Bayesian inverse problems

• 1. The Bayesian Framework
  • Bayesian Approaches
• 2. Applications
• 3. Common Structure
• 4. Algorithms
• 5. Probabilities

Notes from: Stuart, Andrew M. “Inverse problems: a Bayesian perspective.” Acta numerica 19 (2010): 451-559.

1. The Bayesian Framework

Bayesian Approaches

• Inverse Problem: determining $u$ from $y$ \(y = \mathcal{G}(u)\)
  • $\mathcal{G}: X\rightarrow Y$ observation operation ($X$,$Y$ Banach)
  • unknow input $u\in X$ (parameter)
  • an observation $y\in Y$ of the solution of $\mathcal{G}$ (data)
• Regularized minimization: subspace $E\in X$, $m_0\in E$ \(\arg\min_{u\in E}\left(\frac{1}{2}\lVert y-\mathcal{G}(u)\rVert^2_Y + \frac{1}{2}\lVert u-m_0\rVert^2_E\right)\)
• Statistical Approach:
  • find probability measure $\mu^y = P(u|y)$ on $X$
  • observation with noise: $y = \mathcal{G}(u) + \eta, \mathbb{E}(\eta) = 0$
  • posterior measure (e.g., Gaussian): \(\pi^y(u)\propto \exp\left(\frac{1}{2}\lVert y-\mathcal{G}(u)\rVert^2_Y + \frac{1}{2}\lVert u-m_0\rVert^2_E\right)\)
  • Formally:

    2. Applications

3. Common Structure

4. Algorithms

5. Probabilities