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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.