Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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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
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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.


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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
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
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

DOI: https://doi.org/10.1090/qam/1458
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


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