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In the statistical inversion setting it is common to adopt a Gaussian Process (GP) Prior with a Gaussian kernel to preferentially treat smooth parameter fields. With covariance matrix, $$S_{ij}=a \; exp\bigg\{\frac{-\|\mathbf{x}_i-\mathbf{x}_j\|_2^2}{2b^2}\bigg\}+\delta_{ij}$$ this translates to a term akin to regularisation of deterministic inverse problems in the formulation of the posterior density function, i.e. $ \| \mathbf{p}\|^2_{S^{-1}}=\mathbf{p}^T\mathbf{S}^{-1}\mathbf{p}$ for vector of parameter values $\mathbf{p}$.

Question 1 Could someone please explain how i would calculate this norm in the continuous setting? For example, with covariance function $k(x,y)$, $$ \int_{y}\int_{x} p(x) \: k(x,y) \: p(y) \: dxdy $$

Question 2 I have been told that the Helmholtz equation can be used as a proxy for the covariance function here which makes the problem easier to solve numerically (where the PDE would be solved using the FEM). Could someone please explain how to do this?

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