# Bayesian Inverse Approach

The object is to find an unknown function $f$ in a Banach space $W$, possibly the space of continuous functions.

• A Gaussian prior distribution $\mathcal{N}(m,C)$ where $C:W\rightarrow W$ is the covariance operator is a natural choice to model an unknown function. For example, we could take $C=(-\Delta)^{-1}$ and $W=H[0,1]$
• Karhunen-Loeve is used to sample from this distribution

Karhunen-Loeve Theorem

Let $U\colon K\times \Omega \rightarrow \mathbb{R}$ be a square integral mean-zero stochastic process with continuous covariance function $C_{U}(x,y)=\mathbb{E}_{\mu}[U(x)U(y)]$ satisfying the following: $C_{U}$ is continuous, symmetric and positive definite.

Then $U$ can be decomposed as
$U=\sum_{n\in\mathbb{N}}Z_{n} \phi_{n}$
where $\{\phi_{n}\}_{n\in\mathbb{N}}$ are the orthonormal eigenfunctions of the covariance operator and
$Z_n=\int_{K} U(x)\phi_{n}(x)\dd x.$
The convergence of $U$ is uniform in $x$.

• Implement a Markov Chain Monte Carlo (MCMC) algorithm to sample from the posterior.

Here is an example of how MCMC can be implemented on a function that maps the input of the pressure upistream to the corresonding output.