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I am dealing with a parameter identification problem, where I have a system of differential equations $$\frac{dy}{dt}=\mathbf{F}(t,\vec{y}(t,\vec{p}),\vec{p}) \tag1$$ where I need to find the parameters contained in the vector $\vec{p}$, that minimise the objective function $$C=\min_p\left(||\vec{y}(t,\vec{p})-\vec{y}_m||^2_2\right) \tag2$$ where $\vec{y}_m$ is some measured data.

As I am performing the optimisation using a gradient-based algorithm, I need to find the gradient of the objective function. For this as I understand I need to employ the sensitivity (or variational) equations where (for notational simplicity $\vec{\cdot}$ is ommitted): $$\frac{d}{dt}\frac{\partial y}{\partial p}=\frac{\partial \mathbf{F}}{\partial y}\cdot\frac{\partial y}{\partial p}+\frac{\partial \mathbf{F}}{\partial p} \tag3$$

This way the gradient of my objective function would be: $$\vec{G}=\left[\frac{\partial y}{\partial p}\right]^T\cdot\left(y-y_m\right)$$

There are three questions I have:

  1. Is the expression for the gradient $G$ in equation (4) correct?
  2. How is equation (3) derived? This post seems to do it with the chain rule, however Hemker (1971) on page 75 seems to be performing a linear approximation. These seem fairly different, with different assumptions made particularly in Hemker's work, which also seems to result to a different form of the objective function's gradient (which I do not comprehend really well).
  3. For my case I have the differential equations in the form of

$$\mathbf{ M(\mathit{y})\mathit{y''} +C\mathit{y'}+K\mathit{y}+ A(\mathit{y,y',p})} = \mathbf{0}\\ \Rightarrow \mathbf{F} = \begin{bmatrix} y' \\y'' \end{bmatrix}=\begin{bmatrix} \mathbf{0} & \mathbf{I} \\ \mathbf{-[M(\mathit{y})]^{-1}K} & \mathbf{-[M(\mathit{y})]^{-1}C} \end{bmatrix}\begin{bmatrix} y \\ y' \end{bmatrix}+ \\ \begin{bmatrix} \mathbf{0} \\ \mathbf{-[M(\mathit{y})]^{-1}A(\mathit{y,y',p})} \end{bmatrix} \tag4$$

where $\mathbf{M(\mathit{y})}$ is a diagonal variable matrix, $\mathbf{A(\mathit{y,y',p})}$ is a variable vector, and $\mathbf{C,K}$ are constant tri-diagonal matrices. So $$\begin{align} \frac{\partial \mathbf{F}}{\partial y} = & \begin{bmatrix} \mathbf{0} & \mathbf{0} \\ [\mathbf{M(\mathit{y})}]^{-1}\frac{\partial M}{\partial y}[\mathbf{M(\mathit{y})}]^{-1}(\mathbf{K\mathit{y}+ C\mathit{y'}})-\mathbf{[M(\mathit{y})]^{-1}K} & \mathbf{0} \end{bmatrix}+\\ & \begin{bmatrix} \mathbf{0} & \mathbf{0} \\ [\mathbf{M(\mathit{y})}]^{-1}\frac{\partial M}{\partial y}[\mathbf{M(\mathit{y})}]^{-1}\text{diag}\mathbf{\left(A(\mathit{y,y',p})\right)-[\mathbf{M(\mathit{y})}]^{-1}}\frac{\mathbf{\partial A(\mathit{y,y',p})}}{\partial y} & \mathbf{0} \end{bmatrix} \end{align} \tag5$$

However when employing equation (5) to solve equation (3) and subsequently find $G$, this does not give rise to the gradient of the objective function (I know since I have the correct answer which is numerically derived, and my $C$ is very smooth). Instead what returns an answer extremely close to the correct one (and I can't explain that) is if I use the following:

$$\begin{align} \frac{\partial \mathbf{F}}{\partial y} = & \begin{bmatrix} \mathbf{0} & \mathbf{0} \\ -[\mathbf{M(\mathit{y})}]^{-1}\frac{\partial M}{\partial y}[\mathbf{M(\mathit{y})}]^{-1}(\mathbf{K\mathit{y}+ C\mathit{y'}})-\mathbf{[M(\mathit{y})]^{-1}K} & \mathbf{0} \end{bmatrix}-\\ & \begin{bmatrix} \mathbf{0} & \mathbf{0} \\ [\mathbf{M(\mathit{y})}]^{-1}\frac{\partial M}{\partial y}[\mathbf{M(\mathit{y})}]^{-1}\text{diag}\mathbf{\left(A(\mathit{y,y',p})\right)} & \mathbf{0} \end{bmatrix} \end{align} \tag6$$

So to conclude with my 3rd question, based on the above, am I missing part of the theory here?

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