Sensitivity Analysis of Uncertain Dynamic Systems Using Set-Valued Integration

We present an extension of set-valued integration to enable efficient sensitivity analysis of parameter-dependent ordinary differential equation (ODE) systems, using both the forward and adjoint methods. The focus is on continuous-time set-valued integration, whereby auxiliary ODE systems are derived whose solutions describe high-order inclusions of the parametric trajectories in the form of polynomial models. The forward and adjoint auxiliary ODE systems treat the parameterization error of the original differential variables as a time-varying uncertainty, and propagate the sensitivity bounds forward and backward in time, respectively. This construction enables building on the sensitivity analysis capabilities of state-of-the-art solvers, such as CVODES in the SUNDIALS suite. Several numerical case studies are presented to assess the performance and accuracy of these set-valued sensitivity integrators.