Wilkins A Katharina, Tidor Bruce, White Jacob, Barton Paul I
Department of Chemical Engineering, 77 Massachusetts Avenue, Cambridge MA 02139 ; Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge MA 02139.
SIAM J Sci Comput. 2009 Jun;31(4):2706-2732. doi: 10.1137/070707129.
Boundary value formulations are presented for exact and efficient sensitivity analysis, with respect to model parameters and initial conditions, of different classes of oscillating systems. Methods for the computation of sensitivities of derived quantities of oscillations such as period, amplitude and different types of phases are first developed for limit-cycle oscillators. In particular, a novel decomposition of the state sensitivities into three parts is proposed to provide an intuitive classification of the influence of parameter changes on period, amplitude and relative phase. The importance of the choice of time reference, i.e., the phase locking condition, is demonstrated and discussed, and its influence on the sensitivity solution is quantified. The methods are then extended to other classes of oscillatory systems in a general formulation. Numerical techniques are presented to facilitate the solution of the boundary value problem, and the computation of different types of sensitivities. Numerical results are verified by demonstrating consistency with finite difference approximations and are superior both in computational efficiency and in numerical precision to existing partial methods.
针对不同类型的振荡系统，提出了关于模型参数和初始条件的精确且高效的灵敏度分析的边值公式。首先针对极限环振荡器，开发了计算振荡的派生量（如周期、振幅和不同类型的相位）灵敏度的方法。特别地，提出了一种将状态灵敏度新颖地分解为三个部分的方法，以直观地分类参数变化对周期、振幅和相对相位的影响。证明并讨论了时间参考选择（即锁相条件）的重要性，并量化了其对灵敏度解的影响。然后，以一般形式将这些方法扩展到其他类型的振荡系统。提出了数值技术以促进边值问题的求解以及不同类型灵敏度的计算。通过证明与有限差分近似的一致性来验证数值结果，并且在计算效率和数值精度方面均优于现有的部分方法。
