adjoint sensitivitites wrong?

Hello, I tried using the adjoint-family integrators to obtain sensitivities of an arbitrary cost functional $\begin{align*} \Psi(\vec{p}) = y_1(10)+ \int_0^{10} y_1^2(t) + y_2^2(t)\ \text{d}t \end{align*}$ w.r.t. interpolation parameters of a linear system $\begin{align*} \dot{\vec{y}}=\left[\begin{array}{c}y_2\\ -y_2 + u\end{array}\right], \quad\vec{y}(0) = \left[\begin{array}{c} 0\\2\end{array} \right] \end{align*}$ with a linearly interpolated input (using three points) and compared it to the finite difference approximation of the problem. (I attached some commented matlab code as a .zip-file. Use Matlab->Publish->Publish to get a LaTeX-formatted file of the problem and its Jacobians etc.) The sensitivities differ so much that I suspected the finite difference approximation to be wrong. Because of this, I tried implementing a "brute-force" forward sensitivity analysis aswell as an adjoint sensitivity analysis in ode45 by solving the forward sensitivity equations simultaneously and the backward problem after I obtained the forward solution, respectively (this is not included in the minimal working example in the zip file). They both gave the same solution as the finite difference approximation. Interestingly, though, the quadrature value in MATLODE is the same as int_ryp (in provided file) which I calculated numerically using trapz. In short: $\begin{align*} y_1(10) &= 3.7375\\ \int_0^{10}y_1^2(t)+y_2^2(t)\ \text{d}t &= 64.3645 \end{align*}$ MATLODE output Quadrature after the solver call is Quad = 64.3645 => same as above with trapz!

Finite difference sensitivity $\begin{align*} \frac{\partial \Psi}{\partial \vec{p}}\bigg\vert_{\text{FD}}=\frac{\Psi(\vec{p}+\epsilon)-\Psi(\vec{p})}{\epsilon} = \left[\begin{array}{ccc} 101.9459 & 119.6831 & 18.6014\end{array}\right] \end{align*}$
MATLODE sensitivity $\begin{align*} \frac{\partial \Psi}{\partial \vec{p}}\bigg\vert_{\text{MATLODE}}=\vec{\mu}_N^T = \left[\begin{array}{ccc} 109.8392 &142.5333 &27.3191\end{array}\right] \end{align*}$

=> differ significantly?!

Am I missing something here? Thank you in advance.

mweMatloteIssue.zip

EDIT: included formated LaTeX here additionally to the matlab comments.

ComputationalScienceLaboratory / MATLODE

adjoint sensitivitites wrong? #4