In any inverse retrieval problem, the forward model can be stated in matrix form as:
$$Ax = y.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (1)$$
Here, $A$ is the operator acting on the variables $x$ and $y$ is the measurement. The aim of the problem is to evaluate the unknown variables $x$, given $y$ is known and the action of operator $A$ on $x$ is known.
In my problem, the operator $A$ acts on the $i$th element of $x$, namely $x_i$, to give the corresponding element $y_i$ in the following manner:
$$y_i = A(x_i) = {x_i[\exp(a_1 x_i)+\exp(a_2 x_i)+\exp(a_3 x_i)+\dots+\exp(a_n x_i)]}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (2)$$
Here, $\{a_1,a_2,...a_n\}$ are a set of constants. I want to solve this problem through the gradient descent method and so, I need to evaluate the adjoint of the operator $A$. And so, my question is how do I obtain the adjoint? I understand that to evaluate the adjoint, I need to call the inner product definition where, any adjoint of operator $A$, namely $A^H$ should fulfil the inner product equality:
$$<A^Hu,v>\,\,=\,\,<u,Av>.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (3)$$
But I am unable to come up with an expression for the adjoint. Can anyone suggest anything?