Sensitivity Analysis of Stiff and Non-Stiff Initial-Value Problems
The solution y(t, t 0, y 0) of an initial-value problem (IVP) y(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 the derivatives of y(t, t 0, y 0) plays an important role in the efficient numerical solution of optimal-control problems and in parameter identification.
Different implementations of numerical algorithms for the computation of ∂y(t)/∂y 0 already exist, but the techniques are often adapted to the special implementations and cannot easily be transferred to other integration methods. Here we chose a new approach that allows understanding of most of the existing implementations and may serve as theoretical basis of many more.
The basic idea is to regard the numerical approximation of ∂y(t)/∂y 0 as the solution of the variational differential equation of a linearised linear IVP with approximation of the linear right-hand side by the difference quotient of the original non-linear f.
This approach shows how integration codes can in general be extended so that the simultaneous and accurate computation of the sensitivity matrix is possible.
KeywordsDifference Approximation Integration Step Sensitivity Matrix Implicit Integration Parameter Identification Problem
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