EFFICIENT DELAY AND PARAMETER ESTIMATION FOR NONLINEAR SYSTEMS BASED ON LIFTED IRK INTEGRATORS WITH INEXACT JACOBIANS

. For continuous nonlinear systems with multiple state delays, this paper proposes an efficient optimization algorithm for delay and parameter estimation based on a novel implicit Runge-Kutta integrator. As the integrator is in implicit form, the integral step is not restricted to be less than all the delays. To improve computational efficiency, a lifting scheme is employed and integrated with nonlinear programming solver such that the number of Newton iterations for solving collocation equations can be reduced. Moreover, simplified-Newton and single-Newton methods are applied to reduce the cost of LU decomposition in Newton iterations. Complexity analysis shows that, for problems of middle or large scale, this algorithm requires much less computational cost for function evaluation than a sequential algorithm based on a traditional integrator. A delay and parameter estimation problem for a gene regulatory network demonstrates that this algorithm can obtain the estimates efficiently with satisfactory accuracy. The speedup to a traditional sequential algorithm is of factor 2-3.