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A unified approach to integration and optimization of parametric ordinary differential equations. (English) Zbl 1337.65082

Carraro, Thomas (ed.) et al., Multiple shooting and time domain decomposition methods. MuS-TDD, Heidelberg, Germany, May 6–8, 2013. Cham: Springer (ISBN 978-3-319-23320-8/hbk; 978-3-319-23321-5/ebook). Contributions in Mathematical and Computational Sciences 9, 305-314 (2015).
Summary: Parameter estimation in ordinary differential equations, although applied and refined in various fields of the quantitative sciences, is still confronted with a variety of difficulties. One major challenge is finding the global optimum of a log-likelihood function that has several local optima, e.g. in oscillatory systems. In this publication, we introduce a formulation based on continuation of the log-likelihood function that allows to restate the parameter estimation problem as a boundary value problem. By construction, the ordinary differential equations are solved and the parameters are estimated both in one step. The formulation as a boundary value problem enables an optimal transfer of information given by the measurement time courses to the solution of the estimation problem, thus favoring convergence to the global optimum. This is demonstrated explicitly for the fully as well as the partially observed Lotka-Volterra system.
For the entire collection see [Zbl 1333.65003].

MSC:

65L09 Numerical solution of inverse problems involving ordinary differential equations
34A55 Inverse problems involving ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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