random_poisson2d.tex

\documentclass{article} 

\usepackage{amssymb}
\usepackage{amsmath}


% expectation
\newcommand{\cE}[1]{\mathbb{E}[#1]}
% parametric objective function
\newcommand{\pobj}{J}

\begin{document}

We define the real-valued function $\pobj$ by
\begin{align*}
	\pobj(y) = 
	(1/2)(y-y_d)^T N (y-y_d).
\end{align*}
The solution operator $y(u,\xi)$ 
is defined as the solution to
the state equation
\begin{align*}
	\kappa(\xi) A y(u,\xi) = Gu + g(\xi),
\end{align*}
where 
$\kappa : \Xi \to (0,\infty)$ is a real-valued random variable
such that $1/\kappa(\xi)$ has finite moments of all orders, 
and $g(\xi)$ is a random vector with finite moments
of all orders.
Moreover, $G$, $N$, and $A$ are deterministic matrices, and
$N$ and $A$ are symmetric and positive semidefinite.


Using the adjoint approach, we find that
the derivative of $u \mapsto \pobj(y(u,\xi))$
with respect to $u$ is given by
$-G^Tz(u,\xi)$, where $z(u,\xi)$ solves
the adjoint equation
\begin{align*}
	\kappa(\xi) A z(u,\xi) = - (Ny(u,\xi)-\widetilde{y}_d).
\end{align*}
Here $\widetilde{y}_d = Ny_d$.


We formally show that $\cE{z(u,\xi)} = \bar{z}$, 
where $\bar{z}$ solves
\begin{align*}
	A \bar{z} &
	= - (\widetilde{N}\bar{y} - \cE{1/\kappa(\xi)}\widetilde{y}_d)
	\quad 
	\text{where}
	\quad 
	\widetilde{N} = \cE{1/\kappa(\xi)^2} N. 
\end{align*}
and $\bar{y}$ solves
\begin{align*}
	A \bar{y}
	&= G u + \widetilde{g}
	\quad \text{where}
	\quad \widetilde{g}
	= \cE{1/\kappa(\xi)^2}^{-1} \cE{g(\xi)/\kappa(\xi)^2}.
\end{align*}


Dividing the state and adjoint equation by $\kappa(\xi)$, we find that
\begin{align*}
	A z(u,\xi) 
	& =  (1/\kappa(\xi)) \widetilde{y}_d  - Ny(u,\xi) \\
	& =  (1/\kappa(\xi)) \widetilde{y}_d 
	- N A^{-1}\Big[(g(\xi)/\kappa(\xi)^2)+
	(1/\kappa(\xi)^2)G u\Big]. 
\end{align*}
Taking expectations, we obtain 
\begin{align*}
	A \cE{z(u,\xi)}
	& =\cE{1/\kappa(\xi)} \widetilde{y}_d  
	- N A^{-1}\Big[\cE{g(\xi)/\kappa(\xi)^2}+
	\cE{1/\kappa(\xi)^2}Gu\Big] \\
	& =\cE{1/\kappa(\xi)} \widetilde{y}_d  
	- \underbrace{\cE{1/\kappa(\xi)^2} N}_{=\widetilde{N}} 
	A^{-1} \Big[
	\underbrace{\cE{1/\kappa(\xi)^2}^{-1}
		\cE{g(\xi)/\kappa(\xi)^2}}_{=\widetilde{g}}+
	Gu\Big] \\
	& = - (\widetilde{N}\bar{y} - \cE{1/\kappa(\xi)}\widetilde{y}_d)
	\\
	& = A \bar{z}.
\end{align*}

\end{document}