delay equation models, J. Math. Biol. 50 (2005), no. 6, 607–625. MR 2211638, DOI https://doi.org/10.1007/s00285-004-0299-x
- H. T. Banks, D. M. Bortz, and S. E. Holte, Incorporation of variability into the modeling of viral delays in HIV infection dynamics, Math. Biosci. 183 (2003), no. 1, 63–91. MR 1965457, DOI https://doi.org/10.1016/S0025-5564%2802%2900218-3
- A. Cintrón-Arias, H. T. Banks, A. Capaldi, and A. L. Lloyd, A sensitivity matrix based methodology for inverse problem formulation, J. Inverse Ill-Posed Probl. 17 (2009), no. 6, 545–564. MR 2547126, DOI https://doi.org/10.1515/JIIP.2009.034
- H. T. Banks, Ariel Cintrón-Arias, and Franz Kappel, Parameter selection methods in inverse problem formulation, Mathematical modeling and validation in physiology, Lecture Notes in Math., vol. 2064, Springer, Heidelberg, 2013, pp. 43–73. MR 3024538, DOI https://doi.org/10.1007/978-3-642-32882-4_3
- H. T. Banks, Shuhua Hu, and W. Clayton Thompson, Modeling and inverse problems in the presence of uncertainty, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2014. MR 3203115
- H. T. Banks, S. Dediu, and S. L. Ernstberger, Sensitivity functions and their uses in inverse problems, J. Inverse Ill-Posed Probl. 15 (2007), no. 7, 683–708. MR 2374978, DOI https://doi.org/10.1515/jiip.2007.038
- H. T. Banks and K. L. Rehm, Experimental design for vector output systems, Inverse Probl. Sci. Eng. 22 (2014), no. 4, 557–590. MR 3173636, DOI https://doi.org/10.1080/17415977.2013.797973
- H. T. Banks and K. L. Rehm, Parameter estimation in distributed systems: Optimal design, CRSC TR14-06, N. C. State University, Raleigh, NC, May, 2014; Eurasian Journal of Mathematical and Computer Applications, 2 (2014), 70–79.
- H. Thomas Banks, Danielle Robbins, and Karyn L. Sutton, Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations, Math. Biosci. Eng. 10 (2013), no. 5-6, 1301–1333. MR 3103332
- H. T. Banks and H. T. Tran, Mathematical and experimental modeling of physical and biological processes, Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2009. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2488750
- Differential equations and applications in ecology, epidemics, and population problems, edited by Stavros N. Busenberg and Kenneth L. Cooke, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 645184
- Mathematics in biology and medicine, edited by V. Capasso, E. Grosso and S. L. Paveri-Fontana, Lecture Notes in Biomathematics, vol. 57, Springer-Verlag, Berlin, 1985. MR 812870
- Jim M. Cushing, Integrodifferential equations and delay models in population dynamics, Springer-Verlag, Berlin-New York, 1977. Lecture Notes in Biomathematics, Vol. 20. MR 0496838
- K. P. Hadeler, Delay equations in biology, Functional differential equations and approximation of fixed points (Proc. Summer School and Conf., Univ. Bonn, Bonn, 1978) Lecture Notes in Math., vol. 730, Springer, Berlin, 1979, pp. 136–156. MR 547986
- G. E. Hutchinson, Circular causal systems in ecology, Annals of the NY Academy of Sciences, 50 (1948), 221–246.
- G. Evelyn Hutchinson, An introduction to population ecology, Yale University Press, New Haven, Conn., 1978. MR 492532
- Yang Kuang, Delay differential equations with applications in population dynamics, Mathematics in Science and Engineering, vol. 191, Academic Press, Inc., Boston, MA, 1993. MR 1218880
- J. N. Lyness, Numerical algorithms based on the theory of complex variables, Proc. ACM 22nd Nat. Conf., 4 (1967), 124â-134.
- J. N. Lyness and C. B. Moler, Numerical differentiation of analytic functions, SIAM J. Numer. Anal. 4 (1967), 202–210. MR 0214285
- Joaquim R. R. A. Martins, Ilan M. Kroo, and Juan J. Alonso. An automated method for sensitivity analysis using complex variables. AIAA Paper 2000-0689 (Jan.), 2000.
- Joaquim R. R. A. Martins, Peter Sturdza, and Juan J. Alonso, The complex-step derivative approximation, Journal ACM Transactions on Mathematical Software (TOMS), 2003.
- N. Minorsky, Self-excited oscillations in dynamical systems possessing retarded actions, Journal of Applied Mechanics, 9 (1942) A65–A71.
- N. Minorsky, On non-linear phenomenon of self-rolling, Proc. Nat. Acad. Sci. U. S. A. 31 (1945), 346–349. MR 0013473
- Nicolas Minorsky, Nonlinear oscillations, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1962. MR 0137891
- L. F. Shampine and S. Thompson, Solving DDEs in MATLAB, Appl. Numer. Math. 37 (2001), no. 4, 441–458. MR 1831793, DOI https://doi.org/10.1016/S0168-9274%2800%2900055-6
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Additional Information
H. T. Banks
Affiliation:
Center for Research in Scientific Computation, Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8212
MR Author ID:
194993
Email:
htbanks@ncsu.edu
Kidist Bekele-Maxwell
Affiliation:
Center for Research in Scientific Computation, Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8212
Email:
ktzeleke@ncsu.edu
Lorena Bociu
Affiliation:
Center for Research in Scientific Computation, Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8212
MR Author ID:
811365
Email:
lvbociu@ncsu.edu
Chuyue Wang
Affiliation:
Center for Research in Scientific Computation, Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8212
Email:
cwang31@ncsu.edu
Keywords:
Inverse problems,
sensitivity with respect to parameters,
complex-step method,
delay differential equations
Received by editor(s):
September 6, 2016
Published electronically:
November 2, 2016
Article copyright:
© Copyright 2016
Brown University