The inverse problem I solve has the following basic outline: a number $n_g$ of Gaussian kernels ($\textbf{x}$) of some width $\sigma$ and each with an amplitude $s_j$ is propagated in n-d space to $\textbf{x}+\textbf{v}(\textbf{x})*dt$ by some n-d polynomial function $\textbf{v}(\textbf{x})$ with coefficients $\boldsymbol{\beta}$, where $R^{n_g}\rightarrow R^{n_p}$ function $f$ then generates an image consisting of $n_p$ pixels, which is compared to a measurement $\textbf{y}$. The coefficients $\boldsymbol{\beta}$ are then found as that which minimizes $S=\sum_i (y_i-f_i(\textbf{x},\boldsymbol{\beta}))^2$

The usual least squares formulation is to minimize $S=\sum_i (y_i-f(x_i,\boldsymbol{\beta}))^2$, where $f:R\rightarrow R$. This has an associated covariance matrix $F_{ij}=\frac{\partial f(x_i,\boldsymbol{\beta})}{\beta_j}$ and variance $\text{var}(\beta_j)\approx\frac{S}{n-m}([F^TF]^{-1})_{jj}$, where $n$ is the dimension of $\textbf{y}$ and $m$ is the dimension of $\boldsymbol{\beta}$.

When attempting to estimate the variance in the $\beta_j$ obtained by the solution in my inverse problem, I was wondering if it would be valid to define the covariance as $F_{ij}=\frac{\partial f_i(\textbf{x}, \boldsymbol{\beta})}{\beta_j}$, and to keep $n$ as the dimension of $\textbf{y}$. I was also wondering if there were any useful reading references for inverse problems of that form?


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