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
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.