I have a vector $\bf{b}$, and elements of this vector are generated by evaluating a rather complicated function $f(x)$ for $f(x_0), f(x_1),...,f(x_N)$.
Here are the equations that constitute $f(x)$. Let $x=Q(\tau)$ in the following so:
$f(x) =\tilde U(\tau , \omega, Q(\tau)) = \tilde U(\tau ,\omega, x) = \frac{1}{{\Lambda (\tau ,\omega )}}\exp \left[ {i\int\limits_0^\tau {\left( {{{\left( {\frac{\omega }{{{\omega _h}}}} \right)}^{\frac{1}{{\pi Q(\tau ')}}}} - 1} \right)\omega } d\tau '} \right]$
$\Lambda (\tau ,\omega ) = \frac{{\beta (\tau ,\omega ) + {\sigma ^2}}}{{{{\left( {\beta (\tau ,\omega )} \right)}^2} + {\sigma ^2}}}$
$\beta (\tau ,\omega, Q(\tau)) = \exp \left[ { - \int\limits_0^\tau {\frac{\omega }{{2Q(\tau ')}}{{\left( {\frac{\omega }{{{\omega _h}}}} \right)}^{\frac{{ - 1}}{{\pi Q(\tau ')}}}}} d\tau '} \right]$
All variables other than $Q(\tau)$ are known.
However, due to the cumulative integration, the evaluation of $f(x_1)$ is dependent on $f(x_0)$, and so on, such that $f(x_N)$ is dependent on $f(x_{N-1})$.
Note that inside of $f(x)$ is a integration numerically implemented as a running sum using the trapezoidal rule. For each element $f(x_k)$, the integration is from $x_0$ to $x_k$.
Suppose that I do not know $f(x_0)$, but I do know the numerical form of $f(x)$.
Furthermore, suppose that all I know are evaluations of function $f(x)$ for $f(x_0), f(x_1),...,f(x_N)$. These are elements of vector $\bf{b}$.
Is there a numerical procedure that I can use (i.e. non-linear curve fitting or solving a system of equations) to determine $x_0,...,x_N$ from the vector $\bf{b}$?