(456e) Sensitivity Analysis of Uncertain Dynamic Systems Using Set-Valued Integration with Application to Complete-Search Optimization
The main contribution of this paper is an extension of existing set-valued integrators to enable sensitivity analysis. Given an uncertain dynamic system in the form of a nonlinear parametric ODE,
dx/dt = f(t,x(t),p),
with initial condition x(0)=h(p) and uncertain parameters pâ??Pâ??IRn, we consider the problem to compute bounds on the gradients of state-dependent functions such as Î¦(p):=Ï?(x(T),p), with T>0. Our focus is on continuous-time set-valued integration, and we consider both forward and reverse (adjoint) sensitivity analysis. A derivation of auxiliary ODEs describing enclosures of the sensitivity or adjoint trajectories of the parametric ODE is presented, and we discuss efficient numerical integration procedures for these ODEs based on SUNDIALS-CVODES  in order to exploit their underlying structures.
An application of this new sensitivity/adjoint bounding capability for parametric ODEs is presented in the context of parameter estimation in dynamic systems using experimental data. Numerical experiments are carried out by introducing cuts derived from the first-order optimality conditions in order to tighten the relaxations and reduce clustering in branch-and-bound search for global optimization.
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