Sensitivity analysis of ODEs and DAEs — theory and implementation guide

Abstract

The solution y(t,t 0,y 0) of an initial-value problem (IVP)[ydot](t)=f(t,y,p) with initial value y(t 0)=y 0 at a point t is a differentiable function of the initial value y 0 and the parameter vector p, provided f y and f p are continuous. The computation of y(t,t 0,y 0) and the sensitivity matrix play an important role in the efficient numerical solution of boundary-value problems, optimal-control problems and for parameter identification in dynamical systems. There is a wide variety of algorithms for the solution of IVPs but till now, there are just a few efficient implementations for the computation of , which is the even more important part. Here a number of implementations are introduced, treating non-stiff, stiff and differential algebraic equations. The basic new idea is to regard the numerical approximation of as the solution of the variational differential equation of a linearised IVP with approximation of the linear right-hand side by the difference quotient of the original non-linear f. For fur...