# Bayesian Inverse Approach

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

- A Gaussian prior distribution where is the covariance operator is a natural choice to model an unknown function. For example, we could take and

- Karhunen-Loeve is used to sample from this distribution

**Karhunen-Loeve Theorem**

Let be a square integral mean-zero stochastic process with continuous covariance function satisfying the following: is continuous, symmetric and positive definite.

Then can be decomposed as

where are the orthonormal eigenfunctions of the covariance operator and

The convergence of is uniform in .

- 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.